Solutions to H C Verma's Concepts of Physics 2

CHAPTER – 23 HEAT AND TEMPERATURE EXERCISES 1.

Ice point = 20° (L0) L1 = 32° Steam point = 80° (L100) L1  L 0 32  20  100 = T=  100 = 20°C L100  L 0 80  20

2.

Ptr = 1.500 × 10 Pa 4 P = 2.050 × 10 Pa We know, For constant volume gas Thermometer

4

3.

2.050  10 4 P  273.16 K =  273.16 = 373.31 Ptr 1.500  10 4 Pressure Measured at M.P = 2.2 × Pressure at Triple Point 2.2  Ptr P  273.16 =  273.16 = 600.952 K  601 K T= Ptr Ptr

4.

Ptr = 40 × 10 Pa, P = ?

T=

3

T = 100°C = 373 K,

T=

P  273 .16 K Ptr

T  Ptr 373  49  10 3 3 = = 54620 Pa = 5.42 × 10 pa ≈ 55 K Pa 273.16 273.16 P1 = 70 K Pa, P2 = ? T2 = 373K T1 = 273 K, P=

5.

T= T2 =

6.

7.

8.

9.

P1  273.16 Ptr P2  273.16 Ptr

 273 =  373 =

70  10 3  273.16 Ptr

P2  273

70  273.16  10 Pice point = P0° = 80 cm of Hg Psteam point = P100° 90 cm of Hg P0 = 100 cm P  P0 80  100  100 =  100 = 200°C t= 90  100 P100  P0

V T0 T0 = 273, V  V V = 1800 CC, V = 200 CC 1800  273 = 307.125  307 T = 1600 Rt = 86; R0° = 80; R100° = 90 R t  R0 86  80 t=  100 =  100 = 60°C R100  R 0 90  80

T =

R at ice point (R0) = 20 R at steam point (R100) = 27.5 R at Zinc point (R420) = 50 2 R = R0 (1+  +  ) 2  R100 = R0 + R0  +R0   R0 R 2  100 =  +  R0 23.1

3

 Ptr

70  273.16  10 3 273

 P2 =

373  70  10 3 = 95.6 K Pa 273

23.Heat and Temperature

27.5  20 =  × 100 +  × 10000 20 7 .5  = 100  + 10000  20 50  R 0 2 2 R420 = R0 (1+  +  )  =  +  R0 

50  20 3 = 420 ×  + 176400 ×     420  + 176400  20 2 7 .5 3  = 100  + 10000     420  + 176400  20 2 –5 L1 = ?, L0 = 10 m,  = 1 × 10 /°C, t= 35 –5 –4 L1 = L0 (1 + t) = 10(1 + 10 × 35) = 10 + 35 × 10 = 10.0035m t1 = 20°C, t2 = 10°C, L1 = 1cm = 0.01 m, L2 =? –5 steel = 1.1 × 10 /°C –5 –4 L2 = L1 (1 + steelT) = 0.01(1 + 101 × 10 × 10) = 0.01 + 0.01 × 1.1 × 10 4 –6 –6 –6 = 10 × 10 + 1.1 × 10 = 10 (10000 + 1.1) = 10001.1 –2 =1.00011 × 10 m = 1.00011 cm –5  = 11 × 10 /°C L0 = 12 cm, tw = 18°C ts = 48°C –5 Lw = L0(1 + tw) = 12 (1 + 11 × 10 × 18) = 12.002376 m –5 Ls = L0 (1 + ts) = 12 (1 + 11 × 10 × 48) = 12.006336 m L12.006336 – 12.002376 = 0.00396 m  0.4cm –2 d1 = 2 cm = 2 × 10 t1 = 0°C, t2 = 100°C –5 al = 2.3 × 10 /°C –2 –5 2 d2 = d1 (1 + t) = 2 × 10 (1 + 2.3 × 10 10 ) = 0.02 + 0.000046 = 0.020046 m = 2.0046 cm –5 Lst = LAl at 20°C Al = 2.3 × 10 /°C –5 st = 1.1 × 10 /°C So, Lost (1 – st × 20) = LoAl (1 – AI × 20) 

10. 11.

12.

13.

14.

(a) 

Lo st (1   Al  20) 1  2.3  10 5  20 0.99954 = = = = 0.999 Lo Al (1   st  20) 0.99978 1  1.1 10  5  20

(b) 

Lo 40st (1   AI  40) 1  2.3  10 5  20 0.99954 = = = = 0.999 Lo 40 Al (1   st  40) 0.99978 1  1.1 10  5  20

=

Lo Al 1  2.3  10 5  10 0.99977  1.00092  = = 1.0002496 ≈1.00025 Lo st 273 1.00044

Lo100 Al (1   Al  100 ) 0.99977  1.00092 = = = 1.00096 Lo100St (1   st  100 ) 1.00011 15. (a) Length at 16°C = L L=? T1 =16°C, –5  = 1.1 × 10 /°C –5 L = L = L × 1.1 × 10 × 30

T2 = 46°C

   L  L % of error =   100 % =   100 % = 1.1 × 10–5 × 30 × 100% = 0.033%    L  2 (b) T2 = 6°C

 L  L   –5  100 % =   100 % = – 1.1 × 10 × 10 × 100 = – 0.011% % of error =   L  L   23.2

23.Heat and Temperature –3

16. T1 = 20°C, L = 0.055mm = 0.55 × 10 m –6 st = 11 × 10 /°C t2 = ? We know, L = L0T In our case, –3 –6 0.055 × 10 = 1 × 1.1 I 10 × (T1 +T2) –3 –3 0.055 = 11 × 10 × 20 ± 11 × 10 × T2 T2 = 20 + 5 = 25°C or 20 – 5 = 15°C The expt. Can be performed from 15 to 25°C 3 3 ƒ4°C = 1 g/m 17. ƒ0°C=0.098 g/m , ƒ 4 C 1 1 ƒ0°C =  0.998 =  1 + 4 = 1  T 1   4 0.998

1  1   = 0.0005 ≈ 5 × 10–4 0.998 -4 As density decreases  = –5 × 10 18. Iron rod Aluminium rod LAl LFe –8 –8 Fe = 12 × 10 /°C Al = 23 × 10 /°C Since the difference in length is independent of temp. Hence the different always remains constant. …(1) LFe = LFe(1 + Fe × T) LAl = LAl(1 + Al × T) …(2) LFe – LAl = LFe – LAl + LFe × Fe × T – LAl × Al × T L Fe  23 = Al = = 23 : 12 L Al  Fe 12 4+=

2

2

19. g1 = 9.8 m/s , T1 = 2

l1

T2 = 2

g1

Steel = 12 × 10 T1 = 20°C T1 = T2  2

g2 = 9.788 m/s

l1 g1

–6

= 2

l2 g2

= 2

l1(1  T ) g

/°C T2 = ?

l1(1  T ) g2

1  12  10 6  T 1 = 9 .8 9.788 9.788 –6   1 = 12 × 10 T 9 .8 



l1 l (1  T ) = 1 g1 g2

9.788 –6 = 1+ 12 × 10 × T 9 .8 0.00122  T = 12  10  6  T2 = – 101.6 + 20 = – 81.6 ≈ – 82°C  

 T2 – 20 = – 101.6 20. Given dAl = 2.000 cm dSt = 2.005 cm, –6 –6 S = 11 × 10 /°C Al = 23 × 10 /°C ds = 2.005 (1+ s T) (where T is change in temp.) –6  ds = 2.005 + 2.005 × 11 × 10 T –6 dAl = 2(1+ Al T) = 2 + 2 × 23 × 10 T The two will slip i.e the steel ball with fall when both the diameters become equal. So, –6 –6  2.005 + 2.005 × 11 × 10 T = 2 + 2 × 23 × 10 T -6  (46 – 22.055)10 × T = 0.005  T =

0.005  10 6 = 208.81 23.945 23.3

Steel Aluminium

23.Heat and Temperature Now T = T2 –T1 = T2 –10°C [ T1 = 10°C given] T2 = T + T1 = 208.81 + 10 = 281.81 21. The final length of aluminium should be equal to final length of glass. Let the initial length o faluminium = l l(1 – AlT) = 20(1 – 0) –6 –6  l(1 – 24 × 10 × 40) = 20 (1 – 9 × 10 × 40)  l(1 – 0.00096) = 20 (1 – 0.00036) 20  0.99964 l= = 20.012 cm 0.99904 Let initial breadth of aluminium = b b(1 – AlT) = 30(1 – 0) b =

30  (1  9  10 6  40)

(1  24  10 22. Vg = 1000 CC, VHg = ?

6

 40)

=

30  0.99964 = 30.018 cm 0.99904

T1 = 20°C –4 Hg = 1.8 × 10 /°C –6 g = 9 × 10 /°C

T remains constant Volume of remaining space = Vg – VHg Now Vg = Vg(1 + gT) …(1) VHg = VHg(1 + HgT) …(2) Subtracting (2) from (1) Vg – VHg = Vg – VHg + VggT – VHgHgT Vg  Hg 1.8  10 4 1000  =  = VHg g VHg 9  10  6  VHG =

9  10 3

= 500 CC. 1.8  10  4 3 23. Volume of water = 500cm 2 Area of cross section of can = 125 m Final Volume of water –4 3 = 500(1 + ) = 500[1 + 3.2 × 10 × (80 – 10)] = 511.2 cm The aluminium vessel expands in its length only so area expansion of base cab be neglected. 3 Increase in volume of water = 11.2 cm 3 Considering a cylinder of volume = 11.2 cm 11.2 Height of water increased = = 0.089 cm 125 24. V0 = 10 × 10× 10 = 1000 CC 3 T = 10°C, VHG – Vg = 1.6 cm –6 –6 g = 6.5 × 10 /°C, Hg = ?, g= 3 × 6.5 × 10 /°C …(1) VHg = vHG(1 + HgT) Vg = vg(1 + gT) …(2) VHg – Vg = VHg –Vg + VHgHg T – Vgg T –6  1.6 = 1000 × Hg × 10 – 1000 × 6.5 × 3 × 10 × 10

1.6  6.3  3  10 2 –4 –4 = 1.789 × 10  1.8 × 10 /°C 10000 3 3 25. ƒ = 880 Kg/m , ƒb = 900 Kg/m –3 T1 = 0°C,  = 1.2 × 10 /°C, –3 b = 1.5 × 10 /°C The sphere begins t sink when, (mg)sphere = displaced water  Hg =

23.4

23.Heat and Temperature  Vƒ g = Vƒb g ƒb ƒ    1     1   b 

880 900 = 1  1.2  10 3  1  1.5  10 3  –3 –3  880 + 880 × 1.5 × 10 () = 900 + 900 × 1.2 × 10 () –3 –3  (880 × 1.5 × 10 – 900 × 1.2 × 10 ) () = 20 –3  (1320 – 1080) × 10 () = 20   = 83.3°C ≈ 83°C L = 100°C A longitudinal strain develops if and only if, there is an opposition to the expansion. Since there is no opposition in this case, hence the longitudinal stain here = Zero. 1 = 20°C, 2 = 50°C –5 steel = 1.2 × 10 /°C Longitudinal stain = ? L L Stain = = =  L L –5 –4 = 1.2 × 10 × (50 – 20) = 3.6 × 10 2 –6 2 A = 0.5mm = 0.5 × 10 m T1 = 20°C, T2 = 0°C –5 11 2 s = 1.2 × 10 /°C, Y = 2 × 2 × 10 N/m Decrease in length due to compression = L …(1) Stress F L FL Y=  =  L = …(2) Strain A L AY Tension is developed due to (1) & (2) Equating them, FL  F = AY L = AY -5 –5 11 = 1.2 × 10 × (20 – 0) × 0.5 × 10 2 × 10 = 24 N 1 = 20°C, 2 = 100°C 2 –6 2 A = 2mm = 2 × 10 m –6 11 2 steel = 12 × 10 /°C, Ysteel = 2 × 10 N/m Force exerted on the clamps = ? 

26.

27.

28.

29.

F    A  = Y  F = Y  L  L = YLA = YA L Strain L 11 –6 –6 = 2 × 10 × 2 × 10 × 12 × 10 × 80 = 384 N 30. Let the final length of the system at system of temp. 0°C = ℓ Initial length of the system = ℓ0 When temp. changes by .  Strain of the system =  1  0 

total stress of system But the total strain of the system = total young' s mod ulusof of system Now, total stress = Stress due to two steel rod + Stress due to Aluminium = ss + s ds  + al at  = 2% s  + 2 Aℓ  Now young’ modulus of system = s + s + al = 2s + al

23.5



1m

Steel Aluminium Steel

23.Heat and Temperature  Strain of system = 

2 s  s    s  al  2  s   al

  0 2 s  s    s  al  = 0 2  s   al

1   al  al  2 s  s    ℓ = ℓ0    al  2 s   31. The ball tries to expand its volume. But it is kept in the same volume. So it is kept at a constant volume. So the stress arises P V =BP= B = B × V  V     v  11

–6

–6

7

8

= B × 3 = 1.6 × 10 × 10 × 3 × 12 × 10 × (120 – 20) = 57.6 × 19  5.8 × 10 pa.  32. Given 0 = Moment of Inertia at 0°C  = Coefficient of linear expansion To prove,  = 0 = (1 + 2) Let the temp. change to  from 0°C T =  Let ‘R’ be the radius of Gyration, 2 0 = MR where M is the mass. Now, R = R (1 + ), 2 2 2 2 Now,  = MR = MR (1 + )  = MR (1 + 2) 2 2 [By binomial expansion or neglecting   which given a very small value.] (proved) So,  = 0 (1 + 2) 33. Let the initial m.. at 0°C be 0

 K  = 0 (1 + 2) T = 2

(from above question)

At 5°C,

T1 = 2

 0 (1  2) = 2 K

At 45°C,

T2 = 2

 (1  90 )  0 (1  2 45) = 2 0 K K

T2 = T1

1  90 = 1  10

1  90  2.4  10 5 1  10  2.4  10

5

 0 (1  25)  (1  10 ) = 2 0 K K

1.00216 1.00024

 T –2 % change =  2  1  100 = 0.0959% = 9.6 × 10 % T   1 T2 = 50°C, T = 30°C 34. T1 = 20°C, 5  = 1.2 × 10 /°C  remains constant V V (II)  = (I)  = R R –5 Now, R = R(1 + ) = R + R × 1.2 × 10 × 30 = 1.00036R From (I) and (II) V V V  = R R 1.00036R V = 1.00036 V (1.00036 V  V ) –2 % change = × 100 = 0.00036 × 100 = 3.6 × 10 V      23.6

CHAPTER 24

KINETIC THEORY OF GASES 1.

Volume of 1 mole of gas RT 0.082  273 –3 –2 3 PV = nRT  V = = = 22.38 ≈ 22.4 L = 22.4 × 10 = 2.24 × 10 m P 1

2.

n=

3.

1  1 10 3 10 3 PV 1 = = = RT 0.082  273 22.4 22400 1 23 19 No of molecules = 6.023 × 10 × = 2.688 × 10 22400 3 –5 V = 1 cm , T = 0°C, P = 10 mm of Hg 1.36  980  10 6  1 PV ƒgh  V –13 = = = 5.874 × 10 RT RT 8.31  273 23 –13 11 No. of moluclues = No × n = 6.023 × 10 × 5.874 × 10 = 3.538 × 10

n=

4.

n=

1  1 10 3 10 3 PV = = RT 0.082  273 22.4

5.

10

3



 32 –3 g = 1.428 × 10 g = 1.428 mg 22.4 Since mass is same n1 = n2 = n nR  300 nR  600 P1 = , P2 = V0 2V0 mass =

2V0

2V0 P1 nR  300 1  = = =1:1 P2 V0 nR  600 1 6.

600 K

–3

V = 250 cc = 250 × 10 –3 –3 –3 –6 –3 P = 10 mm = 10 × 10 m = 10 × 13600 × 10 pascal = 136 × 10 pascal T = 27°C = 300 K

136  10 3  250 PV 136  250 =  10  3 =  10  6 8.3  300 RT 8.3  300 136  250 No. of molecules =  10  6  6  10 23 = 81 × 1017 ≈ 0.8 × 1015 8.3  300 5 6 P1 = 8.0 × 10 Pa, P2 = 1 × 10 Pa, T1 = 300 K, Since, V1 = V2 = V

n=

7.

8.

9.

T2 = ?

P1V1 PV 1  10 6  V 8  10 5  V 1 10 6  300 = 2 2  =  T2 = = 375° K T1 T2 300 T2 8  10 5 3 6 3 T = 300 K, P=? m = 2 g, V = 0.02 m = 0.02 × 10 cc = 0.02 × 10 L, M = 2 g, m 2 PV = nRT  PV = RT  P × 20 =  0.082  300 M 2 0.082  300 5 5 P= = 1.23 atm = 1.23 × 10 pa ≈ 1.23 × 10 pa 20 nRT m RT ƒRT P= =  = V M V M –3 3 ƒ  1.25 × 10 g/cm 7 R  8.31 × 10 ert/deg/mole T  273 K M=

1.25  10 3  8.31  10 7  273 ƒRT 4 = = 0.002796 × 10 ≈ 28 g/mol P 13.6  980  76 24.1

V0 300 K

Kinetic Theory of Gases 10. T at Simla = 15°C = 15 + 273 = 288 K –2 P at Simla = 72 cm = 72 × 10 × 13600 × 9.8 T at Kalka = 35°C = 35 + 273 = 308 K –2 P at Kalka = 76 cm = 76 × 10 × 13600 × 9.8 PV = nRT m m PM  PV = RT  PM = RT  ƒ = M V RT PSimla  M RTKalka ƒSimla  = ƒKalka RTSimla PKalka  M =

72  10 2  13600  9.8  308 2

288  76  10  13600  9.8 ƒKalka 1 = = 0.987 ƒSimla 1.013 11. n1 = n2 = n nRT nRT , P2 = P1 = V 3V P1 nRT 3 V =  =3:1 P2 V nRT

=

72  308 = 1.013 76  288

V

V

3V

V

PT

PT

P2T

P1 -

12. r.m.s velocity of hydrogen molecules = ? –3 T = 300 K, R = 8.3, M = 2 g = 2 × 10 Kg

3RT C= M

C=

3  8.3  300 2  10  3

= 1932. 6 m/s ≈1930 m/s

Let the temp. at which the C = 2 × 1932.6 is T 2 × 1932.6 =

3  8 .3  T  2  10

3

2

 (2 × 1932.6) =

3  8 .3  T  2  10  3

(2  1932.6)2  2  10 3 = T 3  8 .3  T = 1199.98 ≈ 1200 K. 

