Measurement of Magnetic Susceptibility of Manganese (II) Sulphate solution by Quincke's Method

Indian Institute of Science Education and Research, Kolkata

27th Jan – 3rd Feb 2010

Measurement of Susceptibility of Manganese (II) Sulphate solution by Quincke’s Method Harsh Purwar (07MS – 76) Amit Nag (07MS – 19) Piyush Pushkar (07MS – 33) Sibhasish Banerjee (07MS – 55) th VI Semester (3rd Year), Integrated M.S. Indian Institute of Science Education and Research, Kolkata Abstract: Magnetization in material arises from microscopic phenomena which can best explained by quantum mechanical means. But its manifestation at macroscopic level makes it directly accessible in the realm of theoretical classical electro-magnetism. Due to this, easily measurable field quantities give us a method to predict microscopic properties of material which otherwise remains elusive to elementary direct measurement. In this experiment we have measured the magnetic susceptibility of MnSO4 solution at various concentrations by Quincke's method which is based on the force experienced by a magnetized material in a nonuniform magnetic field.

Introduction & Theory: The magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. In free space (vacuum) there is no magnetization, 𝑀, the magnetization 𝑀 is defined as the magnetic moment per unit volume. The magnetic field can be described by the vector fields 𝐵 and 𝐻 which are linearly related by 𝐵 = 𝜇0 𝐻 The two magnetic fields 𝐵 and 𝐻 are just scaled versions of each other, the former measured in Tesla (abbreviated to 𝑇) and the latter measured in 𝐴𝑚−1 . In a magnetic solid the relation between 𝐵 and 𝐻 is more complicated and the two vector fields may be very different in magnitude and direction. The general vector relationship is 𝐵 = 𝜇0 𝐻 + 𝑀 . In the special case that the magnetization 𝑀 is linearly related to the magnetic field 𝐻, the solid is called a linear material and we write 𝑀 = 𝜒𝐻 where 𝜒 is a dimensionless quantity called the magnetic susceptibility. In this special case there is still a linear relationship between 𝐵 and 𝐻, namely where 𝜇𝑟 = 1 + 𝜒 is the relative permeability of the material. 1|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata 27th Jan – 3rd Feb 2010 If 𝜒 is positive, then 1 + 𝜒 > 1, and the material can be paramagnetic, ferromagnetic, ferri-magnetic, or anti-ferromagnetic. In this case, the magnetic field is strengthened by the presence of the material. Alternatively, if 𝜒 is negative, then 1 + 𝜒 < 1, and the material is diamagnetic. As a result, the magnetic field is weakened in the presence of the material. By Quincke’s Method magnetic susceptibility is measured by the force change felt by a magnetized material in the form of a liquid or an aqueous solution upon the application of a magnetic field gradient. The force on unit volume of the sample is negative of the gradient of the change in energy density, 1 𝐵 . 𝐻 . Thus force, 𝑓, acting on an element of area 𝐴 and length 𝑑𝑥 of the column of liquid is 2 𝑓 = 𝐴. 𝑑𝑥.

𝑑 1 𝐴 𝜇0 𝜇𝑟 − 𝜇𝑟𝑎 𝐻 2 = 𝜇0 χ − χ𝑎 𝑑 𝐻 2 𝑑𝑥 2 2

Where 𝜇𝑟 , 𝜒𝑟 and 𝜇𝑟𝑎 , 𝜒𝑎 are respectively relative permeability and susceptibility of the sample and the air which the sample displaces. So, the total force, 𝐹, on the liquid is 𝐹=

𝐴 𝜇 χ − χ𝑎 (𝐻 2 − 𝐻02 ) 2 0 𝑟

where 𝐻 is equal to the field at the liquid surface between the poles of the magnet and 𝐻0 is the field at the other surface away from the magnet. The liquid (density 𝜌) moves under the action of this force until it is balanced by the pressure exerted over the area 𝐴 due to a height difference 2𝑕 between the liquid surfaces in the two arms of the U-tube. It follows, assuming density of air to be very small which the case is usually: 𝐹 = 2𝐴𝑕𝜌𝑔 χ = χ𝑎 +

