Thermodynamics Challenges - Luis Eduardo Physics Challenges

09/10/13

Thermodynamics Challenges - Luis Eduardo Physics Challenges

Thermodynamics and Gases Challenging Problems Physics Challenges - Home ALL REFERENCES

P1) A monatomic gas in a PV diagram performs a rectangular thermodynamic cycle. Determine the maximum efficiency of the cycle. Answer: 40% P2) (Professor Renato Brito – Brazil) A thermodynamic transformation is called isoclinal when the ratio V²/T remains constant during the process. Consider the thermodynamic cycle ABCA represented in a plane Volume versus Temperature covered by n moles of a monatomic ideal gas. Knowing that the transformation AB is isoclinal, calculate:

a) the work done by the gas in the cycle; b) the efficiency of that cycle. Answer: a)

b)

P3) A vessel of volume V = 30 L contains ideal gas at the temperature 0 °C. After a portion of the gas has been let out, the pressure in the vessel decreased by Δp = 0.78 atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions p = 1.3 g/L. Answer: 30 g P4) Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the www.luiseduardo.com.br/thermology/thermodynamics/thermodynamicsproblems.htm

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other if the pressure difference ΔP 1.10 atm. Initially, there was a vacuum in one vessel while the other contained ideal gas at a temperature t1 = 27 °C and pressure p1 = 1.00 atm. Then both vessels were heated to a temperature t2 = 107 °C. Up to what value will the pressure in the first vessel (which had vacuum initially) increase? Answer: 0.10 atm P5) A sample of an ideal non linear tri-atomic gas has a pressure P0 and temperature T0 taken through the cycle as shown starting from A. Pressure for process C cycle and work done.

D is 3 times P0. Calculate heat absorbed in the

Answer: 31P0V0; -5P0V0 P6) A fixed mass of a gas is taken through a process A adiabatic and C

B

C

A. Here A

B is isobaric. B

C is

A is isothermal. Find efficiency of the process. (take γ = 1.5).

Answer:

P7) A cylinder containing a gas is closed by a movable piston. The cylinder is submerged in an ice-water mixture. The piston is quickly pushed down from position 1 to position 2. The piston is held at position 2 until the gas is again at 0°C and then slowly raised back to position 1. Represent the whole process on P-V diagram. If m = 100 g of ice are melted during the cycle, how much work is done on the gas. Latent heat of ice = 80 cal/g.

Answer:

P8) An ideal gas at NTP is enclosed in a adiabatic vertical cylinder having area of cross section A = 27 cm², between two light movable pistons as shown in the figure. Spring with force constant k = 3700 N/m is in a relaxed state initially. Now the lower piston is moved upwards a height h/2, h being the initial length of gas column. It is observed that the upper piston moves up by a distance h/16. Find h taking γ for the gas to be 1.5. Also find the final temperature of the gas. www.luiseduardo.com.br/thermology/thermodynamics/thermodynamicsproblems.htm

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Answer: 1.6 m and 364K P9) An adiabatic vessel containing n moles of a ideal diatomic gas is fitted with a light conducting piston. The cross-sectional area, thickness and thermal conductivity of piston are A, l and K respectively. The other side of the piston is open to atmosphere of temperature T0. Heat is supplied to the gas by means of an electric heater at a small constant rate q. Initial temperature of gas is T0.

a) Find the temperature of the gas as a function of time b) Find the maximum temperature of the gas c) What is the ratio of the maximum volume to the minimum volume? Answer:

P10) The figure shows an insulated cylinder divided into three parts A, B and C. Pistons I and II are connected by a rigid rod and can move without friction inside the cylinder. Piston I is perfectly conducting while piston II is perfectly insulating. The initial state of the gas (γ = 1.5) present in each compartment A, B and C is as shown. Now, compartment A is slowly given heat through a heater H such that the final volume of C becomes 4V0/9. Assume the gas to be ideal and find:

a) Final pressures in each compartment A, B and C b) Final temperatures in each compartment A, B and C c) Heat supplied by the heater d) Work done by gas in A and B e) Heat flowing across piston I. Answer:

P11) Figure shows the variation of the internal energy U with the density ρ of one mole of ideal monatomic gas for a thermodynamic cycle ABCA. Here process AB is a part of rectangular hyperbola.

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a) Draw the P-V diagram for the above process. b) Find the net amount of heat absorbed by the system for the cyclic process. c) Find the work done in the process AB. Answer:

P12) An ideal monatomic gas undergoes a process where its pressure is inversely proportional to its temperature. Calculate the specific heat for the process and find the work done by two moles of gas if the temperature changes from T1 to T2. Answer: 7R/2M and 4R(T2 – T1) P13) An ideal diatomic gas undergoes a process in which its internal energy relates to the volume as: Where a is a constant. a) Find the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by 100 J. b) Find the molar specific heat of the gas for this process. Answer: a) 80 J, 180 J b) 4.5R P14) Two rectangular boxes shown in figure has a partition which can slide without friction along the length of the box. Initially each of the two chambers of the box has one mole of a monatomic ideal gas (γ = 5/3) at a pressure P0 volume V0 and temperature T0. The chamber on the left is slowly heated by an electric heater. The walls of the box and the partitions are thermally insulated. Heat loss through the lead wires of the heater is negligible. The gas in the left chamber expands, pushing the partition until the final pressure in both chambers becomes 243P0/32.