13. Vrms = =

3P ƒ

5

P = 10 Pa = 1 atm,

ƒ=

1.77  10 4 10  3

3  10 5  10 3

= 1301.8 ≈ 1302 m/s. 1.77  10  4 14. Agv. K.E. = 3/2 KT –19 3/2 KT = 0.04 × 1.6 × 10 –23 –19  (3/2) × 1.38 × 10 × T = 0.04 × 1.6 × 10 T=

2  0.04  1.6  10 19 3  1.38  10  23

4

= 0.0309178 × 10 = 309.178 ≈ 310 K

8RT 8  8.3  300 = M 3.14  0.032 Dis tan ce 6400000  2 T= = = 445.25 m/s Speed 445 .25

15. Vavg =

28747 .83 km = 7.985 ≈ 8 hrs. 3600 –3 16. M = 4 × 10 Kg =

8  8.3  273 8RT = = 1201.35 M 3.14  4  10  3 –27 –24 –24 Momentum = M × Vavg = 6.64 × 10 × 1201.35 = 7.97 × 10 ≈ 8 × 10 Kg-m/s.

Vavg =

24.2

Kinetic Theory of Gases

8RT 8  8.3  300 = M 3.14  0.032 8RT1 8RT2 Now, = 2  4

17. Vavg =

T1 1 = T2 2 8RT M

18. Mean speed of the molecule = Escape velocity =

8RT = M T=

VavgN2



2gr

8RT = 2gr M

2  9.8  6400000  3.14  2  10 3 2grM = = 11863.9 ≈ 11800 m/s. 8R 8  8 .3 8RT M

19. Vavg =

VavgH2

2gr

=

8RT   28 =  8RT 2

28 = 2

14 = 3.74

20. The left side of the container has a gas, let having molecular wt. M1 Right part has Mol. wt = M2 Temperature of both left and right chambers are equal as the separating wall is diathermic

3RT = M1 21. Vmean =

3RT 8RT 8RT  = M1 M2 M2 8RT = M



M1 M 3 3 =  1 = = 1.1775 ≈ 1.18 M2 M2 8 8

8  8.3  273

= 1698.96 3.14  2  10  3 Total Dist = 1698.96 m 1698.96 10 = 1.23 × 10 No. of Collisions = 1.38  10  7 5 22. P = 1 atm = 10 Pascal –3 T = 300 K, M = 2 g = 2 × 10 Kg

8  8.3  300 8RT = = 1781.004 ≈ 1780 m/s M 3.14  2  10  3 (b) When the molecules strike at an angle 45°,

(a) Vavg =

Force exerted = mV Cos 45° – (–mV Cos 45°) = 2 mV Cos 45° = 2 m V No. of molecules striking per unit area = =

10

5

2  2  10

3

 1780

=

3 2  1780

Force 2mv  Area

=

1 2

=

2 mV

Pr essure 2mV

 10 31 = 1.19 × 10–3 × 1031 = 1.19 × 1028 ≈ 1.2 × 1028

6  10 23 PV PV 23. 1 1 = 2 2 T1 T2 5

P1  200 KPa = 2 × 10 pa T1 = 20°C = 293 K 102  V1 V2 = V1 + 2% V1 = 100 

P2 = ? T2 = 40°C = 313 K

P  102  V1 2  10 5  V1 2  10 7  313 = 2  P2 = = 209462 Pa = 209.462 KPa 293 100  313 102  293 24.3

Kinetic Theory of Gases –3

3

5

24. V1 = 1 × 10 m , P1V1 = n1R1T1 n=

P1 = 1.5 × 10 Pa,

P1V1 1.5  10 5  1 10 3 = R1T1 8.3  400

T1 = 400 K

n=

1 .5 8 .3  4

1 .5 1 .5 M =  32 = 1.4457 ≈ 1.446 8 .3  4 8 .3  4 5 –3 3 P2 = 1 × 10 Pa, V2 = 1 × 10 m , P2V2 = n2R2T2  m1 =

 n2 =

T2 = 300 K

P2 V2 10 5  10 3 1 = = = 0.040 R 2 T2 8.3  300 3  8 .3

 m2 = 0.04 × 32 = 1.285 m = m1 – m2 =1.446 – 1.285 = 0.1608 g ≈ 0.16 g 5 5 5 25. P1 = 10 + ƒgh = 10 + 1000 × 10 × 3.3 = 1.33 × 10 pa 4 5 –3 3 T1 = T2 = T, V1 = (2 × 10 ) P2 = 10 , 3 4 3 V2 = r , r=? 3 P1V1 PV = 2 2 T1 T2 

1.33  10 5 

4 4 10 5   r 2    (2  10  3 )3 3 3 = T1 T2 5

–9

5

3

 1.33 × 8 × 10 × 10 = 10 × r 5 26. P1 = 2 atm = 2 × 10 pa 3 T1 = 300 K V1 = 0.002 m , P1V1 = n1RT1 n=

r=

3

–3 10.64  10 3 = 2.19 × 10 ≈ 2.2 mm

P1V1 2  10 5  0.002 4 = = = 0.1606 RT1 8.3  300 8 .3  3 5

P2 = 1 atm = 10 pa 3 V2 = 0.0005 m , P2V2 = n2RT2  n2 =

T2 = 300 K

P2 V2 10 5  0.0005 5 1 = =  = 0.02 RT2 8.3  300 3  8.3 10

n = moles leaked out = 0.16 – 0.02 = 0.14 27. m = 0.040 g, T = 100°C, MHe = 4 g 3 3 m T = ? U = nRt =   RT 2 2 M 3 m 3 m Given   RT  12 =   RT 2 M 2 M  1.5 × 0.01 × 8.3 × 373 + 12 = 1.5 × 0.01 × 8.3 × T 58.4385 = 469.3855 K = 196.3°C ≈ 196°C  T = 0.1245 2 28. PV = constant 2 2  P1V1 = P2V2 nRT1 nRT2   V12 =  V2 2 V1 V2  T1 V1 = T2 V2 = TV = T1 × 2V  T2 =

T 2 24.4

Kinetic Theory of Gases 29. PO2 =

nO2

no2 RT

,

PH2 =

nH2 RT

V V 1.60 m = = = 0.05 32 MO2

 nO  nH2  RT Now, Pmix =  2  V   2.80 m nH2 = = = 0.1 MH2 28 (0.05  0.1)  8.3  300 2 = 2250 N/m 0.166 30. P1 = Atmospheric pressure = 75 × ƒg V1 = 100 × A P2 = Atmospheric pressure + Mercury pessue = 75ƒg + hgƒg (if h = height of mercury) V2 = (100 – h) A P1V1 = P2V2  75ƒg(100A) = (75 + h)ƒg(100 – h)A 2  75 × 100 = (74 + h) (100 – h)  7500 = 7500 – 75 h + 100 h – h 2 2  h – 25 h = 0  h = 25 h  h = 25 cm Height of mercury that can be poured = 25 cm 31. Now, Let the final pressure; Volume & Temp be After connection = PA  Partial pressure of A PB  Partial pressure of B  P V P  2V Now, A = A TA T Pmix =

 P P Or A = A T 2TA

A

…(1)

PA

Similarly,

P PB = B 2TB T

: TA V



…(2)

Adding (1) & (2) P P PA  PB P  1P = A  B =  A  B   2TA 2TB T T 2  TA TB 

P P  1P =  A  B  T 2  TA TB  32. V = 50 cc = 50 × 10–6 cm3 5 P = 100 KPa = 10 Pa (a) PV = nrT1 

 PV =

[ PA + PB = P]

M = 28.8 g

m 50  28.8  10 1 PMV 10 5  28.8  50  10 6 RT1  m = = = = 0.0635 g. M RT1 8.3  273 8.3  273

(b) When the vessel is kept on boiling water PV =

10 5  28.8  50  10 6 50  28.8  10 1 PVM m RT2  m = = = = 0.0465 M RT2 8.3  373 8.3  373

(c) When the vessel is closed 0.0465 –6  8.3  273 P × 50 × 10 = 28.8 0.0465  8.3  273 6 P= = 0.07316 × 10 Pa ≈ 73 KPa 28.8  50  10  6 24.5

B PB

: TB V

Kinetic Theory of Gases 33. Case I  Net pressure on air in volume V = Patm – hƒg = 75 × ƒHg – 10 ƒHg = 65 × ƒHg × g Case II  Net pressure on air in volume ‘V’ = Patm + ƒHg × g × h P1V1 = P2V2  ƒHg × g × 65 × A × 20 = ƒHg × g × 75 + ƒHg × g × 10 × A × h 65  20  62 × 20 = 85 h  h = = 15.2 cm ≈ 15 cm 85 34. 2L + 10 = 100  2L = 90  L = 45 cm Applying combined gas eqn to part 1 of the tube ( 45 A )P0 ( 45  x )P1 = 300 273 273  45  P0  P1 = 300( 45  x ) Applying combined gas eqn to part 2 of the tube 45 AP0 ( 45  x )AP2 = 300 400 400  45  P0  P2 = 300( 45  x ) P1 = P2 273  45  P0 400  45  P0  = 300( 45  x ) 300( 45  x )





V

20 cm

P  20 A P  A = …(1) 400 T P(30  x ) P  10 A  = …(2) 100 T Equating (1) and (2) 1 x  =  30 – x = 2x  3x = 30  x = 10 cm 2 30  x The separator will be at a distance 10 cm from left end. 24.6

h

V

27°C

L

l

1

2

P0

10

P0

L-x

L+x

P1

P2

0°C

 (45 – x) 400 = (45 + x) 273  18000 – 400 x = 12285 + 273 x  (400 + 273)x = 18000 – 12285  x = 8.49 273  46  76 P1 = = 85 % 25 cm of Hg 300  36.51 Length of air column on the cooler side = L – x = 45 – 8.49 = 36.51 35. Case I Atmospheric pressure + pressure due to mercury column Case II Atmospheric pressure + Component of the pressure due to mercury column 20cm P1V1 = P2V2 43cm  (76 × ƒHg × g + ƒHg × g × 20) × A × 43 = (76 × ƒHg × g + ƒHg × g × 20 × Cos 60°) A × ℓ  96 × 43 = 86 × ℓ 96  43 ℓ= = 48 cm 86 36. The middle wall is weakly conducting. Thus after a long 10 cm 20 cm time the temperature of both the parts will equalise. The final position of the separating wall be at distance x 400 K 100 K P from the left end. So it is at a distance 30 – x from the right P end Putting combined gas equation of one side of the separating wall, P1  V1 P  V2 = 2 T1 T2 

10 cm

10 cm

27°C

10

0°C

60° ℓ

x T P

30 – x T P

Kinetic Theory of Gases 37.

dV = r  dV = r dt dt Let the pumped out gas pressure dp Volume of container = V0 At a pump dv amount of gas has been pumped out. Pdv = –V0df  PV df = –V0 dp P





P

dp =  p

t

dtr

V 0

 P = P e rt / V0

0

Half of the gas has been pump out, Pressure will be half =  ln 2 = 38. P =

rt V0

 t = ln2

1  vt / V0 e 2

0 r

P0  V 1    V0



nRT = V



RT = V



RT = V0

  

2

P0  V 1    V0 P0

  

 V 1    V0 P0

  

2

 V 1    V0

  

2

2

[PV = nRT according to ideal gas equation]

[Since n = 1 mole]

[At V = V0]

 P0V0 = RT(1 +1)  P0V0 = 2 RT  T =

P0 V0 2R

39. Internal energy = nRT Now, PV = nRT PV Here P & V constant nT = R  nT is constant  Internal energy = R × Constant = Constant 40. Frictional force =  N Let the cork moves to a distance = dl  Work done by frictional force = Nde Before that the work will not start that means volume remains constant P P P 1  1 = 2  = 2  P2 = 2 atm T1 T2 300 600  Extra Pressure = 2 atm – 1 atm = 1 atm Work done by cork = 1 atm (Adl) Ndl = [1atm][Adl]

1 10 5  (5  10 2 )2 1 10 5    25  10 5 = 2 2 dN N Total circumference of work = 2r = dl 2r N=

=

1 10 5    25  10 5 1  10 5  25  10 5 4 = = 1.25 × 10 N/M 0.2  2r 0.2  2  5  10 5

24.7

Kinetic Theory of Gases 41.

P1V1 PV = 2 2 T1 T2 

2P0

P0

P0 V PV =  P = 2 P0 T0 2T0

Net pressure = P0 outwards  Tension in wire = P0 A Where A is area of tube. [ Since liquid at the same level have same pressure] 42. (a) 2P0x = (h2 + h0)ƒg  2P0 = h2 ƒg + h0 ƒg h2  h2 ƒg = 2P0 – h0 ƒg 2P0 2P0 2P0 h 0 ƒg  =  h0 h2 = ƒg ƒg ƒg (b) K.E. of the water = Pressure energy of the water at that layer P 1 2  mV = m  2 ƒ 2

V =

  2P 2 =   ƒ  ƒP0  ƒg(h1  h 0  1/ 2

  2 V=   ƒ  P ƒ g ( h h    0 1 0   (c) (x + P0)ƒh = 2P0  2P0 + ƒg (h –h0)= P0 + ƒgx P0 X= = h2 + h1 ƒg  h1  h0

 i.e. x is h1 meter below the top  x is –h1 above the top 2 –3 43. A = 100 cm = 10 m m = 1 kg, P = 100 K Pa = 105 Pa ℓ = 20 cm Case I = External pressure exists Case II = Internal Pressure does not exist P1V1 = P2V2

1  9 .8 1  9 .8    10 5  × V V = 3 10 10  3   5

3

3

 (10 + 9.8 × 10 )A × ℓ = 9.8 × 10 × A × ℓ 5 –1 2 3  10 × 2 × 10 + 2 × 9.8 × 10 = 9.8 × 10 × ℓ  ℓ =

2  10 4  19.6  10 2 9.8  10 3

= 2.24081 m

44. P1V1 = P2V2

 mg   P0  A P0 Aℓ   A  

  1  9. 8    10 5 0.2 = 105 ℓ   10  10  4 3

5

5

 (9.8 × 10 + 10 )× 0.2 = 10 ℓ 3 5  109.8 × 10 × 0.2 = 10 ℓ 109.8  0.2 = 0.2196 ≈ 0.22 m ≈ 22 cm  ℓ = 10 2 24.8

h0

h1

Kinetic Theory of Gases 45. When the bulbs are maintained at two different temperatures. The total heat gained by ‘B’ is the heat lost by ‘A’ Let the final temp be x So, m1 St = m2 St  n1 M × s(x – 0) = n2 M × S × (62 – x)  n1 x = 62n2 – n2 x x=

V

V

A

B

62n 2 62n 2 = = 31°C = 304 K n1  n 2 2n 2

For a single ball

Initial Temp = 0°C

P = 76 cm of Hg

P1V1 PV = 2 2 T1 T2

V1 = V2

Hence n1 = n2

P V 76  V 403  76 = 2  P2 = = 84.630 ≈ 84°C 273 304 273 46. Temp is 20° Relative humidity = 100% So the air is saturated at 20°C Dew point is the temperature at which SVP is equal to present vapour pressure So 20°C is the dew point. 47. T = 25°C P = 104 KPa 

VP [SVP = 3.2 KPa, RH = 0.6] SVP 3 3 3 VP = 0.6 × 3.2 × 10 = 1.92 × 10 ≈ 2 × 10 When vapours are removed VP reduces to zero 3 3 3 Net pressure inside the room now = 104 × 10 – 2 × 10 = 102 × 10 = 102 KPa 48. Temp = 20°C Dew point = 10°C The place is saturated at 10°C Even if the temp drop dew point remains unaffected. The air has V.P. which is the saturation VP at 10°C. It (SVP) does not change on temp. RH =

VP SVP The point where the vapour starts condensing, VP = SVP We know P1V1 = P2V2 3 RH SVP × 10 = SVP × V2  V2 = 10RH  10 × 0.4 = 4 cm 50. Atm–Pressure = 76 cm of Hg When water is introduced the water vapour exerts some pressure which counter acts the atm pressure. The pressure drops to 75.4 cm Pressure of Vapour = (76 – 75.4) cm = 0.6 cm

49. RH =

VP 0 .6 = = 0.6 = 60% SVP 1 51. From fig. 24.6, we draw r, from Y axis to meet the graphs. Hence we find the temp. to be approximately 65°C & 45°C 52. The temp. of body is 98°F = 37°C At 37°C from the graph SVP = Just less than 50 mm B.P. is the temp. when atmospheric pressure equals the atmospheric pressure. Thus min. pressure to prevent boiling is 50 mm of Hg. 53. Given SVP at the dew point = 8.9 mm SVP at room temp = 17.5 mm Dew point = 10°C as at this temp. the condensation starts Room temp = 20°C R. Humidity =

RH =

SVP at dew po int 8 .9 = = 0.508 ≈ 51% SVP at room temp 17.5 24.9

Kinetic Theory of Gases 3

54. 50 cm of saturated vapour is cooled 30° to 20°. The absolute humidity of saturated H2O vapour 30 g/m 3 Absolute humidity is the mass of water vapour present in a given volume at 30°C, it contains 30 g/m 3 at 50 m it contains 30 × 50 = 1500 g at 20°C it contains 16 × 50 = 800 g Water condense = 1500 – 800 = 700 g. 55. Pressure is minimum when the vapour present inside are at saturation vapour pressure As this is the max. pressure which the vapours can exert. Hence the normal level of mercury drops down by 0.80 cm  The height of the Hg column = 76 – 0.80 cm = 75.2 cm of Hg. [ Given SVP at atmospheric temp = 0.80 cm of Hg] 56. Pressure inside the tube = Atmospheric Pressure = 99.4 KPa = Atmospheric pressure – V.P. Pressure exerted by O2 vapour = 99.4 KPa – 3.4 KPa = 96 KPa No of moles of O2 = n 3 –6 96 × 10 ×50 × 10 = n × 8.3 × 300

96  50  10 3 –3 –3 = 1.9277 × 10 ≈ 1.93 × 10 8.3  300 57. Let the barometer has a length = x Height of air above the mercury column = (x – 74 – 1) = (x – 73) Pressure of air = 76 – 74 – 1 = 1 cm nd For 2 case height of air above = (x – 72.1 – 1 – 1) = (x – 71.1) Pressure of air = (74 – 72.1 – 1) = 0.99 n=