2 2𝑕 𝑔𝜌 2 𝜇0 (𝐻 − 𝐻0 2 )

In actual practice 𝜒𝑎 is negligible and can be ignored and the above expression simplifies to χ=

4ρgh μ0 (𝐻 2 − 𝐻0 2 )

Now 𝐻 can be approximated as 𝐻 = 𝐵 𝜇0 , since 𝜒 is very small and 1 + 𝜒 ≈ 1, thus χ=

4μ0 ρgh

(𝐵 2 − 𝐵0 2 )

In the case of a solution this determined susceptibility, 𝜒 , is total susceptibility, summation of susceptibilities of both the solute and solvent, this assumes that the number of solvent molecules per unit volume is not very different in the solution from the that in pure. Also note that 𝜒 here is the volume susceptibility of the solution which is related to the mass susceptibility by the following relation, 𝜒𝑣𝑠𝑜𝑙 =

𝜒𝑚 𝑠𝑜𝑙 𝜌

Here 𝜌 is the density of the solution. As mentioned earlier 𝜒𝑣 is a dimensionless quantity whereas 𝜒𝑚 is measured in 𝑚3 /𝐾𝑔 in S.I. units. 2|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata

Procedure:

27th Jan – 3rd Feb 2010

1. Solutions of Manganese sulphate with different concentrations were prepared and their densities were calculated separately as mentioned below. 2. Calibrate the magnetic field was calibrated against magnet current using the digital Hall probe. The probe was positioned so that it gives positive values of B and the stand was used so that probe remains in the same position throughout your calibration. 3. Before the actual experiment, magnetic field against magnet current was calibrated by using Hall Probe and it was fitted linear. 4. The tube was thoroughly cleaned and then dried using a drier. Now some of the solution was transferred to the U-tube. 5. The U tube was placed in between the pole pieces of the magnetic field such that the length of the horizontal connecting limb should be sufficient to keep the wide limb out of the magnetic field. 6. The liquid in the tube is filled and the meniscus is set centrally within the pole pieces as shown. Microscope is focussed on the meniscus and reading is taken. 7. And with different applied magnetic field, difference in the two levels of the solution in U-tube was measured by a travelling microscope. 8. By using above mentioned expression for the magnetic susceptibility of the solution was calculated and it was corrected for Manganese by subtracting the susceptibility of water = −0.90 × 10−5 𝑚3/𝐾𝑔.

Preparation of Manganese sulphate monohydrate solutions: Manganese (II) Sulphate (MnSO4.H2O) Solubility: 37.3 g in 100mL of water at 24 deg C. Solution 1: 15.72 grams of Manganese (II) sulphate monohydrate was dissolved in 50 mL of water. Density of Solution 1: Weight of solution: 63.7 grams Volume of solution: 53 mL Density of the solution: 1.20 gm/mL Solution 2: 11.79 grams of Manganese (II) sulphate monohydrate was dissolved in 50 mL of water. Density of Solution 2: Weight of solution: 59.59 grams Volume of solution: 52 mL Density of the solution: 1.14 gm/mL Solution 3: 13.055 grams of Manganese (II) sulphate monohydrate was dissolved in 50 mL of water. Density of Solution 3: Weight of solution: 61.01 grams Volume of solution: 52 mL Density of the solution: 1.17 gm/mL

3|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata

27th Jan – 3rd Feb 2010

Observations: Table 1: For calibration of the magnetic field with the current flowing through the electromagnets. S. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Current through electromagnetic coil (A) 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00

Magnetic Field (Gauss) 169 590 1046 1505 2020 2520 3040 3540 4010 4500 5000 5540 5850 6350 6630 6930 7210 7440 7680 7880 8050

Following is the graph for calibration of the magnetic field at the centre of the two coils with current through the electromagnets.