Then, determine the final temperature of the gas in each chamber and the work-done by the gas in the right chamber. Answer:

P15) An adiabatic cylinder of length 2l and cross-sectional area A is closed at both ends. A freely moving non-conducting this piston divides the cylinder in two parts. The piston is connected with right end by a spring having force constant K and natural length l. Left part of the cylinder contains one mole of helium and right part contains 0.5 mole of each of helium and oxygen. If initial pressure of gas in each part is P0, calculate heat supplied by the heating coil, connected to left part, to compress the spring through half of its natural length. Answer:

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P16) 0.01 moles of an ideal diatomic gas is enclosed in an adiabatic cylinder of cross-sectional area A = 10– 4 m². In the arrangement shown, a block of mass M = 0.8 kg is placed on a horizontal support, and another block of mass m = 1 kg is suspended from a spring of stiffness constant k = 16 N/m. Initially, the spring is relaxed and the volume of the gas is V = 1.4*10−4 m³.

(a) Find the initial pressure of the gas. (b) If block m is gently pushed down and released it oscillates harmonically, find its angular frequency of oscillation. (c) When the gas in the cylinder is heated up the piston starts moving up and the spring gets compressed so that the block M is just lifted up. Determine the heat supplied. Take atmospheric pressure P0 = 105 Nm−2, g = 10m/s2. Answer:

P17) A thermally insulated vessel is divided into two parts by a heat-insulating piston which can move in the vessel without the friction. The left part of the vessel contains one mole of an ideal monatomic gas, & the right part is empty. The piston is connected to the right wall of the vessel through a spring whose length in free state is equal to the length of the vessel as shown in the figure. Determine the heat capacity C of the system, neglecting the heat capacities of the vessel, piston and spring.

Answer: C = 2R P18) A gaseous mixture enclosed in a vessel of volume V consists of one gram mole of a gas A with γ = Cp/Cv = 5/3 & another gas B γ = 7/5 with at a certain temperature T. The gram molecular weights of the gases A & B are 4 & 32 respectively. The gases A & B do not react with each other and are assumed to be ideal. The gaseous mixture follows the equation; PV19/13 = const. in adiabatic processes. a) Find the number of gram moles of the gas B in the gaseous mixture. b) Compute the speed of sound in the gaseous mixture at T = 300 K. c) If T is raised by 1 K from 300 K, find the percentage change in the speed of sound in the gaseous mixture. d) The mixture is compressed adiabatically to 1/5 its initial volume V. Find the change in its adiabatic compressibility in terms of the given quantities. Answer:

P19) At 27 °C two moles of an ideal monatomic gas occupy a volume V. The gas expands adiabatically to a volume 2V. Calculate: (i) the final temperature of the gas (ii) change in its internal energy (iii) the work done by the gas during the process . Answer:

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P20) One mole of a diatomic ideal gas (γ = 1.4) is taken through a cyclic process starting from point A. The process A

B is an adiabatic compression, B

C is isobaric expansion. C

D is an adiabatic expansion,

and D A is isochoric. The volume ratios are VA/VB = 16 and VC/VB= 2 and the temperature at A is TA = 300 K. Calculate the temperature of the gas at the points B and D and find the efficiency of the cycle. Answer: 909 K, 791.4 K, 0.614 P21) A weightless piston divides a thermally insulated cylinder into two parts of volumes V and 3V. 2 moles of an ideal gas at pressure P = 2atm are confined to the part with volume V = 1 L. The remainder of the cylinder is evacuated. The piston is now released and the gas expands to fill the entire space of the cylinder. The piston is then pressed back to the initial position. Find the increase of internal energy in the process and final temperature of the gas. The ratio of the specific heat of the gas γ = 1.5. Answer:

P22) The piston cylinder arrangement shown contains a diatomic gas at temperature 300 K. The crosssectional area of the cylinder is 1 m². Initially the height of the piston above the base of the cylinder is 1 m. The temperature is now raised to 400 K at constant pressure. Find the new height of the piston above the base of the cylinder. If the piston is now brought back to its original height without any heat loss, find the new equilibrium temperature of the gas. You can leave the answer in fraction.