9 (x – 71.1)  10(x – 73) = 9 (x – 71.1) 10  x = 10 × 73 – 9 × 71.1 = 730 – 639.9 = 90.1 Height of air = 90.1 Height of barometer tube above the mercury column = 90.1 + 1 = 91.1 mm 58. Relative humidity = 40% SVP = 4.6 mm of Hg (x – 73)(1) =

VP 4 .6 P1V PV = 2 T1 T2

0.4 =

 VP = 0.4 × 4.6 = 1.84 

P 1.84 1.84 = 2  P2 =  293 273 273 293

Relative humidity at 20°C

VP 1.84  293 = = 0.109 = 10.9% SVP 273  10 VP 59. RH = SVP VP Given, 0.50 = 3600  VP = 3600 × 0.5 Let the Extra pressure needed be P m RT m 8.3  300  =  So, P = M V 18 1 m Now,  8.3  300  3600  0.50 = 3600 18 =

[air is saturated i.e. RH = 100% = 1 or VP = SVP]

 36  18  m=    6 = 13 g  8 .3  24.10

3

Kinetic Theory of Gases 3

60. T = 300 K, Rel. humidity = 20%, V = 50 m 3 SVP at 300 K = 3.3 KPa, V.P. = Relative humidity × SVP = 0.2 × 3.3 × 10 PV =

m m 3 RT  0.2 × 3.3 × 10 × 50 =  8.3  300 M 18

0.2  3.3  50  18  10 3 = 238.55 grams ≈ 238 g 8.3  300 Mass of water present in the room = 238 g. m=

61. RH =

VP VP 3  0.20 =  VP = 0.2 × 3.3 × 10 = 660 SVP 3.3  10 3

nRT m RT 500 8.3  300  =  = = 1383.3 V M V 18 50 2034.3 = 0.619 ≈ 62% Net P = 1383.3 + 660 = 2043.3 Now, RH = 3300 VP VP 3 62. (a) Rel. humidity =  0.4 =  VP = 0.4 × 1.6 × 10 SVP at 15C 1.6  10 3 PV = nRT P =

The evaporation occurs as along as the atmosphere does not become saturated. 3 3 3 3 Net pressure change = 1.6 × 10 – 0.4 × 1.6 × 10 = (1.6 – 0.4 × 1.6)10 = 0.96 × 10 3

Net mass of water evaporated = m  0.96 × 10 × 50 =

m  8.3  288 18

0.96  50  18  10 3 = 361.45 ≈ 361 g 8.3  288 (b) At 20°C SVP = 2.4 KPa, At 15°C SVP = 1.6 KPa 3 3 Net pressure charge = (2.4 – 1.6) × 10 Pa = 0.8 × 10 Pa m 3  8.3  293 Mass of water evaporated = m = 0.8 × 10 50 = 18 m=

 m =

0.8  50  18  10 3 = 296.06 ≈ 296 grams 8.3  293



24.11

CHAPTER – 25 CALORIMETRY 1.

Mass of aluminium = 0.5kg, Mass of water = 0.2 kg Mass of Iron = 0.2 kg Temp. of aluminium and water = 20°C = 297°k Sp heat o f Iron = 100°C = 373°k. Sp heat of aluminium = 910J/kg-k Sp heat of Iron = 470J/kg-k Sp heat of water = 4200J/kg-k Heat again = 0.5 × 910(T – 293) + 0.2 × 4200 × (343 –T) = (T – 292) (0.5 × 910 + 0.2 × 4200) Heat lost = 0.2 × 470 × (373 – T)  Heat gain = Heat lost  (T – 292) (0.5 × 910 + 0.2 × 4200) = 0.2 × 470 × (373 – T)  (T – 293) (455 + 8400) = 49(373 – T)

 1295   (T – 293)   = (373 – T)  94   (T – 293) × 14 = 373 – T

4475 = 298 k 15  T = 298 – 273 = 25°C. The final temp = 25°C. mass of Iron = 100g water Eq of caloriemeter = 10g mass of water = 240g Let the Temp. of surface = 0C Total heat gained = Total heat lost. Siron = 470J/kg°C 100 250 So, × 470 ×( – 60) = × 4200 × (60 – 20) 1000 1000  47 – 47 × 60 = 25 × 42 × 40 T=

2.

2820 44820 = = 953.61°C 47 47 The temp. of A = 12°C The temp. of B = 19°C The temp. of C = 28°C The temp of  A + B = 16° The temp. of  B + C = 23° In accordance with the principle of caloriemetry when A & B are mixed 3 …(1) MCA (16 – 12) = MCB (19 – 16)  CA4 = CB3  CA = CB 4 And when B & C are mixed   = 4200 +

3.

4 CB 5 When A & c are mixed, if T is the common temperature of mixture MCA (T – 12) = MCC (28 – T) MCB (23 – 19)= MCC (28 – 23)  4CB = 5CC  CC =

3 4    CB(T – 12) =   CB(28 – T) 4 5  15T – 180 = 448 – 16T T=

628 = 20.258°C = 20.3°C 31     

25.1

…(2)

CHAPTER 26

LAWS OF THERMODYNAMICS QUESTIONS FOR SHORT ANSWER 1. 2. 3.

4.

5.

6. 7. 8. 9. 10.

11.

12. 13. 14. 15.

No in isothermal process heat is added to a system. The temperature does not increase so the internal energy does not. Yes, the internal energy must increase when temp. increases; as internal energy depends upon temperature U  T Work done on the gas is 0. as the P.E. of the container si increased and not of gas. Work done by the gas is 0. as the gas is not expanding. The temperature of the gas is decreased. W = F × d = Fd Cos 0° = Fd F Change in PE is zero. Change in KE is non Zero. 1 d 1 So, there may be some internal energy. The outer surface of the cylinder is rubbed vigorously by a polishing machine. The energy given to the cylinder is work. The heat is produced on the cylinder which transferred to the gas. No. work done by rubbing the hands in converted to heat and the hands become warm. When the bottle is shaken the liquid in it is also shaken. Thus work is done on the liquid. But heat is not transferred to the liquid. Final volume = Initial volume. So, the process is isobaric. Work done in an isobaric process is necessarily zero. No word can be done by the system without changing its volume. Internal energy = U = nCVT Now, since gas is continuously pumped in. So n2 = 2n1 as the p2 = 2p1. Hence the internal energy is also doubled. When the tyre bursts, there is adiabatic expansion of the air because the pressure of the air inside is sufficiently higher than atmospheric pressure. In expansion air does some work against surroundings. So the internal energy decreases. This leads to a fall in temperature. ‘No’, work is done on the system during this process. No, because the object expands during the process i.e. volume increases. No, it is not a reversible process. Total heat input = Total heat out put i.e., the total heat energy given to the system is converted to mechanical work. Yes, the entropy of the body decreases. But in order to cool down a body we need another external sink which draws out the heat the entropy of object in partly transferred to the external sink. Thus once nd entropy is created. It is kept by universe. And it is never destroyed. This is according to the 2 law of thermodynamics OBJECTIVE – 

1.

3.

(d) Dq = DU + DW. This is the statement of law of conservation of energy. The energy provided is utilized to do work as well as increase the molecular K.E. and P.E. (b) Since it is an isothermal process. So temp. will remain constant as a result ‘U’ or internal energy will also remain constant. So the system has to do positive work. AQ1  (a) In case of A W 1 > W 2 (Area under the graph is higher for A than for B). P Q = u + dw. du for both the processes is same (as it is a state function) BQ2  Q1 > Q2 as W 1 > W2 V

4.

(b) As Internal energy is a state function and not a path function. U1 = U2

2.

A P B

26.1

V

Laws of thermodynamics 5.

(a) In the process the volume of the system increases continuously. Thus, the work done increases continuously.

P V

6.

(c) for A  In a so thermal system temp remains same although heat is added. for B  For the work done by the system volume increase as is consumes heat.

7.

(c) In this case P and T varry proportionally i.e. P/T = constant. This is possible only when volume does not change.  pdv = 0 

P

f T

8.

(c) Given : VA = VB. But PA < PB Now, WA = PA VB; WB = PB VB; So, W A < WB.

P

B A T

9.

 1. 2.

 (b) As the volume of the gas decreases, the temperature increases as well as the pressure. But, on passage of time, the heat develops radiates through the metallic cylinder thus T decreases as well as the pressure. OBJECTIVE –  (b), (c) Pressure P and Volume V both increases. Thus work done is positive (V increases). Heat must be added to the system to follow this process. So temperature must increases. (a) (b) Initial temp = Final Temp. Initial internal energy = Final internal energy. i.e. U = 0, So, this is found in case of a cyclic process.

3.

(d) U = Heat supplied, W = Work done. (Q – W) = du, du is same for both the methods since it is a state function.

4.

(a) (c) Since it is a cyclic process. So, U1 = – U2, hence U1 + U2 = 0

P

Q – W = 0

A B

5.

V (a) (d) Internal energy decreases by the same amount as work done. du = dw,  dQ = 0. Thus the process is adiabatic. In adiabatic process, dU = – dw. Since ‘U’ decreases nR T1  T2  is +ve. T1 > T2  Temperature decreases. U2 – U2 is –ve. dw should be +ve   1

1.

t1 = 15°c t2 = 17°c t = t2 – t1 = 17 – 15 = 2°C = 2 + 273 = 275 K mw = 200 g = 0.2 kg mv = 100 g = 0.1 kg cug = 420 J/kg–k W g = 4200 J/kg–k (a) The heat transferred to the liquid vessel system is 0. The internal heat is shared in between the vessel and water. (b) Work done on the system = Heat produced unit –3 –3  dw = 100 × 10 × 420 × 2 + 200 × 10 × 4200 × 2 = 84 + 84 × 20 = 84 × 21 = 1764 J. (c)dQ = 0, dU = – dw = 1764. [since dw = –ve work done on the system] (a) Heat is not given to the liquid. Instead the mechanical work done is converted to heat. So, heat given to liquid is z. (b) Work done on the liquid is the PE lost by the 12 kg mass = mgh = 12 × 10× 0.70 = 84 J We know, 84 = mst (c) Rise in temp at t 12 kg

EXERCISES

2.

 84 = 1 × 4200 × t (for ‘m’ = 1kg)  t =

84 = 0.02 k 4200

26.2

Laws of thermodynamics 3.

mass of block = 100 kg u = 2 m/s, m = 0.2 v = 0 dQ = du + dw In this case dQ = 0

1 1  1  – du = dw  du =   mv 2  mu 2  =  100  2  2 = 200 J 2 2  2 4.

5.

6.

7.

8.

9.

Q = 100 J We know, U = Q – W Here since the container is rigid, V = 0, Hence the W = PV = 0, So, U = Q = 100 J. 3 3 P1 = 10 kpa = 10 × 10 pa. P2 = 50 × 10 pa.

v1 = 200 cc.

v2 = 50 cc

1 (i) Work done on the gas = (10  50)  10 3  (50  200 )  10  6 = – 4.5 J 2 (ii) dQ = 0  0 = du + dw  du = – dw = 4.5 J initial State ‘I’ Final State ‘f’ P P Given 1 = 2 T1 T2 where P1  Initial Pressure ; P2  Final Pressure. T2, T1  Absolute temp. So, V = 0 Work done by gas = PV = 0 In path ACB, 3 –6 W AC + WBC = 0 + pdv = 30 × 10 (25 – 10) × 10 = 0.45 J 3 –6 In path AB, W AB = ½ × (10 + 30) × 10 15 × 10 = 0.30 J 3 –6 In path ADB, W = W AD + WDB = 10 × 10 (25 – 10) × 10 + 0 = 0.15 J

V 25 cc 10 cc

Q = U + W In abc, Q = 80 J W = 30 J So, U = (80 – 30) J = 50 J Now in adc, W = 10 J So, Q = 10 + 50 = 60 J [U = 50 J]

D

B C

A 10 kpa

30 kpa P

d

b

V

c

a

P

In path ACB, dQ = 50 0 50 × 4.2 = 210 J 3 –6 dW = W AC + WCB = 50 × 10 × 200 × 10 = 10 J dQ = dU + dW  dU = dQ – dW = 210 – 10 = 200 J In path ADB, dQ = ? dU = 200 J (Internal energy change between 2 points is always same) 3 –6 dW = W AD + WDB = 0+ 155 × 10 × 200 × 10 = 31 J dQ = dU + dW = 200 + 31 = 231 J = 55 cal

P 155 kpa 50 kpa

D

B C

A 200 cc

400 cc V

(cc)

10. Heat absorbed = work done = Area under the graph In the given case heat absorbed = area of the circle 4 –6 3 =  × 10 × 10 × 10 = 3.14 × 10 = 31.4 J

V

300 100

100 300

26.3

P

(kpa)

Laws of thermodynamics 11. dQ = 2.4 cal = 2.4 J Joules dw = WAB + WBC + W AC 3 –6 3 –6 = 0 + (1/2) × (100 + 200) × 10 200 × 10 – 100 × 10 × 200 × 10 3 –6 = (1/2) × 300 × 10 200 × 10 – 20 = 30 – 20 = 10 joules. du = 0 (in a cyclic process) dQ = dU +dW  2.4 J = 10

10 J= ≈ 4.17 J/Cal. 2 .4 12. Now, Q = (2625 × J) J U = 5000 J 3 From Graph W = 200 × 10 × 0.03 = 6000 J. Now, Q = W + U  2625 J = 6000 + 5000 J

V 700 cc 600 cc

C

B

A

c 300 kpa 200 kpa

11000 = 4.19 J/Cal 2625 dQ = 70 cal = (70 × 4.2) J 3 –6 dW = (1/2) × (200 + 500) × 10 × 150 × 10 –3 = (1/2) × 500 × 150 × 10 –1 = 525 × 10 = 52.5 J dU = ? dQ = du + dw  – 294 = du + 52.5  du = – 294 – 52.5 = – 346.5 J 5 U = 1.5 pV P = 1 × 10 Pa 3 3 –4 3 dV = (200 – 100) cm = 100 cm = 10 m 5 –4 dU = 1.5 × 10 × 10 = 15 dW = 105 × 10–4 = 10 dQ = dU + dW = 10 + 15 = 25 J dQ = 10 J 3 3 –6 3 dV = A × 10 cm = 4 × 10 cm = 40 × 10 cm 3 –6 3 dw = Pdv = 100 × 10 × 40 × 10 = 4 cm du = ? 10 = du + dw  10 = du + 4  du = 6 J. (a) P1 = 100 KPa 3 V1 = 2 m 3 V1 = 0.5 m P1 = 100 KPa From the graph, We find that area under AC is greater than area under than AB. So, we see that heat is extracted from the system. (b) Amount of heat = Area under ABC.

14.

15.

16.

1 5   10 5 = 25000 J 2 10 17. n = 2 mole dQ = – 1200 J dU = 0 (During cyclic Process) dQ = dU + dwc  – 1200 = W AB + WBC + WCA  – 1200 = nRT + WBC + 0  – 1200 = 2 × 8.3 × 200 + WBC  W BC = – 400 × 8.3 – 1200 = – 4520 J.

b

a 0.02 m3

J=

13.

200 kpa P

100 kpa

0.05 m3

250 cc

100 cc 200 kpa 500 kpa

P 100 kpa 2.5 m3 V

2 m3

=

T 500 k

300 k O

26.4

C

B

A V

Laws of thermodynamics 18. Given n = 2 moles dV = 0 in ad and bc. Hence dW = dQ dW = dW ab + dWcd 2V0 V0 = nRT1Ln  nRT2Ln V0 2V0 = nR × 2.303 × log 2(500 – 300) = 2 × 8.314 × 2.303 × 0.301 × 200 = 2305.31 J 19. Given M = 2 kg 2t = 4°c Sw = 4200 J/Kg–k 3 3 5 0 = 999.9 kg/m 4 = 1000 kg/m P = 10 Pa. Net internal energy = dv dQ = DU + dw  msQ = dU + P(v0 – v4) 5  2 × 4200 × 4 = dU + 10 (m – m)

V a

b

c

d V0

500 k

200 k

2V0

V

m 5 m  = dU + 105(0.0020002 – 0.002) = dU + 105 0.0000002  33600 = dU + 10   V v 4   0  33600 = du + 0.02  du = (33600 – 0.02) J 20. Mass = 10g = 0.01kg. 5 P = 10 Pa dQ = QH2o 0° – 100° + QH2o – steam 6

= 0.01 × 4200 × 100 + 0.01 × 2.5 × 10 = 4200 + 25000 = 29200 dW = P × V

0.01 0.01  = 0.01699 0.6 1000 5 dW = PV = 0.01699 × 10 1699J 4 dQ = dW + dU or dU = dQ – dW = 29200 – 1699 = 27501 = 2.75 × 10 J 21. (a) Since the wall can not be moved thus dU = 0 and dQ = 0. Hence dW = 0. (b) Let final pressure in LHS = P1 In RHS = P2 V/2 ( no. of mole remains constant) P1V PV = 1 2RT1 2RT =

 P1 =

P1T P (P  P2 )T1T2 = 1 1 T1 

(P1  P2 )T1T2  P T (P  P2 ) Simillarly P2 = 2 1 1  (c) Let T2 > T1 and ‘T’ be the common temp. PV PV Initially 1 = n1 rt1  n1 = 1 2 2RT1 As, T =

n2 =

P2 V Hence dQ = 0, dW = 0, Hence dU = 0. 2RT2

In case (LHS) RHS u1 = 1.5n1 R(T - T1) But u1 -u2 = 0 u2 = 1.5n2 R(T2 –T)  1.5 n1 R(T -T1) = 1.5 n2 R(T2 –T)  n2 T – n1 T1 = n2 T2 – n2 T  T(n1 + n2) = n1 T1 + n2 T2 26.5

P1 T1

P2 T2 V/2

U = 1.5nRT

Laws of thermodynamics 

n1T1  n 2 T2  n1  n 2

P1  P2 P1V PV  T1  2  T2 P1T2  P2 T1 2RT1 2RT2    P1V PV T1T2  2 2RT1 2RT2 

(P1  P2 )T1T2 (P  P2 )T1T2  1 as P1 T2 + P2 T1 =  P1T2  P2 T1 

(d) For RHS dQ = dU (As dW = 0)

= 1.5 n2 R(T2 – t)

=

1.5P2 V  P1t 2 2  P1T1T2  1.5P2 V  T2  (P1  P2 )T1T2  R  =  2T2   2RT2  P1T2  P2 T1  

=

3P1P2 (T2  T1 )V 1.5P2 V T2P1(T2  T1 )  =  2T2 4

22. (a) As the conducting wall is fixed the work done by the gas on the left part during the process is Zero. (b) For left side For right side Pressure = P Let initial Temperature = T2 Volume = V No. of moles = n(1mole) Let initial Temperature = T1

PV = nRT1 2 PV  = (1)RT1 2  T1 =

PV = n2 RT2 2 PV  T2 = 1 2n 2R

PV 2(moles )R

 T2 =

PV 4(moles )R

(c) Let the final Temperature = T Final Pressure = R No. of mole = 1 mole + 2 moles = 3 moles  PV = nRT  T =

PV PV = nR 3(mole )R

(d) For RHS dQ = dU [as, dW = 0]

 PV  PV  = 1.5 n2 R(T - T2) = 1.5 × 2 × R ×    3(mole )R 4(mole )R  = 1.5 × 2 × R ×

4PV  3PV 3  R  PV PV = = 4  3(mole 3 4R 4

(e) As, dQ = –dU  dU = – dQ =

PV 4 

26.6

V/2

V/2

PT1

PT2

T V = 1.5nRT

V = 3nRT

T

CHAPTER – 27

SPECIFIC HEAT CAPACITIES OF GASES 1.