4|Magnetic Susceptibility, Experiment No. - 2

27th Jan – 3rd Feb 2010

Indian Institute of Science Education and Research, Kolkata Fitting the above graph linearly gives, 𝐵 = 2348 × I + 181.1

Table 2: For Magnetic Field versus rise in the solution's meniscus (Relative Height) for Solution # 1. Current through the electromagnet (A) For Solution # 1 1 0.00 2 0.50 3 1.00 4 1.50 5 2.00 6 2.50 7 3.00 8 3.50 9 4.00 For Solution # 2 1 0.00 2 0.50 3 1.00 4 1.50 5 2.00 6 2.50 7 3.00 8 3.50 9 4.00 For Solution # 3 1 0.00 2 1.01 3 1.50 4 2.00 5 2.50 6 3.00 7 3.50 8 4.00 Obs. No.

Corresponding Magnetic Field (Gauss)

Main Scale Reading (cm)

Vernier Scale Division

Final Reading (cm)

Relative Height {h} (cm)

181.1 1355.1 2529.1 3703.1 4877.1 6051.1 7225.1 8399.1 9573.1

2.95 2.95 3.00 3.05 3.15 3.25 3.35 3.45 3.50

6 21 24 32 28 39 25 0 7

2.956 2.971 3.024 3.082 3.178 3.289 3.375 3.450 3.507

0 0.015 0.068 0.126 0.222 0.333 0.419 0.494 0.551

181.1 1355.1 2529.1 3703.1 4877.1 6051.1 7225.1 8399.1 9573.1

3.30 3.30 3.35 3.40 3.45 3.50 3.60 3.65 3.65

31 47 24 16 29 46 5 3 47

3.331 3.347 3.374 3.416 3.479 3.546 3.605 3.653 3.697

0 0.016 0.043 0.085 0.148 0.215 0.274 0.322 0.366

181.1 2552.58 3703.1 4877.1 6051.1 7225.1 8399.1 9573.1

3.25 3.35 3.35 3.35 3.40 3.45 3.50 3.55

37 2 19 42 24 37 34 21

3.287 3.352 3.369 3.392 3.424 3.487 3.534 3.571

0 0.065 0.082 0.105 0.137 0.200 0.247 0.284

*Relative height field above in the table is the rise in height of the solution’s meniscus taking its height at zero current i.e. at 181.1 gauss of magnetic field to be zero. This relative height is plotted versus square of the applied magnetic field below. The data points were then fitted linearly. Following are the graphs of relative height versus square of the magnetic field for the three solutions of Manganese sulphate of concentrations and densities as mentioned above.

5|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata

For Solution # 1

For Solution # 2

6|Magnetic Susceptibility, Experiment No. - 2

27th Jan – 3rd Feb 2010

27th Jan – 3rd Feb 2010

Indian Institute of Science Education and Research, Kolkata

For Solution # 3 Fitting the above three graphs linearly gives their slopes as, For Solution 1: 𝑆𝑙𝑜𝑝𝑒 = 6.319 × 10−9 ,

𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 3.919 × 10−2

𝑆𝑙𝑜𝑝𝑒 = 4.139 × 10−9 ,

𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 2.676 × 10−2

𝑆𝑙𝑜𝑝𝑒 = 2.943 × 10−9 ,

𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 3.146 × 10−2

For Solution 2:

For Solution 3:

Calculations: 𝜒𝑣𝑠𝑜𝑙 = 4𝜇0 𝜌𝑔 × 𝑆𝑙𝑜𝑝𝑒 ⇒ 𝜒𝑚 𝑠𝑜𝑙 = 4𝜇0 𝑔 × 𝑆𝑙𝑜𝑝𝑒 Solution No. 1 2 3

Density 𝝆 𝒈 𝒄𝒄 1.20 1.14 1.17

𝝌𝒗𝒔𝒐𝒍 3.735 × 10−5 2.324 × 10−5 1.696 × 10−5

𝝌𝒎𝒔𝒐𝒍 𝒄𝒎𝟑 𝒈 3.113 × 10−5 2.039 × 10−5 1.450 × 10−5

7|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata 27th Jan – 3rd Feb 2010 Mass magnetic susceptibility due to Manganese (II) sulphate monohydrate is given by, 𝜒𝑚 𝑠𝑜𝑙 =