Answer:

P23) In a thermodynamic process temperature of the gas is inversely proportional to its volume. Starting from pressure p0 and temperature T0, two moles of gas is cooled to T0/2. Find the final pressure of the gas and work done by gas. Answer:

P24) One mole of an ideal gas with adiabatic exponent γ, follows the thermodynamic process p = αV, where α is a constant. The gas expands to increase its volume η times. Find the change in internal energy and heat capacity of the gas. Answer:

P25) During a thermodynamic process in an ideal gas with adiabatic exponent γ, one third of heat supplied is used to raise internal energy of gas. Find molar specific heat of the gas in this process. Answer:

P26) A gaseous mixture enclosed in a vessel consists of 1 g mole of a gas A with (γ = 5/3) and another gas B with (γ = 7/5) at a temperature T. The gas A and B are mutually inert and can be considered ideal. Find the number of g moles of gas B in the mixture if γ of the mixture is 19/13. Answer: 2 P27) Find the amount work done to increase the temperature of one mole of an ideal gas by 30 °C, if it www.luiseduardo.com.br/thermology/thermodynamics/thermodynamicsproblems.htm

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expands following the process Answer: 166.2 J

Thermodynamics Challenges - Luis Eduardo Physics Challenges

.

P28) Find the specific heat of gas as the function of its volume if one mole of it follows the thermodynamic process T = T0 + αp, where T0 and α are constants. Answer:

P29) Find the equation of the thermodynamic process in which specific heat of the gas is given by:

Answer:

P30) Find the efficiency of a heat engine in the form of a cyclic process bounded by two isochoric and two isobaric curves, in which ratio of the maximum and minimum volume of gas, as well as the ratio of maximum and minimum pressure of gas is n. Answer:

P31) Two ideal monatomic gasses at absolute temperature T1 and T2 are mixed. There is no loss of energy. The masses of the molecules are m 1 and m 2 and the number of molecules in the gases are n1 and n2 respectively. Find the temperature of mixture. Answer:

P32) An ideal gas has adiabatic exponent γ. In some process its molar heat capacity varies as C = α/T, where α is a constant. Find the work performed by one mole of gas during its heating from T0 to nT0. Answer:

P33) A movable heavy piston is supported by a spring inside a vertical cylindrical container as shown. When all air is pumped out of the container, the piston is in equilibrium as shown in Diagram 1, with only a tiny gap between the piston and the bottom of the container. When a portion of gas at temperature T is introduced under the piston, the latter rises to a height h as shown in Diagram 2. What would be the height of the piston above the bottom of the container if the gas is then heated to temperature 2T? Assume that the piston moves without friction and that the spring obeys Hooke’s law.

Answer: www.luiseduardo.com.br/thermology/thermodynamics/thermodynamicsproblems.htm

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h P34) One mole of a certain ideal gas is contained under a weightless piston of a vertical cylinder at a temperature T. The space over the piston opens into the atmosphere. What work has to be performed in order to increase isothermally the gas volume under the piston n times by slowly raising the piston? The friction of the piston against the cylinder walls is negligibly small. Answer:

P35) Each of the two isolated vessels, A and B of fixed volumes, contains N molecules of a perfect monatomic gas at pressure P. The temperatures of A and B are T1 and T2, respectively. The two vessels are brought into thermal contact. At equilibrium, the change in entropy is:

Answer: C P36) An ideal gas with the adiabatic exponent y undergoes a process in which its internal energy relates to the volume as U = aVb , where a and b are constants. Find the work performed by the gas and the amount of heat to be transferred to this gas to increase its internal energy by ΔU. Answer:

P37) Two moles of an ideal monatomic gas is taken through a cycle ABCA as shown in the P–T diagram. During the process AB, pressure and temperature of the gas vary such that PT = constant. If T1 = 300 K, calculate the work done on the gas in the process AB.

Answer: -1200R P38) Consider free expansion of one mole of an ideal gas in an adiabatic container from volume V1 to V2. The entropy change of the gas, calculated by considering a reversible process between the original state (V1, T) to the final state (V2, T) where T is the temperature of the system, is denoted ΔS1. The corresponding change in the entropy of the surrounding is ΔS2. Which of the following combinations is correct?

Answer: C P39) Two vessels A and B, thermally insulated, contain an ideal monatomic gas. A small tube fitted with a valve connects these vessels. Initially the vessel A has 2 L of gas at 300 K and 2*105 Nm–2 pressure while vessel B has 4 L of gas at 350 K and 4*105 Nm–2 pressure. The valve is now opened and the system www.luiseduardo.com.br/thermology/thermodynamics/thermodynamicsproblems.htm

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reaches equilibrium in pressure and temperature. Calculate the new pressure and temperature. Answer: 10*105/3 N/m² and 339K P40) A portion of helium gas in a vertical cylindrical container is in thermodynamic equilibrium with the surroundings. The gas is confined by a movable heavy piston. The piston is slowly elevated a distance H from its equilibrium position and then kept in the elevated position long enough for the thermodynamic equilibrium to be reestablished. After that, the container is insulated and then the piston is released. After the piston comes to rest, what is the new equilibrium position of the piston? Answer: 0.6H

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