N = 1 mole, W = 20 g/mol, V = 50 m/s K.E. of the vessel = Internal energy of the gas = (1/2) mv2 = (1/2) × 20 × 10–3 × 50 × 50 = 25 J

3 3 50 r(T)  25 = 1 × × 8.31 × T  T= ≈ 2 k. 2 2 3  8 .3 m = 5 g, t = 25 – 15 = 10°C CV = 0.172 cal/g-°CJ = 4.2 J/Cal. dQ = du + dw Now, V = 0 (for a rigid body) So, dw = 0. So dQ = du. Q = msdt = 5 × 0.172 × 10 = 8.6 cal = 8.6 × 4.2 = 36.12 Joule. 2  = 1.4, w or piston = 50 kg., A of piston = 100 cm 2 Po = 100 kpa, g = 10 m/s , x = 20 cm. 25 = n

2.

3.

 50  10  mg   5 –4  10 5 100  10  4  20  10  2 = 1.5 × 10 × 20 × 10 = 300 J.  Po  Adx =  dw = pdv =  4 A  100  10   

300 nR nR300 300 300  1.4 dQ = nCpdT = nCp × = = = 1050 J. nR (   1)nR 0 .4

nRdt = 300  dT =

4.

5.

2

CPH = 3.4 Cal/g°C CVH2 = 2.4 Cal/g°C, 7 M = 2 g/ Mol, R = 8.3 × 10 erg/mol-°C We know, CP – CV = 1 Cal/g°C So, difference of molar specific heats = CP × M – CV × M = 1 × 2 = 2 Cal/g°C 7 7  J = 4.15 × 10 erg/cal. Now, 2 × J = R  2 × J = 8.3 × 10 erg/mol-°C

CP = 7.6, n = 1 mole, CV

T = 50K

(a) Keeping the pressure constant, dQ = du + dw, T = 50 K,  = 7/6, m = 1 mole, dQ = du + dw  nCVdT = du + RdT  du = nCpdT – RdT 7 R R 6 dT  RdT  dT  RdT = = 1 7  1 1 6 = DT – RdT = 7RdT – RdT = 6 RdT = 6 × 8.3 × 50 = 2490 J. (b) Kipping Volume constant, dv = nCVdT

R 1  8 .3  dt =  50 7  1 1 6 = 8.3 × 50 × 6 = 2490 J (c) Adiabetically dQ = 0, du = – dw = 1

n  R T1  T2  = 1 8.3 T2  T1  = 8.3 × 50 × 6 = 2490 J =  7   1   1 6 27.1

Specific Heat Capacities of Gases 6.

3

m = 1.18 g,

3

V = 1 × 10 cm = 1 L

T = 300 k,

5

P = 10 Pa

PV 5 = 10 = atm. RT 1 PV 1 1 N= = = =  2 2 RT 8 .2  3 24.6 8.2  10  3  10

PV = nRT or

n=

1 Q  = 24.6 × 2 = 49.2 n dt Cp = R + Cv = 1.987 + 49.2 = 51.187 Now, Cv =

1  51.187  1 = 2.08 Cal. 24.6 3 5 V1 = 100 cm3, V2 = 200 cm P = 2 × 10 Pa, Q = 50J 5 –6 (a) Q = du + dw  50 = du + 20× 10 (200 – 100 × 10 )  50 = du + 20  du = 30 J Q = nCpdT =

7.

3 3 × 8.3 × 300 [ U = nRT for monoatomic] 2 2 2 2 n= = = 0.008 3  83 249 dndTu 30 (c) du = nCv dT  Cv = = = 12.5 0.008  300

(b) 30 = n ×

8.

Cp = Cv + R = 12.5 + 8.3 = 20.3 (d) Cv = 12.5 (Proved above) Q = Amt of heat given Work done =

Q , 2

Q = W +  U

for monoatomic gas  U = Q –

Q Q = 2 2

3 Q 3 RT = = nT× R = 3R × nT 2 2 2 Again Q = n CpdT Where CP > Molar heat capacity at const. pressure. 3RnT = ndTCP  CP = 3R V=n

nRT RT 2 = KV  RT = KV  R T = 2KV U  = dv V 2KV PRdF dQ = du + dw  mcdT = CVdT + pdv  msdT = CV dT+ 2KV RKV R  ms = CV +  CP + 2KV 2 CP r R 10. = , CP – CV = R, CV = , CP = CV  1  1 9.

P = KV 

1 Rdt  b 1 1 Rdt   1 = C V  0 = CVdT + b 1 R b 1 CP  C V  R b+1= = = – +1  b = – CV CV Pdv =

11. Considering two gases, in Gas(1) we have, , Cp1 (Sp. Heat at const. ‘P’), Cv1 (Sp. Heat at const. ‘V’), n1 (No. of moles)

Cp1  & Cp1 – Cv1 = R Cv1 27.2

Specific Heat Capacities of Gases  Cv1 – Cv1 = R  Cv1 ( – 1) = R  Cv1 =

R R & Cp1 =  1  1

In Gas(2) we have, , Cp2 (Sp. Heat at const. ‘P’), Cv2 (Sp. Heat at const. ‘V’), n2 (No. of moles)

Cp 2 R R  & Cp2 – Cv2 = R  Cv2 – Cv2 = R  Cv2 ( – 1) = R  Cv2 = & Cp2 = Cv 2  1  1 Given n1 : n2 = 1 :2 dU1 = nCv1 dT & dU2 = 2nCv2 dT = 3nCvdT R 2R  Cv1  2Cv 2 3R R  1  1  CV = = = = 3 3 3(   1)  1 &Cp = Cv =

…(1)

r …(2)  1

Cp =  [from (1) & (2)] Cv 12. Cp = 2.5 RCp = 3.5 R Cv = 2.5 R Cv = 1.5 R (n1 + n2)CVdT = n1 CvdT + n2 CvdT n1 = n2 = 1 mol n Cv   n 2Cv  1.5R  2.5R  CV = 1 = 2R n1  n 2 2 So,

CP = CV + R = 2R + R = 3R Cp 3R = = = 1.5 CV 2R

1 25 mole, R= J/mol-k, 2 3 (a) Temp at A = Ta, PaVa = nRTa

13. n =

=

5 3 200 KPa

d

Pa Va 5000  10 6  100  10 3 = = 120 k. a Ta 1 25 nR 100 KPa  2 3 Similarly temperatures at point b = 240 k at C it is 480 k and at D it is 240 k. 5000 cm3 (b) For ab process, dQ = nCpdT [since ab is isobaric] 35 5  1 R 1 125 3 1 3 3 Tb  Ta  =  5 =    120 = 1250 J  (240  120) =  2  1 2 9 2 2 1 3 For bc, dQ = du + dw [dq = 0, Isochorie process] 25 nR 1 Tc  Ta  =  3 (240) = 1  25  3  240 = 1500 J  dQ = du = nCvdT = 2 3 2 2 5  1    1  3  Ta =

(c) Heat liberated in cd = – nCpdT =

1 nR Td  Tc  = 1  125  3  240 = 2500 J  2 3 2 2  1

Heat liberated in da = – nCvdT =

1 R Ta  Td  = 1  25  (120  240) = 750 J  2  1 2 2 27.3

Tc

Td c

b

Tb

10000 cm3

Specific Heat Capacities of Gases 14. (a) For a, b ’V’ is constant So,

P1 P2 200 100 200  300   =  T2 = = 600 k T1 T2 300 T2 100

c

150 cm3

For b,c ‘P’ is constant

100 cm3

a

V V 150 100 600  150 So, 1  2  =  T2 = = 900 k T1 T2 T2 600 100

100 KPa 200 KPa

(b) Work done = Area enclosed under the graph 50 cc × 200 kpa = 50 × 10 (c) ‘Q’ Supplied = nCvdT Now, n = Cv = Qbc =

[Cp=

( = 1.67)

R ]  1

200  10 3  150  10 6 1.67  8.3 PV R   300 = 24.925  dT = 8.3  900 0.67 RT   1

(d) Q = U + w Now, U = Q – w = Heat supplied – Work done = (24.925 + 14.925) – 1 = 29.850  15. In Joly’s differential steam calorimeter Cv =

m 2L m1( 2  1 )

m2 = Mass of steam condensed = 0.095 g, L = 540 Cal/g = 540 × 4.2 J/g 1 = 20°C, 2 = 100°C m1 = Mass of gas present = 3 g,  Cv =

0.095  540  4.2 = 0.89 ≈ 0.9 J/g-K 3(100  20 )

16.  = 1.5 Since it is an adiabatic process, So PV = const. (a) P1V1 = P2V2 



P2 =? P1

Given V1 = 4 L, V2 = 3 L, 1.5

V  P2 4 =  1  =   P1 V 3  2

= 1.5396 ≈ 1.54

(b) TV–1 = Const. T1V1–1

=

T2V2–1

V  T  2 =  1  T1  V2 

 1

4 =   3

5

17. P1 = 2.5 × 10 Pa, V1 = 100 cc, (a) P1V1 = P2V2

0.5

= 1.154

T1 = 300 k

1.5

5

 2.5 × 10 × V 1.5

1.5

V =   2

 P2

5

5

5

 P2 = 2 × 2.5 × 10 = 7.07 × 10 ≈ 7.1 × 10 1.5 – 1 1.5 – 1 (b) T1V1–1 = T2V2–1  300 × (100) = T2 × (50)  T2 =

3000 = 424.32 k ≈ 424 k 7.07 27.4

3

× 200 × 10 J = 10 J

R dT = 300 a, b.  1

Q supplied to be nCpdT =

–6

PV considering at pt. ‘b’ RT

200  10 3  100  10 6 PV R  300 = 14.925  dT = 600  0.67 RT   1

b

Specific Heat Capacities of Gases (c) Work done by the gas in the process W= =

mR T2  T1 = P1V1 T2  T1   1 T(   1)

2.5  10 2.5  10 5  100  10 6 [ 424  300] =  124 = 20.72 ≈ 21 J 300  0.5 300(1,5  1)

18.  = 1.4, T1 = 20°C = 293 k, P1 = 2 atm, p2 = 1 atm We know for adiabatic process, 1– 1– 1.4 1–1.4 1.4 1–1.4 P1 × T1 = P2 × T2or (2) × (293) = (1) × T2 1.4 1.4 0.4 1.4 1/1.4  (2) × (293) = T2  2153.78 = T2  T2 = (2153.78) = 240.3 K 5 3 –6 3 19. P1 = 100 KPa = 10 Pa, V1 = 400 cm = 400 × 10 m , T1 = 300 k, CP = = 1.5 CV 3

(a) Suddenly compressed to V2 = 100 cm 5 1.5 1.5 P1V1 = P2V2  10 (400) = P2 × (100) 5 1.5  P2 = 10 × (4) = 800 KPa

300  20 = 600 K 10 (b) Even if the container is slowly compressed the walls are adiabatic so heat transferred is 0. Thus the values remain, P2 = 800 KPa, T2 = 600 K. CP = P0 (Initial Pressure), V0 (Initial Volume) 20. Given CV T1V1–1 = T2V2–1  300 × (400)

1.5–1

= T2 × (100)

1.5-1

(a) (i) Isothermal compression, P1V1 = P2V2 or, P0V0 =

 T2 =

P2 V0  P2 = 2P0 2

V (ii) Adiabatic Compression P1V1 = P2V2 or 2P0  0   2   P =

Vo 2





 2P0 

4 V0 



 V0     4 

= P1



= 2 × 2 P0  P02+1 

V  (b) (i) Adiabatic compression P1V1 = P2V2 or P0V0 = P 0   P = P02  2  (ii) Isothermal compression P1V1 = P2V2 or 2 P0 × 21. Initial pressure = P0 Initial Volume = V0 =

CP CV

(a) Isothermally to pressure P0V0 =

P0 2

P0 V1  V1 = 2 V0 2

Adiabetically to pressure =

P0 4

P0 V1  = P0 V2   P0 (2V0 )  = P0 ( V2 )  2 4 2 4 (+1)/  2+1 V0 = V2  V2 = 2 V0 (+1)/  Final Volume = 2 V0 27.5

V0 V = P2 × 0  P2 = P02+1 2 4

Specific Heat Capacities of Gases (b) Adiabetically to pressure

P0 to P0 2

P0  ( V )  2 P Isothermal to pressure 0 4 P0 P  21 /  V0 = 0  V   V = 2(+1)/ V0  2 4 (+1)/ V0  Final Volume = 2 22. PV = nRT 3 Given P = 150 KPa = 150 × 10 Pa, P0 × (2+1 V0 ) =

3

(a) n = (b)

PV 150  10  150  10 = RT 8.3  300

6

–3

= 9.036 × 10

–6

3

m,

T = 300 k

= 0.009 moles.

 R   CP    1  

CP R = = CV (   1)C V

 CV =

3

V = 150 cm = 150 × 10

R 8 .3 8 .3 = = = 2R = 16.6 J/mole  1 1 .5  1 0 .5 3

P2 =? (c) Given P1 = 150 KPa = 150 × 10 Pa, 3 –6 3 V1 = 150 cm = 150× 10 m ,  = 1.5 3 –6 3 T1 = 300 k, T2 = ? V2 = 50 cm = 50 × 10 m ,  Since the process is adiabatic Hence – P1V1 = P2V2 3 –6 –6  150× 10 (150 × 10 ) = P2 × (50 × 10 )

 150  10  6 3  P2 = 150 × 10 ×   50  10  6 

1. 5

   

= 150000 × 3

1.5

3

= 779.422 × 10 Pa ≈ 780 KPa

(d) Q = W + U or W = –U [U = 0, in adiabatic] = – nCVdT = – 0.009 × 16.6 × (520 – 300) = – 0.009 × 16.6 × 220 = – 32.8 J ≈ – 33 J (e) U = nCVdT = 0.009 × 16.6 × 220 ≈ 33 J 23. VA = VB = VC For A, the process is isothermal PAVA = PAVA  PA = PA

VA 1 = PA   2 V A

For B, the process is adiabatic, 

1.5 V  P  1 PA(VB) = PA(VB) = PB = PB  B  = PB ×   = 1B.5 V  2 2  B  



For, C, the process is isobaric   VC V V 2VC 2  C  C  TC =    TC T T C C TC TC Final pressures are equal.

pA P 1.5 = 1B.5 = PC  PA : PB : PC = 2 : 2 : 1 = 2 : 2 2 : 1 2 2 V1 = Initial Volume P2 = Final Pressure 24. P1 = Initial Pressure =

V  Given, V2 = 2V1, Isothermal workdone = nRT1 Ln  2   V1 

27.6

V2 = Final Volume

Specific Heat Capacities of Gases Adiabatic workdone =

P1V1  P2 V2  1

Given that workdone in both cases is same.