𝑚𝑠 𝑚𝑤 𝜒𝑚 𝑀𝑛𝑆 𝑂 4 .𝐻 2 𝑂 − 𝜒 𝑚𝑠 + 𝑚𝑤 𝑚𝑠 + 𝑚𝑤 𝑚 𝐻 2 𝑂

𝜒𝑚 𝑀𝑛𝑆 𝑂 4 .𝐻 2 𝑂 = 𝜒𝑚 𝑠𝑜𝑙 −

𝑚𝑤 𝑚𝑠 + 𝑚𝑤 𝜒𝑚 𝐻 2 𝑂 𝑚𝑠 + 𝑚𝑤 𝑚𝑠

Mass of Solute Mass of Solvent Solution No. 𝒎𝒔 (𝒈) 𝒎𝒘 (𝒈) 1 2 3

15.72 11.79 13.06

50.0 50.0 50.0

𝝌𝒎𝑴𝒏𝑺𝑶

𝝌𝒎𝒔𝒐𝒍 𝒄𝒎𝟑 𝒈

𝟒 .𝑯𝟐 𝑶

𝟑

3.113 × 10−5 2.039 × 10−5 1.450 × 10−5

𝒄𝒎 𝒈 1.324 × 10−5 1.099 × 10−5 7.278 × 10−5

Now we have mass solubility of manganese (II) sulphate monohydrate 𝑀𝑛𝑆𝑂4 . 𝐻2 𝑂 three different solutions as listed in the above table. ′′ Now 𝜒𝑚 is given by, 𝑀𝑛𝑆 𝑂 4 .𝐻 2 𝑂 ′′ 𝜒𝑚 = 𝜒𝑚 𝑀𝑛𝑆 𝑂 4 .𝐻 2 𝑂 × 𝑀. 𝑤𝑡. 𝑜𝑓 𝑀𝑛𝑆𝑂4 . 𝐻2 𝑂 = 169 𝑔 𝑀𝑛𝑆 𝑂 4 .𝐻 2 𝑂

𝝌′′𝒎𝑴𝒏𝑺𝑶

𝟒 .𝑯𝟐 𝑶

𝟑

𝒄𝒎

0.0224 0.0186 0.0123

𝜒𝑚 𝑀𝑛𝑆 𝑂 4 .𝐻2 𝑂 for

This value is also calculated in the above table.

Conclusions: 

The volume susceptibility and mass susceptibility for the three solutions of manganese sulphate monohydrate 𝑀𝑛𝑆𝑂4 . 𝐻2 𝑂 in water are as summarized in the following table.

Solution No.

Molarities 𝑴

Volume Susceptibility of solution

1 2 3

1.76 1.34 1.49

3.735 × 10−5 2.324 × 10−5 1.696 × 10−5



Mass Susceptibility of solution 𝒄𝒄 𝒈 3.113 × 10−5 2.039 × 10−5 1.450 × 10−5

Mass Susceptibility for 𝑴𝒏𝑺𝑶𝟒 . 𝑯𝟐 𝑶 𝒄𝒄 𝒈

𝝌′′𝒎 for 𝑴𝒏𝑺𝑶𝟒 . 𝑯𝟐 𝑶 𝒄𝒄

1.324 × 10−5 1.099 × 10−5 7.278 × 10−5

0.0224 0.0186 0.0123

′′ As quoted in the manual the standard value of mass susceptibility 𝜒𝑚 for manganese (II) sulphate monohydrate is 0.0142 𝑐𝑐. Hence the above found results are valid under the experimental regime.

Discussion & Sources of Error: In above experiment the magnetic field was calibrated using the hall probe which involved constantly varying current in one direction across the electromagnet; hence invariably we have incorporated error due to hysteresis. Moreover the hysteresis effect was present even when the actual experiment was conducted since again no correction was made for hysteresis. The Hall probe was only corrected for zero error via zero adjustment, hence any other complicated systemic error that might have been associated with the hall probe and hence with measurement of magnetic field has been neglected. The result

8|Magnetic Susceptibility, Experiment No. - 2

Indian Institute of Science Education and Research, Kolkata 27th Jan – 3rd Feb 2010 obtained on hinges on theoretical derivation of the above formula which assumes dilute concentration of the solution but under such concentration effect of non uniform magnetic field is not very observable.

9|Magnetic Susceptibility, Experiment No. - 2