V  P V  P2 V2  ( – 1) ln Hence nRT1 Ln  2  = 1 1  1  V1 

 V2  P V  P2 V2  = 1 1  nRT1  V1 

V  nRT1  nRT2 T  T1  ( – 1)ln  2  =  ( – 1) ln 2 = 1 V nRT T1 1 1  

…(i)

[ V2 = 2V1 ]

We know TV–1 = const. in adiabatic Process. T1V1–1 = T2 V2–1, or T1 (V2)–1 = T2 × (2)–1 × (V1)–1 1- Or, T1 = 2–1 × T2 or T2 = T1 …(ii) From (i) & (ii) ( – 1) ln 2 =

T1  T1  21  1–  ( – 1) ln2 = 1 – 2  T1

1 l 2 (a) The process is adiabatic as it is sudden,

25.  = 1.5,

T = 300 k,

V = 1Lv =

1. 5



 1  V P1 V1 = P2 V2  P1 (V0) = P2  0   P2 = P1   2  1/ 2    nR 5 (b) P1 = 100 KPa = 10 Pa W = [T1  T2 ]  1 T1 V1–1 = P2 V2–1  300 × (1) T2 = 300 ×

1.5–1

= T2 (0.5)

1.5

 300 × 1 = T2



P2 1.5 =2 = 2 2 P1

0 .5

1 = 300 2 K 0 .5

P1 V1 =nRT1  n = w=

1.5–1

= P1 (2)

P1V1 10 5  10 3 1 = = RT1 R  300 3R



3

(V in m )







nR 1R 300 [T1  T2 ] = 300  300 2 = 1  2 = –82.8 J ≈ – 82 J.  1 3R(1.5  1) 3  0 .5

(c) Internal Energy,  du = – dw = –(–82.8)J = 82.8 J ≈ 82 J. dQ = 0, (d) Final Temp = 300 2 = 300 × 1.414 × 100 = 424.2 k ≈ 424 k. (e) The pressure is kept constant.  The process is isobaric. Work done = nRdT =

1 × R × (300 – 300 2 ) 3R

Final Temp = 300 K

1 × 300 (0.414) = – 41.4 J. Initial Temp = 300 2 3 V V V 1 1  (f) Initial volume  1 = 1 = V1 = 1  T1 =  300 = L.  T1 T 2  300  2 2 2 1 T1 =–

Final volume = 1L Work done in isothermal = nRTln =

V2 V1

 

 1  1  = 100 × ln 2 2 = 100 × 1.039 ≈ 103  R  300 ln 3R  1/ 2 2 

(g) Net work done = WA + WB + WC = – 82 – 41.4 + 103 = – 20.4 J. 27.7

Specific Heat Capacities of Gases 26. Given  = 1.5 We know fro adiabatic process TV–1 = Const. …(eq) So, T1 V1–1 = T2 V2–1 As, it is an adiabatic process and all the other conditions are same. Hence the above equation can be applied. 1.5 1

 3V  So, T1 ×    4  

T1 V =   T2 4

0.5

1.5 1

V = T2 ×   4

 4     3V 

0.5

=

 3V   T1 ×    4 

1

0. 5

V = T2 ×   4

So, T1 :T2 = 1 :

3

3

27. V = 200 cm , C = 12.5 J/mol-k, (a) No. of moles of gas in each vessel,

T = 300 k,

0.5



P T

P T

V/4 T2

T1

3

3:1

P = 75 cm A

dQ = du  5 = nCVdT  5 = 0.008 × 12.5 × dT  dT =

10 0.008  12.5

V/2

3V/4

PV 75  13.6  980  200 = = 0.008 RT 8.3  10 7  300 (b) Heat is supplied to the gas but dv = 0

For (B) dT =

V/2

B

5 for (A) 0.008  12.5

P PA  [For container A] T TA

P  0.008  12.5 75 75  5 = A  PA = = 12.5 cm of Hg. 300 5 300  0.008  12.5 P  0.008  12.5 75 P P   B [For Container B]  = B  PB = 2 PA = 25 cm of Hg. T TB 300 10



Mercury moves by a distance PB – PA = 25 – 12.5 = 12.5 Cm.  = 1.67,  = 4 g/mol, mH2 =? 28. mHe = 0.1 g,  = 28/mol 2 = 1.4 Since it is an adiabatic surrounding He dQ = nCVdT =

0 .1 R 0 .1 R   dT =   dT 4  1 4 (1.67  1)

…(i) H2

He

m R m R H2 = nCVdT =   dT =   dT [Where m is the rqd. 2  1 2 1 .4  1 Mass of H2] Since equal amount of heat is given to both and T is same in both. Equating (i) & (ii) we get 0 .1 R m R 0 .1 0 .4   dT    dT  m =  = 0.0298 ≈ 0.03 g 4 0.67 2 0 .4 2 0.67 29. Initial pressure = P0, Initial Temperature = T0 Initial Volume = V0 CP = CV

B

A

(a) For the diathermic vessel the temperature inside remains constant P Temperature = To P1 V1 – P2 V2  P0 V0 = P2 × 2V0  P2 = 0 , 2 For adiabatic vessel the temperature does not remains constant. The process is adiabatic

 V T1 V1–1 = T2 V2–1  T0V0–1 = T2 × (2V0)–1  T2 = T0  0  2V0 27.8

  

 1

 1 = T0    2

 1

=

T0

2  1

Specific Heat Capacities of Gases

 V P1 V1 = P2 V2  P0 V0 = p1 (2V0)  P1 = P0  0  2 V0



 P  = 0 2 

(b) When the values are opened, the temperature remains T0 through out n RT n RT P1 = 1 0 , P2 = 2 0 [Total value after the expt = 2V0 + 2V0 = 4V0] 4 V0 4 V0 P = P1 + P 2 =

nRT0 P (n1  n 2 )RT0 2nRT0 = = = 0 4 V0 4 V0 2V 2

30. For an adiabatic process, Pv = Const. There will be a common pressure ‘P’ when the equilibrium is reached 

V  Hence P1  0  = P(V)  2 

V0/2

V0/2

P1 T1

P2 T2



V  For left P = P1 0  ( V )   2 

…(1) V



V  For Right P = P2  0  ( V0  V )   2 

V0–V

1

…(2)

Equating ‘P’ for both left & right 1/ 

P  V0  V  =  2  V  P1 

=

P1 P2 = ( V )  ( V0  V ) 



1/  1/  1/  1/  V0 V VP P P P  V = 1 / 0 1 1 /   1 = 21 /   0 = 2 1 /  1 V V P1 P1 P1  P2

or

1/  V0P2 1/  P1  P21 / 

Similarly V0 – V =

For left …….(3)

For right ……(4)

(b) Since the whole process takes place in adiabatic surroundings. The separator is adiabatic. Hence heat given to the gas in the left part = Zero.

V  P1 0  2 (c) From (1) Final pressure P =   ( V )

Again from (3) V =

2

1/  V0P1 1/  P1  P21 / 

–4

2

31. A = 1 cm = 1 × 10 m , 5 P = 1 atm = 10 pascal, L1 = 80 cm = 0.8 m, The process is adiabatic

or P =

y

P1

V0  2



 V0P11 /      P 1/   P 1/   2  1  –3 M = 0.03 g = 0.03 × 10 kg, L= 40 cm = 0.4 m. P = 0.355 atm

=

P1 V0  2

P(V) = P(V) =  1 × (AL) = 0.355 × (A2L)  1 1 = 0.355 2  



P

1/  1



1/  

 P2

V0  P1

 P 1 /   P21 /  =  1  2 

1 = 2 0.355

 1  =  log 2 = log   = 1.4941  0.355  V=

P = 

1.4941  10 5 = m/v

1.4941  10 5  0.03  10  3     10  4  1 0.4   

=

27.9

1.441  10 5  4  10 5 3  10  5

= 446.33 ≈ 447 m/s 

   



Specific Heat Capacities of Gases 3

oH2 = 0.089 kg/m ,

32. V = 1280 m/s, T = 0°C, 5 At STP, P = 10 Pa, We know

P  1280 = o

Vsound =

rR = 8.3 J/mol-k,

  10 5   10 5 0.089  (1280 )2 2  (1280) = = ≈ 1.458 0.089 0.089 10 5

Again,

R 8 .3 = = 18.1 J/mol-k  1 1.458  1

CV =

Again,

CP =  or CP = CV = 1.458 × 18.1 = 26.3 J/mol-k CV –3

3

V = 22400 cm = 22400 × 10 33.  = 4g = 4 × 10 kg, CP = 5 cal/mol-ki = 5 × 4.2 J/mol-k = 21 J/mol-k

–6

m

3

R   8 .3 =  1  1

CP =

 21( – 1) =  (8.3)  21  – 21 = 8.3    =

21 12.7

5

Since the condition is STP, P = 1 atm = 10 pa

21  10 5 12.7 4  10  3

 = 

V=

21  10 5  22400  10 6

=

12.7  4  10  3

= 962.28 m/s

22400  10  6 34. Given o = 1.7 ×10  = 3.0 KHz.

–3

3

3

g/cm = 1.7 kg/m ,

5

P = 1.5 × 10 Pa,

R = 8.3 J/mol-k,

 –3 = 6 cm,   = 12 cm = 12 × 10 m 2 3 -2 So, V =  = 3 × 10 × 12 × 10 = 360 m/s Node separation in a Kundt’ tube =

We know, Speed of sound = But CV = Again

P   1.5  10 5 (360 )2  1.7 2  (360) = = = 1.4688 o 1 .7 1.5  10 5

R 8 .3 = = 17.72 J/mol-k  1 1.488  1

CP = CV

So, CP = CV = 17.72 × 1.468 = 26.01 ≈ 26 J/mol-k 

 –2 = 3.3 cm   = 6.6 × 10 m 2 3 –2 V =  = 5 × 10 × 6.6 × 10 = (66 × 5) m/s 3

35.  = 5 × 10 Hz,

T = 300 Hz,

V=

P P m RT [Pv = nRT  P = ×Rt  PM = oRT  = ]  o mV m

=

RT (66  5 ) = m

Cv=

  8.3  300 32  10

3

2

 (66 × 5) =

  8.3  300 32  10

R 8 .3 = = 20.7 J/mol-k,  1 0.3995

CP = CV + R = 20.77 + 8.3 = 29.07 J/mol-k. 

27.10

3

=

(66  5)2  32  10 3 = 1.3995 8.3  300

CHAPTER 28

HEAT TRANSFER 1.

2.

3.

4.

t1 = 90°C, t2 = 10°C –3 l = 1 cm = 1 × 10 m 2 –2 2 A = 10 cm × 10 cm = 0.1 × 0.1 m = 1 × 10 m K = 0.80 w/m-°C

10 cm

KA (1   2 ) 8  10 1  1 10 2  80 Q = = = 64 J/s = 64 × 60 3840 J. t l 1  10  2 2 t = 1 cm = 0.01 m, A = 0.8 m 1 = 300, 2 = 80 K = 0.025, KA(1   2 ) Q 0.025  0.8  (30030 ) = = = 440 watt. t l 0.01 2 K = 0.04 J/m-5°C, A = 1.6 m t1 = 97°F = 36.1°C t2 = 47°F = 8.33°C l = 0.5 cm = 0.005 m

10

1 cm

KA (1   2 ) 4  10 2  1.6  27.78 Q = = = 356 J/s t l 5  10  3 2 –4 2 A = 25 cm = 25 × 10 m –3 l = 1 mm = 10 m K = 50 w/m-°C Q = Rate of conversion of water into steam t 10 1  2.26  10 6 100  10 3  2.26  10 6 4 = = = 0.376 × 10 1 min 60 KA (1   2 ) 50  25  10 4  (  100 ) Q 4 =  0.376 ×10 = t l 10  3 =

5.

6.

7.

10 3  0.376  10 4 50  25  10

4

=

10 5  0.376 = 30.1 ≈ 30 50  25

K = 46 w/m-s°C l=1m 2 –6 2 A = 0.04 cm = 4 × 10 m 5 Lfussion ice = 3.36 × 10 j/Kg

46  4  10 6  100 Q –8 –5 = = 5.4 × 10 kg ≈ 5.4 × 10 g. t 1 A = 2400 cm2 = 2400 × 10–4 m2 –3 ℓ = 2 mm = 2 × 10 m K = 0.06 w/m-°C 1 = 20°C 2 = 0°C

0°C

KA(1   2 ) Q 0.06  2400  10 4  20 –1 = = = 24 × 6 × 10 × 10 = 24 × 6 = 144 J/sec 3 t  2  10 144 144  3600 m Q = = Kg/h = Kg/s = 1.52 kg/s. Rate in which ice melts = t t L 3.4  10 5 3.4  10 5 –3 ℓ = 1 mm = 10 m m = 10 kg 2 –2 2 A = 200 cm = 2 × 10 m 6 Lvap = 2.27 × 10 J/kg K = 0.80 J/m-s-°C 28.1

100°C

Heat Transfer 6

dQ = 2.27 × 10 × 10,

2.27  10 7 dQ 2 = = 2.27 × 10 J/s dt 10 5 Again we know

dQ 0.80  2  10 2  ( 42  T ) = dt 1 10  3 So,

8.

3

= 2.27 × 10

2

10  16 × 42 – 16T = 227  T = 27.8 ≈ 28°C K = 45 w/m-°C –2 ℓ = 60 cm = 60 × 10 m 2 –4 2 A = 0.2 cm = 0.2 × 10 m Rate of heat flow, =

9.

8  2  10 3 ( 42  T )

Q2 = 20°

Q1 = 40°

KA (1   2 ) 45  0.2  10 4  20 –3 = = 30 × 10 0.03 w  60  10  2 2

A = 10 cm ,

h = 10 cm

KA(1   2 ) Q 200  10 3  30 = = = 6000 t  1  10  3 Since heat goes out from both surfaces. Hence net heat coming out. Q Q  = = 6000 × 2 = 12000, = MS t t t   –3 –1  6000 × 2 = 10 × 10 × 1000 × 4200 × t  72000  = = 28.57 t 420 So, in 1 Sec. 28.57°C is dropped 1 Hence for drop of 1°C sec. = 0.035 sec. is required 28.57 –2 10. ℓ = 20 cm = 20× 10 m 2 –4 2 A = 0.2 cm = 0.2 × 10 m 1 = 80°C, 2 = 20°C, K = 385 KA(1   2 ) Q 385  0.2  10 4 (80  20) –4 –3 = = = 385 × 6 × 10 ×10 = 2310 × 10 = 2.31 (a) t  20  10  2 (b) Let the temp of the 11 cm point be   Q = 20°C l tKA  2.31 11 cm  = l 385  0.2  10  4   20 2.31  = 2 11  10 385  0.2  10  4

2.31  10 4  11 10  2 = 33 385  0.2   = 33 + 20 = 53 11. Let the point to be touched be ‘B’ No heat will flow when, the temp at that point is also 25°C i.e. QAB = QBC C KA(100  25) KA(25  0) So, = 100  x x  75 x = 2500 – 25 x  100 x = 2500  x = 25 cm from the end with 0°C

80°C

  – 20 =

28.2

100 cm B x

A 100–x

Heat Transfer 12. V = 216 cm a = 6 cm,

3 2

Surface area = 6 a = 6 × 36 m Q = 100 W, t

t = 0.1 cm

2

KA(1   2 ) Q = t   100 = K=

K  6  36  10 4  5 0.1 10  2 100

= 0.9259 W/m°C ≈ 0.92 W/m°C 6  36  5  10 1 13. Given 1 = 1°C, 2 = 0°C –3 K = 0.50 w/m-°C, d = 2 mm = 2 × 10 m –2 2 v = 10 cm/s = 0.1 m/s A = 5 × 10 m , Power = Force × Velocity = Mg × v KA(1   2 ) dQ = Again Power = dt d KA(1   2 ) So, Mgv = d

M

KA(1   2 ) 5  10 1  5  2 1 = = 12.5 kg. dvg 2  10  3  10 1  10 3 14. K = 1.7 W/m-°C ƒw = 1000 Kg/m 5 –2 Lice = 3.36 × 10 J/kg T = 10 cm = 10 × 10 m KA(1   2 ) KA(1   2 )  KA (1   2 ) Q 10 cm =  = = (a) t  t Q mL KA (1   2 ) 1.7  [0  ( 10)] = = At ƒ w L 10  10  2  1000  3.36  10 5 M=

–0°C

0°C

17 –7 –7  10  7 = 5.059 × 10 ≈ 5 × 10 m/sec 3.36 (b) let us assume that x length of ice has become formed to form a small strip of ice of length dx, dt time is required. dQ KA ( ) dmL KA ( ) Adx ƒL KA ( ) =  =  = dt x dt x dt x xdxƒL dx ƒL K (  )  =  dt = x dt x K (  ) =





t

dt =

0

l

ƒ L t xdx K( ) 0

Putting values



t=

ƒ L l 2 ƒ L  x 2    = K( )  2  K 2

dx

o





2

3.36  10 6 1000  3.36  10 5  10  10 2 3.36 =  10 6 sec. = hrs = 27.45 hrs ≈ 27.5 hrs. 1.7  10  2 2  17 2  17  3600 15. let ‘B’ be the maximum level upto which ice is formed. Hence the heat conducted at that point from both the levels is the same. A Let AB = x –10°C K ice  A  10 K water  A  4 Q Q x ice = water  = i.e. 1 cm t t x (1  x ) t=

1.7  10 17 2 5  10 1  4 =  = x 1 x x 1 x 17  17 – 17 x = 2x  19 x = 17  x = = 0.894 ≈ 89 cm 19 

28.3

1–x C

4°C

Heat Transfer 16. KAB = 50 j/m-s-°c A = 40°C B = 80°C KBC = 200 j/m-s-°c KAC = 400 j/m-s-°c C = 80°C –2 Length = 20 cm = 20 × 10 m 2 –4 2 A = 1 cm = 1 × 10 m (a)

Q AB K  A ( B   A ) 50  1 10 4  40 = AB = = 1 W. t l 20  10  2

(b)

Q AC K  A(C   A ) 400  1 10 4  40 –2 = AC = = 800 × 10 = 8 t l 20  10  2

QBC K  A ( B   C ) 200  1 10 4  0 = BC = =0 t l 20  10  2 KA(1   2 ) 17. We know Q = d KA (1   2 ) KA (1   2 ) Q1 = , Q2 = d1 d2 (c)

KA (1  1 ) Q1 2r 2 r = = = [d1 = r, d2 = 2r] KA (1  1 ) Q2 r  2r 18. The rate of heat flow per sec. dQ A d = = KA dt dt The rate of heat flow per sec. dQB d = KA B = dt dt This part of heat is absorbed by the red. Q ms d = where = Rate of net temp. variation t dt dt d d d  msd d  d  = KA A  KA B  ms = KA  A  B  dt  dt dt dt dt  dt

r

r

d –4 = 200 × 1 × 10 (5 – 2.5) °C/cm dt d -4  0 .4  = 200 × 10 × 2.5 dt  0 .4 

200  2.5  10 4 d –2 = °C/m = 1250 × 10 = 12.5 °C/m dt 0.4  10  2 19. Given T2 - T1 = 90°C Krubber = 0.15 J/m-s-°C We know for radial conduction in a Cylinder 2Kl(T2  T1 ) Q = ln(R 2 / R1 ) t 

= 20.

2  3.14  15  10 2  50  10 1  90 = 232.5 ≈ 233 j/s. ln(1.2 / 1)

dQ = Rate of flow of heat dt Let us consider a strip at a distance r from the center of thickness dr. dQ K  2rd  d = [d = Temperature diff across the thickness dr] dt dr 28.4

120°C

50 cm

Heat Transfer C=

K  2rd  d dr

d   c  dr   

dr r

dr = K2d d r Integrating  C

r2

C

 r1

dr = K2d r

r1

2



r2

 Clog r r2 = K2d (2 – 1) r

d

1

1

r   C(log r2 – log r1) = K2d (2 – 1)  C log  2  = K2d (2 – 1)  r1  K 2d( 2  1 ) C= log(r2 / r1 ) 21. T1 > T2 2 2 A = (R2 – R1 ) 2

2

KA (T2  T1 ) KA (R 2  R1 )(T2  T1 ) = l l Considering a concentric cylindrical shell of radius ‘r’ and thickness ‘dr’. The radial heat flow through the shell dQ d = – KA [(-)ve because as r – increases  H= dt dt decreases] d H = –2rl K A = 2rl dt

So, Q =

R2

or



R1

2LK dr =  H r

T2

R1

R2 T1

l

T2

 d

T1

Integrating and simplifying we get 2KL(T2  T1 ) 2KL( T2  T1 ) dQ = =  H= dt Loge(R 2 / R1 ) ln(R 2 / R1 ) 22. Here the thermal conductivities are in series, K 1A(1  2 ) K 2 A(1   2 )  KA (1   2 ) l1 l2  = K 1A(1   2 ) K 2 A(1   2 ) l1  l2  l1 l2

K1 K 2  l l2  1 K1 K 2  l1 l2 

L1 L2

=

K l1  l 2

K 1K 2 (K 1K 2 )(l1  l2 ) K = K= K 1l 2  K 2l1 l1  l2 K 1l2  K 2l1

23. KCu = 390 w/m-°C KSt = 46 w/m-°C Now, Since they are in series connection, So, the heat passed through the crossections in the same. So, Q1 = Q2 K  A  (  0) K  A  (100  ) Or Cu = St l l  390( – 0) = 46 × 100 – 46  436  = 4600 4600 = = 10.55 ≈ 10.6°C 436 28.5

0°C

Cu

Steel °C

100°C

Heat Transfer 24. As the Aluminum rod and Copper rod joined are in parallel

Q Q Q =      t t  1  Al  t  Cu

40°C 80°C

Cu

Al

80°C

KA(1   2 ) K A(1   2 ) K 2 A(1   2 ) = 1  l l l  K = K1 + K2 = (390 + 200) = 590 

KA (1   2 ) 590  1 10 4  (60  20) Q –4 = = = 590 × 10 × 40 = 2.36 Watt t l 1 KCu = 400 w/m-°C 25. KAl = 200 w/m-°C 2 –5 2 A = 0.2 cm = 2 × 10 m –1 l = 20 cm = 2 × 10 m Heat drawn per second K Al  A(80  40) K Cu  A(80  40) 2  10 5  40 =  [200  400] = 2.4 J l l 2  10 1 Heat drawn per min = 2.4 × 60 = 144 J 26. (Q/t)AB = (Q/t)BE bent + (Q/t)BE KA (1   2 ) KA (1   2 ) (Q/t)BE bent = (Q/t)BE = 70 60 (Q / t )BE bent 60 6 = = D (Q / t )BE 70 7 = QAl + QCu =

0°C F

(Q/t)BE bent + (Q/t)BE = 130  (Q/t)BE bent + (Q/t)BE 7/6 = 130

7     1 (Q/t)BE bent = 130 6  27.

 (Q/t)BE bent =

20 cm

A 100°C

B 60 cm

20 cm

130  6 = 60 13

Q 780  A  100 bent = t 70 Q 390  A  100 str = t 60 (Q / t ) bent 780  A  100 60 12 =  = (Q / t ) str 70 390  A  100 7

28. (a)

E

C

KA(1   2 ) Q 1 2  1( 40  32) = = = 8000 J/sec. t  2  10  3

60 cm 5 cm 20 cm

5 cm 20 cm

1 mm

   (b) Resistance of glass = ak g ak g Resistance of air = Net resistance =

g

 ak a

     ak g ak g ak a =

1    2   2k a  k g  =     a  kg ka  a  K gk a 

=

1 10 3  2  0.025  1    2 0.025  

1 10 3  1.05 0.05 8  0.05 = = 380.9 ≈ 381 W 1 10  3  1.05 =

  2 Q = 1 t R

28.6

a

g

Heat Transfer 29. Now; Q/t remains same in both cases K  A  (100  70) K  A  (70  0) = B In Case  : A    30 KA = 70 KB K  A  (100  ) K  A  (   0) In Case  : B = A    100KB – KB  = KA  70 KB   100KB – KB  = 30 7 300 = = 30°C  100 =    3 10 30. 1 – 2 = 100   2 Q 0°C = 1 t R    1   2    R = R1 + R2 +R3 = =   = aK Al aK Cu aK Al a  200 400 

100°C

100°C

Al

70°C A

0°C

B

°C B

0°C

A

Al

Cu

100°C

 1   4  1   = a  400  a 80

100 Q a =  40 = 80 × 100 ×  / a 1 / 80 t  a 1 =  200 For (b) 

l l l   R CuR Al R CuR Al AK Al AK Cu AK Al R = R1 + R2 = R1 + = RAl + = l l R Cu  R Al R Cu  R Al  A Cu A Al =

R2

  2 100 Q 100  600 A 100  600 1 = 1 = = =  = 75 l / A 4 / 600 t R 4 l 4 200 For (c) 1 1 1 1 1 1 1 =   =   l l l R R1 R 2 R 3 aK Al aK Cu aK Al

Al



T  T3 T  T2 T1  T =  l 3l / 2 3l / 2

 – 7T = – 3T1 – 2(T2 + T3)

 3T1 – 3T = 4T – 2(T2 + T3)

3T1  2(T2  T3 ) T= 7 28.7

Al

Al 0°C

a a a (K Al  K Cu  K Al ) = 2  200  400  = 800  l l l l 1 R=  a 800   2 Q 100  800  a  = 1 = t R l 100  800 = = 400 W 200 31. Let the temp. at B be T Q QA Q KA(T1  T ) KA(T  T3 ) KA(T  T2 )  = B  C  = t t t l l  (l / 2) l  (l / 2) =

R 100°C

Cu

R1

l l l l 4 l  1 1    =   =  AK Al A K Al  K Cu A  200 200  400  A 600

Cu

100°C

Al T2

T3

E

F QB

QC B

C

D QA T1

A

T2

T3

E

F QB

QC C

D QA T1

A

Heat Transfer 32. The temp at the both ends of bar F is same Rate of Heat flow to right = Rate of heat flow through left  (Q/t)A + (Q/t)C = (Q/t)B + (Q/t)D K (T  T )A K C (T1  T )A K (T  T2 )A K D (T  T2 )A  A 1 = B   l l l l  2K0(T1 – T) = 2 × 2K0(T – T2)  T1 – T = 2T – 2T2 T  2T2 T= 1 3 y  r1  r2  r1 33. Tan  = = L x  xr2 – xr1 = yL – r1L Differentiating wr to ‘x’ Ldy  r2 – r 1 = 0 dx r r dyL dy  = 2 1  dx = …(1) r2  r1  dx L

Q

(r2 – r1) r2 r1

L

QLdy (r2  r1 )Ky 2

Integrating both side 2



d =

1

QL r2  r1 k

r2

 r1

dy y r

  1 2 QL  (2 – 1) =   r2  r1 K  y  r 1

1 1 QL     (2 – 1) = r2  r1 K  r1 r2   (2 – 1) =

r  r  QL   2 1 r2  r1 K  r1  r2 

Kr1r2 ( 2  1 )  L d 60 = = 0.1°C/sec dt 10  60 dQ KA 1  2  = dt d KA  0.1 KA  0.2 KA  60   .......  = d d d KA KA 600 = (0.1  0.2  ........  60) =   (2  0.1  599  0.1) d d 2 [ a + 2a +……….+ na = n/2{2a + (n – 1)a}]

Q= 34.

=

200  1 10 4

 300  (0.2  59.9) =

20  10  2 = 3 × 10 × 60.1 = 1803 w ≈ 1800 w



x

Ky 2 d Q dx 2 =  = ky d T dx T Ldy 2 = Ky d from(1)  r2r1



200  10 2  300  60.1 20

28.8

d

y

Now

 d

dx

Heat Transfer 35. a = r1 = 5 cm = 0.05 m b = r2 = 20 cm = 0.2 m 1 = T1 = 50°C 2 = T2 = 10°C Now, considering a small strip of thickness ‘dr’ at a distance ‘r’. 2 A = 4 r d 2 H = – 4 r K [(–)ve because with increase of r,  decreases] dr b dr 4K 2 = On integration, = d 2 a r H 1 4ab(1   2 ) dQ H= = K dt (b  a) Putting the values we get



20 cm



K  4  3.14  5  20  40  10 3 K=

15  10  2 15

dr

= 100

a

b

r

= 2.985 ≈ 3 w/m-°C 4  3.14  4  10 1 KA (T1  T2 ) KA (T1  T2 ) Q 36. = Rise in Temp. in T2  t L Lms KA (T1  T2 ) KA(T1  T2 ) Fall in Temp in T1 = Final Temp. T1  T1  Lms Lms KA (T1  T2 ) Final Temp. T2 = T2  Lms KA(T1  T2 ) KA (T1  T2 ) T Final = T1   T2  dt Lms Lms = T1  T2    Ln 37.

2KA(T1  T2 ) 2KA(T1  T2 ) dT = =   Lms dt Lms

(T1  T2 ) / 2 2KAt = (T1  T2 ) Lms

 ln (1/2) =

KA(T1  T2 ) Q = t L Fall in Temp in T1 

( T1  T2 )



( T1  T2 )

2KAt Lms

Rise in Temp. in T2 

KA (T1  T2 ) Lm 2 s 2

Final Temp. T2 = T2 

Final Temp. T1 = T1 

2KA dt = dt Lms (T1  T2 )

 ln2 =

2KAt Lms

 t = ln2

KA (T1  T2 ) Lm1s1 KA(T1  T2 ) Lm1s1

KA (T1  T2 ) Lm1s1

 KA(T1  T2 ) KA (T1  T2 )  KA(T1  T2 ) KA (T1  T2 ) T  T2  = T1  = T1  T2      dt Lm1s1 Lm 2 s 2 Lm 2 s 2   Lm1s1 

KA T1  T2   1 dT 1  =   dt L  m1s1 m 2 s 2

 lnt = 

  



dT KA  m 2 s 2  m1s1  dt  =  T1  T2  L  m1s1m 2 s 2 

KA  m2 s 2  m1s1  t  C  L  m1s1m2 s 2 

At time t = 0, T = T0,  ln

5 cm

T = T0

T T KA  m 2 s 2  m1s1  t   =  = e T0 T0 L  m1s1m 2 s 2 

 T = T0 e



KA  m1s1  m2s2   t L  m1s1m2s2 

= T2  T1  e



 C = lnT0 KA  m1s1  m2s2  t   L  m1s1m2s2 

KA  m1s1  m2s2   t L  m1s1m2s2  

28.9

Lms 2KA

Heat Transfer 38.

KA(Ts  T0 ) nCP dT KA(Ts  T0 ) Q =  = t x dt x KA(Ts  T0 ) 2LA n(5 / 2)RdT dT  =  = (TS  T0 ) 5nRx dt x dt dT 2KAdt 2KAdt  =   ln(TS  T0 )TT0 =  (TS  T0 ) 5nRx 5nRx

ℓ

x

2KAt

 ln

 TS  T 2KAdt =   TS – T = (TS  T0 )e 5nRx TS  T0 5nRx

 T = TS  ( TS  T0 )e



2KAt 5nRx

= TS  (TS  T0 )e

 T = T – T0 = (TS  T0 )  (TS  T0 )e



2KAt 5nRx



2KAt 5nRx

2KAt   = (TS  T0 )  1  e 5nRx  

   

2KAt    Pa AL P AL = (TS  T0 )  1  e 5nRx  [padv = nRdt PaAl = nRdt dT = a ]   nR nR   2KAt    nR L= (TS  T0 )  1  e 5nRx     Pa A   2 –8 2 4 39. A = 1.6 m , T = 37°C = 310 K,  = 6.0 × 10 w/m -K Energy radiated per second 4 –8 4 –4 = AT = 1.6 × 6 × 10 × (310) = 8865801 × 10 = 886.58 ≈ 887 J 2 –4 2 40. A = 12 cm = 12 × 10 m T = 20°C = 293 K –8 2 4 e = 0.8  = 6 × 10 w/m -k Q 4 –4 –8 4 12 –13 = Ae T = 12 × 10 0.8 × 6 × 10 (293) = 4.245 × 10 × 10 = 0.4245 ≈ 0.42 t 41. E  Energy radiated per unit area per unit time Rate of heat flow  Energy radiated (a) Per time = E × A



So, EAl =

eT 4  A 4

=

4r 2 2

4(2r ) eT  A (b) Emissivity of both are same m1S1dT1 =1 = m 2S 2 dT2 

=

1 4

1:4

dT1 m S s 4r 3  S 2 1   900 = 2 2 = 1 13 = =1:2:9 dT2 m1S1 3 . 4  8  390 s 2 4r2  S1

Q 4 = Ae T t 100  4 4 T = T = teA 0.8  2  3.14  4  10  5  1  6  10  8  T = 1697.0 ≈ 1700 K 2 –4 2 43. (a) A = 20 cm = 20× 10 m , T = 57°C = 330 K 4 –4 –8 4 4 E = A T = 20 × 10 × 6 × 10 × (330) × 10 = 1.42 J E 4 4 2 –4 2 (b) = Ae(T1 – T2 ), A = 20 cm = 20 × 10 m t –8  = 6 × 10 T1 = 473 K, T2 = 330 K –4 –8 4 4 = 20 × 10 × 6 × 10 × 1[(473) – (330) ] 10 10 = 20 × 6 × [5.005 × 10 – 1.185 × 10 ] –2 from the ball. = 20 × 6 × 3.82 × 10 = 4.58 w

42.

28.10

Heat Transfer –3

44. r = 1 cm = 1 × 10 m –2 2 –4 2 A = 4(10 ) = 4 × 10 m –8 E = 0.3,  = 6 × 10 E 4 4 = Ae(T1 – T2 ) t –8 –4 4 4 = 0.3 × 6 × 10 × 4 × 10 × [(100) – (300) ] –12 12 = 0.3 × 6 × 4 × 10 × [1 – 0.0081] × 10 –4 = 0.3 × 6 × 4 × 3.14 × 9919 × 10 –5 = 4 × 18 × 3.14 × 9919 × 10 = 22.4 ≈ 22 W 45. Since the Cube can be assumed as black body e=ℓ –8 2 4  = 6 × 10 w/m -k –4 2 A = 6 × 25 × 10 m m = 1 kg s = 400 J/kg-°K T1 = 227°C = 500 K T2 = 27°C = 300 K d 4 4  ms = eA(T1 – T2 ) dt 



4

eA T1  T2 d = dt ms

4



1 6  10 8  6  25  10 4  [(500 ) 4  (300) 4 ] 1 400 36  25  544 –4 =  10  4 = 1224 × 10 = 0.1224°C/s ≈ 0.12°C/s. 400 4 4 46. Q = eA(T2 – T1 ) For any body, 210 = eA[(500)4 – (300)4] For black body, 700 = 1 × A[(500)4 – (300)4] 210 e Dividing =  e = 0.3 700 1 2 2 AB = 80 cm 47. AA = 20 cm , (mS)A = 42 J/°C, (mS)B = 82 J/°C, TA = 100°C, TB = 20°C KB is low thus it is a poor conducter and KA is high. Thus A will absorb no heat and conduct all =

E 4 4   = AA [(373) – (293) ] t  A





 d   mS A   =  dt  A





B A

4

4

AA [(373) – (293) ]



A a (373 ) 4  (293 ) 4 6  10 8 (373 ) 4  (293 ) 4  d     = = = 0.03 °C/S (mS ) A 42  dt  A

 d  Similarly   = 0.043 °C/S  dt B 48.

Q 4 4 = eAe(T2 – T1 ) t Q –8 4 4 –8 8 8  = 1 × 6 × 10 [(300) – (290) ] = 6 × 10 (81 × 10 – 70.7 × 10 ) = 6 × 10.3 At KA (1   2 ) Q = t l K(1   2 ) Q K  17 K  17 6  10.3  0.5  = = = 6 × 10.3 = K= = 1.8 tA l 0 .5 0 .5 17 28.11

Heat Transfer –8

2

4

49.  = 6 × 10 w/m -k L = 20 cm = 0.2 m, K=? KA(1   2 ) 4 4 E= = A(T1 – T2 ) d s(T1  T2 )  d 6  10 8  (750 4  300 4 )  2  10 1 K= = 1  2 50  K = 73.993 ≈ 74. 50. v = 100 cc  = 5°C t = 5 min For water mS  KA =  dt l

100  10 3  1000  4200 KA = 5 l For Kerosene ms KA = at l 



100  10 3  800  2100 KA = t l

100  10 3  800  2100 100  10 3  1000  4200 = t 5 5  800  2100 T= = 2 min 1000  4200 51. 50°C 45°C 40°C Let the surrounding temperature be ‘T’°C 50  45 Avg. t = = 47.5 2 Avg. temp. diff. from surrounding T = 47.5 – T 50  45 Rate of fall of temp = = 1 °C/mm 5 From Newton’s Law 1°C/mm = bA × t 1 1  bA = = …(1) t 47.5  T In second case, 40  45 Avg, temp = = 42.5 2 Avg. temp. diff. from surrounding t = 42.5 – t 45  40 5 Rate of fall of temp = = °C/mm 8 8 From Newton’s Law 5 = bAt B 1 5 =  ( 42.5  T )  8 ( 47.5  T ) By C & D [Componendo & Dividendo method] We find, T = 34.1°C 

28.12

300 K

750 K

800 K 20 cm

Heat Transfer 52. Let the water eq. of calorimeter = m

(m  50  10 3 )  4200  5 = Rate of heat flow 10 (m  100  10 3 )  4200  5 = Rate of flow 18 (m  50  10 3 )  4200  5 (m  100  10 3 )  4200  5  = 10 18 –3 –3  (m + 50 × 10 )18 = 10m + 1000 × 10 –3 –3  18m + 18 × 50 × 10 = 10m + 1000 × 10 –3  8m = 100 × 10 kg –3  m = 12.5 × 10 kg = 12.5 g 53. In steady state condition as no heat is absorbed, the rate of loss of heat by conduction is equal to that of the supplied. i.e. H = P m = 1Kg, Power of Heater = 20 W, Room Temp. = 20°C d = P = 20 watt (a) H = dt (b) by Newton’s law of cooling d = K( – 0) dt –20 = K(50 – 20)  K = 2/3 d 2 20 = K( – 0) =  (30  20) = w Again, 3 dt 3 20 10  dQ   dQ   dQ  = (c)    = 0,  =   3 3  dt  20  dt  30  dt  avg T = 5 min = 300 

10  300 = 1000 J 3 Net Heat absorbed = Heat supplied – Heat Radiated = 6000 – 1000 = 5000 J Now, m = 5000 5000 5000 –1 –1 S= = = 500 J Kg °C m 1  10 54. Given: Heat capacity = m × s = 80 J/°C Heat liberated =

 d  = 2 °C/s    dt  increase  d  = 0.2 °C/s    dt  decrease

 d  = 80 × 2 = 160 W (a) Power of heater = mS   dt  increa sin g  d  = 80 × 0.2 = 16 W (b) Power radiated = mS   dt  decrea sin g  d  = K(T – T0) (c) Now mS   dt  decrea sin g  16 = K(30 – 20)

K=

16 = 1.6 10

d = K(T – T0) = 1.6 × (30 – 25) = 1.6 × 5 = 8 W dt (d) P.t = H  8 × t

Now,

28.13

30°C

T 20°C

t

Heat Transfer 55.

d = – K(T – T0) dt Temp. at t = 0 is 1 (a) Max. Heat that the body can loose = Qm = ms(1 – 0) ( as, t = 1 – 0) (b) if the body loses 90% of the max heat the decrease in its temp. will be (  0 )  9 Q m  9 = 1 10ms 10 If it takes time t1, for this process, the temp. at t1 101  91  90   9 0 9 = 1  (1  0 ) = = 1 1 10 10 10 d Now, = – K( – 1) dt Let  = 1 at t = 0; &  be temp. at time t 

 

t

d   K dt    o

 0

  0 or, ln = – Kt 1  0 –kt

…(2) or,  – 0 = (1 – 0) e Putting value in the Eq (1) and Eq (2) 1  90 –kt  0 (1 – 0) e 10 ln 10  t1 = k

 

28.14

CHAPTER – 29

ELECTRIC FIELD AND POTENTIAL EXERCISES 1.

Coulomb 2

0 =

Newton m

F= 2.

1

–1 –3 4

=l M L T

2

kq1q2

r2 3 q1 = q2 = q = 1.0 C distance between = 2 km = 1 × 10 m kq1q2

so, force =

F=

2

r

(9  10 9 )  1 1 3 2

(2  10 )

=

9  10 9 2 2  10 6

3

= 2,25 × 10 N

The weight of body = mg = 40 × 10 N = 400 N

 2.25  10 3 wt of body =   4  10 2 force between ch arg es 

So,

3.

   

1

= (5.6)

–1

=

1 5 .6

So, force between charges = 5.6 weight of body. q = 1 C, Let the distance be  F = 50 × 9.8 = 490

Kq 2

F=

9  10 9  12

 490 =

2

2

or  =

2

9  10 9 6 = 18.36 × 10 490

3

4.

  = 4.29 ×10 m charges ‘q’ each, AB = 1 m wt, of 50 kg person = 50 × g = 50 × 9.8 = 490 N FC =

kq1q2 r

490  r 2

2

q =

9  10

r2

= 490 N

490  1 1 9  10 9 –5

–4

coulomb ≈ 2.3 × 10

coulomb

–19

Charge on each proton = a= 1.6 × 10 coulomb –15 Distance between charges = 10 × 10 metre = r Force =

6.

=

9

kq2

54.4  10 9 = 23.323 × 10

q= 5.



2

kq2

9  10 9  1.6  1.6  10 38

= 9 × 2.56 × 10 = 230.4 Newton r 10  30 –6 –6 q2 = 1.0 × 10 r = 10 cm = 0.1 m q1 = 2.0 × 10 Let the charge be at a distance x from q1 kqq 2 Kq1q q1 F1 = F2 = 2  (0.1  )2 =

=

2

9.9  2  10 6  10 9  q 2

Now since the net force is zero on the charge q. 

kq1q 2

=

(0.1  )2 2

2

 2(0.1 – ) =  =

 f 1 = f2

kqq 2

0 .1 2 1 2

 2 (0.1 – ) = 

= 0.0586 m = 5.86 cm ≈ 5.9 cm

From larger charge 29.1

q xm

(0.1–x) m 10 cm

q2

Electric Field and Potential 7.

–6

–6

q1 = 2 ×10 c q2 = – 1 × 10 c r = 10 cm = 10 × 10 Let the third charge be a so, F-AC = – F-BC 

kQq1

KQq2

=

r12

r2

2

2

(10  )

2

=

m 10 × 10–10 m

1  10 6 

a

C B –1 × 10–6 c

A 2 × 10–6 c

2

2  = 10 +   ( 2 - 1) = 10   =

10 = 24.14 cm  1.414  1

So, distance = 24.14 + 10 = 34.14 cm from larger charge  Minimum charge of a body is the charge of an electron –19 –2  = 1 cm = 1 × 10 cm Wo, q = 1.6 × 10 c So, F =

9.

2  10 6



2

 2 = (10 + )  8.

–2

kq1q2 r2

=

9  10 9  1.6  1.6  10 19  10 19 10  2  10  2

–38+9+2+2

= 23.04 × 10

–25

= 23.04 × 10

–24

= 2.3 × 10



10  100 = 55.5 Nos Total charge = 55.5 18 23 24 No. of electrons in 18 g of H2O = 6.023 × 10 × 10 = 6.023 × 10 No. of electrons of 100 g water =

6.023  10 24  100 26 25 = 0.334 × 10 = 3.334 × 10 18 25 –19 6 Total charge = 3.34 × 10 × 1.6 × 10 = 5.34 × 10 c 10. Molecular weight of H2O = 2 × 1 × 16 = 16 No. of electrons present in one molecule of H2O = 10 23 18 gm of H2O has 6.023 × 10 molecule 23 18 gm of H2O has 6.023 × 10 × 10 electrons No. of electrons in 100 g of H2O =

6.023  10 24  100 electrons 18

100 gm of H2O has

So number of protons = Charge of protons =

6.023  10 26 protons (since atom is electrically neutral) 18

1.6  10 19  6.023  10 26 1.6  6.023  10 7 coulomb = coulomb 18 18

Charge of electrons = =

1.6  6.023  10 7 coulomb 18

 1.6  6.023  10 7   1.6  6.023  10 7  9  10 9     18 18    Hence Electrical force = 2 2 (10  10 )

   

8  6.023 25  1.6  6.023  10 25 = 2.56 × 10 Newton 18 11. Let two protons be at a distance be 13.8 femi =

F=

9  10 9  1.6  10 38 (14.8)2  10  30

– + – + + – + + –

= 1.2 N

12. F = 0.1 N –2 r = 1 cm = 10 (As they rubbed with each other. So the charge on each sphere are equal) So, F = 1.6 × 10

kq1q2 r

2

–19

c

0.33 × 10

–7

 0.1 =

kq 2 (10

2

2 2

)

q =

Carries by 1 electron

c carries by

1 1.6  10 19

0.1  10 4 9  10 9

2

q =

1 c carried by

1 1  10 14  q =  10  7 9 3

1 1.6  10 19

12 11  0.33  10  7 = 0.208 × 10 = 2.08 × 10

29.2

Electric Field and Potential 13. F =

kq1q2 r

9  10 9  1.6  1.6  10 19  10 19

=

2

(2.75  10

10 2

)

=

23.04  10 29 7.56  10  20

= 3.04 × 10

–9

–27

14. Given: mass of proton = 1.67 × 10 kg = Mp 9 –19 k = 9 × 10 Charge of proton = 1.6 × 10 c = Cp –11 G = 6.67 × 10 Let the separation be ‘r’ Fe =

k(C p ) 2

,

r2

G(Mp )2

fg=

Now, Fe : Fg =

K (C p ) 2 r

2

r2 

r2 G(Mp )

2

9  10 9  (1.6  1`0 19 )2

=

15. Expression of electrical force F = C  e –kr

Since e

= 9 × 2.56 × 10 ≈ 1,24 ×10

38

kr r2

is a pure number. So, dimensional formulae of F =

–2

38

6.67  10 11  (1.67  10  27 )2

2

dim ensional formulae of C dim ensional formulae of r 2

3 –2

Or, [MLT ][L ] = dimensional formulae of C = [ML T ] 2 2 2 Unit of C = unit of force × unit of r = Newton × m = Newton–m Since –kr is a number hence dimensional formulae of k=

1 –1 = [L ] dim entional formulae of r

Unit of k = m

–1

16. Three charges are held at three corners of a equilateral trangle. Let the charges be A, B and C. It is of length 5 cm or 0.05 m force exerted by C on A = F2 Force exerted by B on A = F1 So, force exerted on A = resultant F1 = F2 F=

kq 2

=

9  10 9  2  2  2  10 12

=

F2 A

36  10 = 14.4 25

0.05 m

0.05 m

5  5  10  4 Now, force on A = 2 × F cos 30° since it is equilateral . r2

F1 

60° B

3 = 24.94 N.  2 –6 17. q1 = q2 = q3 = q4 = 2 × 10 C –2 v = 5 cm = 5 × 10 m

0.05 m

C

 Force on A = 2 × 1.44 ×

B

A

so force on c = FCA  FCB  FCD

C

so Force along × Component = FCD  FCA cos 45  0 6 2

= =

k(2  10 ) (5  10

2 2

)



6 2

k(2  10 ) (5  10

2 2

)

1 2 2

9  10 9  4  10 12  1   1  4 24  10 2 2 

So, Resultant R =

 1 1 = kq 2   4 50 2  10  4  25  10

D

  

FCB

= 1.44 (1.35) = 19.49 Force along % component = 19.49

Fx 2  Fy 2 = 19.49 2 = 27.56

18. R = 0.53 A° = 0.53 × 10–10 m

Kq1q2

9  10 9  1.6  1.6  10 38

–9 = 82.02 × 10 N r 0.53  0.53  10 10  10 10 –8 Ve = ? 19. Fe from previous problem No. 18 = 8.2 × 10 N –31 –10 r = 0.53 × 10 m Now, Me = 9.12 × 10 kg

F=

2

Now, Fe =

=

Me v 2 8.2  10 8  0.53  10 10 Fe  r 2 13 12 2 2 v = = = 0.4775 × 10 = 4.775 × 10 m /s r me 9.1 10  31 6

 v = 2.18 × 10 m/s 29.3

FCD FCA

Electric Field and Potential –8

20. Electric force feeled by 1 c due to 1 × 10 F1 = F2 =

k  1  10 (10  10

8

1

-6

= k × 10 N.

2 2

)

k  8  10 8  1

Similarly F3 =

electric force feeled by 1 c due to 8 × 10

k  27  10 8  1 (30  10

2 2

)

= 3k × 10

–6

N

–6

(1 + 2 + 3 +……+10) N

So, F = F1 + F2 + F3 + ……+ F10 = k × 10 –6

= k × 10 ×

–8

c.

k  8  10 8  10 2 28k  10 6 –6 = = 2k × 10 N. 9 4

=

(23  10  2 )2

c.

10  11 –6 9 –6 3 = 55k × 10 = 55 × 9 × 10 × 10 N = 4.95 × 10 N 2 2

21. Force exerted =

r

2

9

=

r=1m

kq1

9  10  2  2  10

= 3.6 × 10

12 –7 22. q1 = q2 = 2 × 10 c –2 l = 50 cm = 5 × 10 m (a) Now Electric force F= K

–6

is the force exerted on the string

m = 100 g –2 d = 5 × 10 m T

9  10 9  4  10 14

q2

q1

q1

16

T Cos 

–2

= N = 14.4 × 10 N = 0.144 N r2 25  10  4 (b) The components of Resultant force along it is zero, because mg balances T cos  and so also. F = mg = T sin  (c) Tension on the string T cos  = mg T sin  = F

F

 T Sin 



90°

90° 

T Sin 

F

T Cos 

F 0.144 = = 0.14693 mg 100  10  3  9.8 2 –3 But T cos  = 10 × 10 × 10 = 1 N Tan  =

T=

1 = sec  cos 

T=

F , sin 



Sin  = 0.145369 ; Cos  = 0.989378; –8

23. q = 2.0 × 10 c

n= ?

T=?



Sin  =

1 20

20 cm

Force between the charges F=

Kq1q2 r

2

=

9  10 9  2  10 8  2  10 8

mg sin  = F  m = Cos =

(3  10  2 )2

= 4 × 10

–3

N

F 4  10 3 –3 = = 8 × 10 = 8 gm g sin  10  (1 / 20)

1  Sin 2  =

1

1 = 400

400  1 = 0.99 ≈ 1 400

So, T = mg cos  –3 –2 Or T = 8 × 10 10 × 0.99 = 8 × 10 M 

5 cm

1 cm mg

29.4

20

T 1 cm

0 T 20 1 cm

Electric Field and Potential 24. T Cos  = mg T Sin  = Fe

…(1) …(2)



9

2

=

1596 2

q = q=

9  10  q

40 cm

Fe kq2 1 =  mg r mg

Solving, (2)/(1) we get, tan  =

 1596

20 g

A

2

B

20 g

4 cm

(0.04) 2  0.02  9.8

(0.04)2  0.02  9.8  2 9  10 9  1596

17  10

16

9  10 9  39.95

–8

= 4.123 × 10

kq

25. Electric force =

6.27  10 4

=

–16 2

c

c

q

=

2

EF

kq2

2

( sin Q   sin Q)

= 17 × 10

2

ℓ   v2

ℓ

FBD for a mass (m) T cos T

q

a

4 sin

mg

2

ℓ sin 

T Sin 

EF

So, T Cos  = ms (For equilibrium) T sin  = Ef

Ef mg

Or tan  =

mg

 mg = Ef cot  = or m =

kq2 4 2 sin2 

q2 cot  16E 0  2 Sin2 g

q2 cot 

cot  =

 2 sin2  16E 0

unit.

26. Mass of the bob = 100 g = 0.1 kg So Tension in the string = 0.1 × 9.8 = 0.98 N. For the Tension to be 0, the charge below should repel the first bob. F=

kq1q2

T – mg + F = 0  T = mg – f

r2

 0.98 =

9  10 9  2  10 4  q2 (0.01)2

 q2 =

10 cm

2 × 10–4 C

T = mg

0.98  1 10 2 9  2  10 5

–9

= 0.054 × 10 N

27. Let the charge on C = q So, net force on c is equal to zero

2

2

 2x = (d – x) 

d

x=

2 1

=

C

q

kqQ

So FAC  FBA = 0, But FAC = FBC 

x2

=

mg

k 2qQ

A

(d  x ) 2

2q d–x

x

B

d

2x=d–x

d ( 2  1)



( 2  1) ( 2  1)

= d( 2  1)

For the charge on rest, FAC + FAB = 0

(2.414)2

kqQ d2



kq(2q) d2

=0

kq d2

[(2.414 )2 Q  2q] = 0

2

 2q = –(2.414) Q Q=

  2 q = –(0.343) q = –(6 – 4 2 ) q =    ( 2  1) 32 2  2

28. K = 100 N/m

2

ℓ = 10 cm = 10

Force between them F =

kq1q2 r

2

–1

m

q = 2.0 × 10

9  10 2  10

8

10

2

9

=

–8

 2  10

c Find ℓ = ?

8

= 36 × 10

36  10 5 F –7 –6 So, F = – kx or x = = = 36 × 10 cm = 3.6 × 10 m K 100 29.5

–5

N

K q1

q2

Electric Field and Potential –6

29. qA = 2 × 10 C Mb = 80 g  =0.2 Since B is at equilibrium, So, Fe = R 

Kq A qB

10 cm R

=  R = m × g

r2

 mg =  R

9  10 9  2  10 6  qB = 0.2 × 0.08 × 9.8 0.01 0.2  0.08  9.8  0.01 –8  qB = = 8.7 × 10 C 9  10 9  2  10  6 –6 30. q1 = 2 × 10 c Let the distance be r unit kq1q2  Frepulsion = r2 kq1q2 For equilibrium = mg sin  r2

Fe





9

9  10  4  10 r

12

= m × 9.8 ×

2

mg

–8

Range = 8.7 × 10 C

q2 x

1 2

30°

q1

18  4  10 3 72  10 3 –2 = = 7.34 × 10 metre 1 m  9 .8 9.8  10 –1  r = 2.70924 × 10 metre from the bottom. 31. Force on the charge particle ‘q’ at ‘c’ is only the x component of 2 forces 2

r =

So, Fon c = FCB Sin  + FAC Sin  = 2 FCB Sin  = 2

KQq 2

x  ( d / 2)



   





x

x 2

2

d /4

2 2

 16 x + d + 8x d = 12 x 2

2

d =0

2

2

( x  d / 4)



( 4 x 2  d 2 )3 4

2kqx

=

 2 2 2 2   ( 4 x  d )  x 3 / 2 4 x  d = 0  K [ 4 x 2  d2 ] 3  

K( 4 x 2  d2 )1 / 2 ( 4 x 2  d2 )3  12x 2 4



1/ 2

dF =0 dx

For maximum force

d  16kQqx dx  ( 4 x 2  d2 )3 / 2

2

But FCB = FAC

 = 0  (4x 4

2

2

2 3

d +8x d =0

2

d =8x d=

2

=

16kQq ( 4 x 2  d 2 )3 / 2

x

FCB

 8 x    =0   

+d ) = 12 x

2

d +8x =0



1/ 2

3/2

FAC C

x 

2

A

d

2

2 2 FBO

So, force on 0 = FAB  FBO

= 2 =

2

2

(d / 2  x )

4  2  kQq (d2  4 x 2 )



F=

x [(d / 2)2  x 2 ]1 / 2

=

( d / 2)  x 2kQq

[(d / 2)2  x 2 ]3 / 2







2KQq 2

O

x

FAO  FBO 

Sin

FOA 

But FAO Cos  = FBO Cos  So, force on ‘0’ in due to vertical component.

KQq

B

d/2

2

32. (a) Let Q = charge on A & B Separated by distance d q = charge on c displaced  to –AB

F = FAO Sin  + FBO Sin  



2

Sin

A

d/2 Q

x = Electric force  F  x

29.6

B

C d

Q

Electric Field and Potential (b) When x > t1/2, the number of active nuclei will become constant. i.e. (dN/dt)present = R = (dN/dt)decay  R = (dN/dt)decay  R = N [where,  = Radioactive decay constant, N = constant number] Rt1/ 2 0.693  R= (N)  Rt1/2 = 0.693 N  N = . t1/ 2 0.693

38. Given : Half life period = t1/2, Rate of radio active decay =

39. Let N0 = No. of radioactive particle present at time t = 0 N = No. of radio active particle present at time t. –t  N = N0 e [ - Radioactive decay constant] –t –t  The no.of particles decay = N0 – N = N0 – N0e = N0 (1 – e ) We know, A0 = N0 ; R = N0 ; N0 = R/ From the above equation R –t (substituting the value of N0) N = N0 (1 – e ) = (1  e t )  23 40. n = 1 mole = 6  10 atoms, t1/2 = 14.3 days t = 70 hours, dN/dt in root after time t = N 0.69370 –t

23

23

23

= 6  10  e 14.324 = 6  10  0.868 = 5.209  10 . 23 5.209  1023  0.693  0.010510 dis/hour. 14.324 3600 –6 23 17 = 2.9  10  10 dis/sec = 2.9  10 dis/sec.  1ci  Fraction of activity transmitted =    100%  2.9  1017 

N = No e

 1 3.7  108  –11    2.9  1011  100  % = 1.275  10 %.   41. V = 125 cm3 = 0.125 L, P = 500 K pa = 5 atm. 8 T = 300 K, t1/2 = 12.3 years = 3.82  10 sec. Activity =   N 5  0.125 23 22 N = n  6.023  10 =  6.023  1023 = 1.5  10 atoms. 8.2  10 2  3  102 0.693 –8 –9 –1 = = 0.1814  10 = 1.81  10 s 3.82  108 –9 22 3 Activity = N = 1.81  10  1.5  10 = 2.7  10 disintegration/sec = 42.

2.7  1013 3.7  1010

Ci = 729 Ci.

212 83 Bi

208  81 Ti  24He( )

212 83 Bi

212 212  84 Bi  84 P0  e 

t1/2 = 1 h. Time elapsed = 1 hour 212 Present = 1 g at t = 0 Bi 212  at t = 1 Bi Present = 0.5 g Probability -decay and -decay are in ratio 7/13.  Tl remained = 0.175 g  P0 remained = 0.325 g 46.6

The Nucleus 108

110

8

43. Activities of sample containing Ag and Ag isotopes = 8.0  10 disintegration/sec. 8 a) Here we take A = 8  10 dis./sec  i) In (A1/ A 01 ) = In (11.794/8) = 0.389 12

ii) In (A2/ A 02 ) = In(9.1680/8) = 0.1362

10

iii) In (A3/ A 03 ) = In(7.4492/8) = –0.072

8

iv) In (A4/ A 04 ) = In(6.2684/8) = –0.244 v) In(5.4115/8) = –0.391 vi) In(3.0828/8) = –0.954 vii) In(1.8899/8) = –1.443 viii) In(1.167/8) = –1.93 ix) In(0.7212/8) = –2.406 b) The half life of 110 Ag from this part of the plot is 24.4 s. 110 c) Half life of Ag = 24.4 s.  decay constant  = 0.693/24.4 = 0.0284  t = 50 sec, –t 8 –0.028450 8 The activity A = A0e = 8  10  e = 1.93  10 d)

6 4 2

20 40 60 80 100 200 300

2 4

400

500

Time

6 4 2 O

20 40 60 80 108

e) The half life period of Ag from the graph is 144 s. 44. t1/2 = 24 h tt 24  6  t1/2 = 1 2  = 4.8 h. t1  t 2 24  6 A0 = 6 rci ; A = 3 rci A 6 rci t  A = t / t0  3 rci = t / 4.8h  = 2  t = 4.8 h. 24.8h 2 2 1/ 2 45. Q = qe  t / CR ; A = A0e

–t

Energy 1q2  e 2t / cR  Activity 2 CA 0 e t Since the term is independent of time, so their coefficients can be equated, 2t 2 1 2  = t or,  = or,  or, R = 2 (Proved) So,  CR CR CR C 46. R = 100  ; L = 100 mH After time t, i = i0 (1  e t / Lr )

i i0 (1  e  tR / L )  N N0 et

–t

N = N0 (e )

i/N is constant i.e. independent of time.

Coefficients of t are equal –R/L = –  R/L = 0.693/t1/2 –3 –4 = t1/2 = 0.693  10 = 6.93  10 sec. 235 23 47. 1 g of ‘I’ contain 0.007 g U So, 235 g contains 6.023  10 atoms. So, 0.7 g contains

6.023  1023  0.007 atom 235

1 atom given 200 Mev. So, 0.7 g contains

6.023  10 23  0.007  200  106  1.6  10 19 –8 J = 5.74 10 J. 235

48. Let n atoms disintegrate per second 6 –19 Total energy emitted/sec = (n  200  10  1.6  10 ) J = Power 6 300 MW = 300  10 Watt = Power 46.7

The Nucleus 6

6

300  10 = n  200  10  1.6  10 3 3  n=  1019 =  1019 2  1.6 3.2 6  10

23

–19

atoms are present in 238 grams

3 238  3  1019 –4 = 3.7  10 g = 3.7 mg.  1019 atoms are present in 3.2 6  1023  3.2 8 49. a) Energy radiated per fission = 2  10 ev 8 7 –12 Usable energy = 2  10  25/100 = 5  10 ev = 5  1.6  10 8 8 Total energy needed = 300  10 = 3  10 J/s

3  108

20 = 0.375  10 5  1.6  1012 20 24 No. of fission per day = 0.375  10  3600  24 = 3.24  10 fissions. 24 b) From ‘a’ No. of atoms disintegrated per day = 3.24  10 23 We have, 6.023  10 atoms for 235 g 235 for 3.24  1024 atom =  3.24  1024 g = 1264 g/day = 1.264 kg/day. 23 6.023  10

No. of fission per second =

50. a)

2 2 1 H  1H

 13H  11H

Q value = 2M(12 H) = [M(13 H)  M(13 H)] = [2  2.014102 – (3.016049 + 1.007825)]u = 4.0275 Mev = 4.05 Mev. b)

2 2 1 H  1H

 32H  n

Q value = 2[M(12 H)  M(32 He)  Mn ] = [2  2.014102 – (3.016049 + 1.008665)]u = 3.26 Mev = 3.25 Mev. c)

2 3 1 H  1H

 24H  n

Q value = [M(12 H)  M(13 He)  M( 24 He)  Mn ] = (2.014102 + 3.016049) – (4.002603 + 1.008665)]u = 17.58 Mev = 17.57 Mev.

Kq1q2 9  109  (2  1.6  1019 )2 = r r –23 1.5 KT = 1.5  1.38  10  T

51. PE =

Equating (1) and (2) 1.5  1.38  10

–23

…(1) …(2)

T=

9  109  10.24  10 38 2  10 15

9  109  10.24  10 38

9 10 = 22.26087  10 K = 2.23  10 K. 2  10 15  1.5  1.38  10 23 4 4 8  Be 52. H + H 2  4.0026 u M( H) 8  8.0053 u M( Be) 2 8 Q value = [2 M( H) – M( Be)] = (2  4.0026 – 8.0053) u = –0.0001 u = –0.0931 Mev = –93.1 Kev. 23 53. In 18 g of N0 of molecule = 6.023  10

 T=

6.023  1026 25 = 3.346  10 18 26  % of Deuterium = 3.346  10  99.985 25 Energy of Deuterium = 30.4486  10 = (4.028204 – 3.016044)  93 5 = 942.32 ev = 1507  10 J = 1507 mJ In 100 g of N0 of molecule =

 46.8

THE SPECIAL THEORY OF RELATIVITY CHAPTER - 47 1.

6

S = 1000 km = 10 m The process requires minimum possible time if the velocity is maximum. 8 We know that maximum velocity can be that of light i.e. = 3  10 m/s.

Distance 106 1   s. Speed 3  108 300 ℓ = 50 cm, b = 25 cm, h = 10 cm, v = 0.6 c a) The observer in the train notices the same value of ℓ, b, h because relativity are in due to difference in frames. b) In 2 different frames, the component of length parallel to the velocity undergoes contraction but the perpendicular components remain the same. So length which is parallel to the x-axis changes and breadth and height remain the same.

So, time = 2.

e = e 1 

3.

V2 C

2

 50 1 

C2

= 50 1  0.36 = 50  0.8 = 40 cm. The lengths observed are 40 cm  25 cm  10 cm. L=1m 5 a) v 3  10 m/s L = 1 1 

9  1010

9  1016 6 b) v = 3 x 10 m/s L = 1 1 

9  1012

9  1016 7 c) v = 3  10 m/s

 1  10 6 = 0.9999995 m

 1  10 4 = 0.99995 m.

9  1014

 1  10 2 = 0.9949 = 0.995 m. 9  1016 v = 0.6 cm/sec ; t = 1 sec 6 8 a) length observed by the observer = vt  0.6  3  10  1.8  10 m L = 1 1 

4.

(0.6)2 C2

8

b) ℓ =  0 1  v 2 / c 2  1.8  10 =  0 1 

(0.6)2 C2 C2

1.8  108 8 = 2.25  10 m/s. 0.8 The rectangular field appears to be a square when the length becomes equal to the breadth i.e. 50 m. i.e. L = 50 ; L = 100 ; v = ? 8 C = 3  10 m/s ℓ0 =

5.

We know, L = L 1  v 2 / c 2 6.

 50 = 100 1  v 2 / c 2  v = 3 / 2C = 0.866 C. 6 L0 = 1000 km = 10 m v = 360 km/h = (360  5) / 18 = 100 m/sec. 2

10 4  100  9 6 a) h = h0 1  v 2 / c 2  10 6 1   = 10 .   10 1  8 6 9  10  3  10  Solving change in length = 56 nm. b) t = L/v = 56 nm / 100 m = 0.56 ns. 47.1

The Special Theory of Relativity 7.

v = 180 km/hr = 50 m/s t = 10 hours let the rest dist. be L.

A

B

L = L 1  v 2 / c 2  L = 10  180 = 1800 k.m.

1802

1800 = L 1 

(3  105 )2 –14

or, 1800  1800 = L(1 – 36  10

3.24  10

or, L =

6

)

= 1800 + 25  10

14

–12

1  36  10 or 25 nm more than 1800 km.

b) Time taken in road frame by Car to cover the dist = 5

8.

1.8  106  25  10 9 50

–8

= 0.36  10 + 5  10 = 10 hours + 0.5 ns. a) u = 5c/13

t

t =

2

1 v / c

2

1y

 1



2

25c 169c 2

y  13 13  y. 12 12

The time interval between the consecutive birthday celebration is 13/12 y. b) The fried on the earth also calculates the same speed. 9.

The birth timings recorded by the station clocks is proper time interval because it is the ground frame. That of the train is improper as it records the time at two different places. The proper time interval T is less than improper. i.e. T = v T Hence for – (a) up train  Delhi baby is elder

(b) down train  Howrah baby is elder.

10. The clocks of a moving frame are out of synchronization. The clock at the rear end leads the one at 2 from by L0 V/C where L0 is the rest separation between the clocks, and v is speed of the moving frame. Thus, the baby adjacent to the guard cell is elder. 11. v = 0.9999 C ; t = One day in earth ; t = One day in heaven v=

1 2

1 v / c

1



2

1

2

(0.9999) C C2

2



1 = 70.712 0.014141782

t = v t ; Hence, t = 70.7 days in heaven. 12. t = 100 years ; V = 60/100 K ; C = 0.6 C. t =

t 2

1 V / C

2

100y

 1

2

(0.6) C C2

2



100y = 125 y. 0.8

13. We know f = f 1  V 2 / C2 f = apparent frequency ; f = frequency in rest frame v = 0.8 C f =

1

0.64C2 C

2

 0.36 = 0.6 s

–1

2

The Special Theory of Relativity 14. V = 100 km/h, t = Proper time interval = 10 hours t 10  3600  t = 2 2 2 1 V / C  1000  1  8   36  3  10 

1   t – t = 10  3600   1 2 1000   1     36  3  108   By solving we get, t – t = 0.154 ns.  Time will lag by 0.154 ns. 15. Let the volume (initial) be V. V = V/2 So, V/2 = v 1  V 2 / C2  C/2 =

C2  V 2  C2/4 = C2 – V2

C2 3 2 3  C V= C. 4 4 2 16. d = 1 cm, v = 0.995 C 2

 V = C2 

a) time in Laboratory frame =

1 10 2

d 1 10 2  v 0.995C

= 33.5  10 0.995  3  108 b) In the frame of the particle =

t =

t 1  V 2 / C2



–12

= 33.5 PS

33.5  10 12 1  (0.995)2

= 335.41 PS.

–2

17. x = 1 cm = 1  10 m ; K = 500 N/m, m = 200 g 2 –4 Energy stored = ½ Kx = ½  500  10 = 0.025 J Increase in mass =

0.025 C

2



Fractional Change of max =

0.025 9  1016 0.025



1

9  10 2  10 1 18. Q = MS   1  4200 (100 – 0) = 420000 J. 2 E = (m)C E Q 420000  m = 2  2  C C (3  108 )2 –12

16

= 0.01388  10

–12

= 4.66  10 = 4.7  10 kg. 19. Energy possessed by a monoatomic gas = 3/2 nRdt. Now dT = 10, n = 1 mole, R = 8.3 J/mol-K. E = 3/2  t  8.3  10

1.5  8.3  10

124.5  C2 9  1015 –16 –15 = 1383  10 = 1.38  10 Kg. 20. Let initial mass be m 2 ½ mv = E Loss in mass =

2

1  12  5  m  50 m   2  18  9 2 m = E/C  E=

3

–16

–8

= 1.4  10 .

The Special Theory of Relativity

m  50

m 50  m 9  9  10 81 1016 –16 –17  0.617  10 = 6.17  10 . 21. Given : Bulb is 100 Watt = 100 J/s So, 100 J in expended per 1 sec. Hence total energy expended in 1 year = 100  3600  24  365 = 3153600000 J Total energy 315360000 Change in mass recorded =  C2 9  1016 8 –16 –8 = 3.504  10  10 kg = 3.5  10 Kg.  m =



16

2

22. I = 1400 w/m 2 Power = 1400 w/m  A 2 11 2 = (1400  4R )w = 1400  4  (1.5  10 ) 2 22 = 1400  4  (1/5)  10 a)

sun

E mC2 m E / t    2 t t t C 2

1400  4   2.25  1022

66

9

9

= 1696  10 = 4.396  10 = 4.4  10 . 9  1016 b) 4.4  109 Kg disintegrates in 1 sec. C =

2  10

30

Kg disintegrates in

2  1030 4.4  109

sec.

  1 1021 –8 21 13 =   = 1.44  10  10 y = 1.44  10 y.  2.2  365  24  3600  –31 23. Mass of Electron = Mass of positron = 9.1  10 Kg Both are oppositely charged and they annihilate each other. Hence, m = m + m = 2  9.1  10–31 Kg 2 Energy of the resulting  particle = m C = 2  9.1  10

–31

 9  10

16

J=

2  9.1 9  10 15

1.6  10 19 4 6 = 102.37  10 ev = 1.02  10 ev = 1.02 Mev. –31 24. me = 9.1  10 , v0 = 0.8 C a) m =

Me 1  V 2 / C2



9.1 1031 1  0.64C2 / C2

–31



ev

9.1 10 31 0.6

–31

= 15.16  10 Kg = 15.2  10 Kg. 2 2 2 b) K.E. of the electron : mC – meC = (m – me) C –31 –31 8 2 –31 18 = (15.2  10 – 9.1  10 )(3  10 ) = (15.2  9.1)  9  10  10 J –15 –14 –14 = 54.6  10 J = 5.46  10 J = 5.5  10 J. c) Momentum of the given electron = Apparent mass  given velocity –31 8 –23 = 15.2  10 – 0.8  3  10 m/s = 36.48  10 kg m/s –22 = 3.65  10 kg m/s 2

25. a) ev – m0C =

m0 C2 2 1

=

2

V C2

9.1 9  1031  1016 2 1

0.36C C2

2

 ev – 9.1  10

–31

 9  10

 eV – 9.1  9  10

–15

4

16

R

The Special Theory of Relativity

9.1 9  10 15 9.1 9  10 15 –15  eV – 9.1  9  10 = 2  0.8 1.6

=

 9.1 9   81.9   9.1 9   1015 = eV   81.9   10 15  eV =   1.6   1.6  –15 4 eV = 133.0875  10  V = 83.179  10 = 831 KV.

m0 C2

2

b) eV – m0C =

2 1  eV – 81.9  10

–15

 eV = 12.237  10  V=

 eV – 9.1  9  10

V2 C2

m0 C2 2 1

2

V C2

+m0C =

 eV = 372.18  10

9.1 9  10 15

= 76.48 kV. 2

2 1

=

2 1

V = 0.99 C = eV – m0C =

2

16

–15

1.6  10 19

m0 C2

 9  10

9.1 9  10 15 2  0.435

=

12.237  10 15

 eV =

–19

–15

V2 C2

9.1 10 31  9  1016 2

2 1  (0.99)

V=

372.18  20 15 1.6  1019

 9.1 1031  9  1016

= 272.6  10

4

6

 V = 2.726  10 = 2.7 MeV. 26. a)

m0 C2 V2 1 2 C

2

– m0C = 1.6  10

–19

1   –19  m0 C2   1 = 1.6  10 2 2  1 V / C  

1 1  V 2 / C2

1 =

1.6  10 19 9.1 10 31  9  1016 8

5

 V = C  0.001937231 = 3  0.001967231  0 = 5.92  10 m/s. b)

m0 C2 1

2

V C2

2

– m0C = 1.6  10

–19

3

 10  10

1   –15  m0 C2   1 = 1.6  10 2 2  1 V / C  

1 1  V 2 / C2

1 =

1.6  10 15 9.1 9  1015 8

7

 V = 0.584475285  10 = 5.85  10 m/s. 6 7 –19 –12 c) K.E. = 10 Mev = 10  10 eV = 10  1.6  10 J = 1.6  10 J 

m0 C2 2

V 1 2 C 2

2

– m0C = 1.6  10

16

 V = 8..999991359  10

–12

J

8

 V = 2.999987038  10 . 5

0.81C2 C2

The Special Theory of Relativity 27. m = m – m0 = 2m0 – m0 = m0 2 –31 16 Energy E = m0c = 9.1  10  9  10 J E in e.v. =

9.1 9  10 15 1.6  10 19

4

= 51.18  10 ev = 511 Kev.

 m0 C2  1  m0 C2   mv 2  2   2 V  1 2  C  28.  = 0.01 1 2 m0 v 2   v2 1 3 V2 1 3 5 V6 2 )  m0C2        m0 C (1  2 2 6 1 2 4 2 4 6 2C C C   mv 2 = 0.1   1 2   2 m0 v 2   1 3 V 4 15 V4 1 m0 v 2  m0 2  m0 2  m0 v 2 2 8 96 2 C C  = 0.01 1 2 m0 v 2 3 V4 15 V 4 = 0.01  2 4C 96  2 C4 4 Neglecting the v term as it is very small 



3 V2 V2 = 0.01  = 0.04 / 3 4 C2 C2 0.2  3  108 1.732 8 7 = 0.346  10 m/s = 3.46  10 m/s.

 V/C = 0.2 / 3 = V = 0.2 / 3 C =



6