Lui Chi Kong
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
28th May, 2003
The magnetic susceptibility of an iron(III) chloride solution of known concentration can be deduced by placing a U-shaped capillary tube containing the solution into magnetic field and measuring the change of height of the liquid level. The order of the magnetic susceptibility of the iron(III) chloride solution is found to be 10-5 while the magnetic moment of iron(III) ion is found to be of the order of 10-23. It was found that the magnetic susceptibility is dependent on the concentration of the solution, but independent from the diameter of the capillary tube.
The electronic configuration of an iron(III) ion is 1s22s22p63s23p63d5. All the free electrons in the 3d subshell have parallel spin, resulting a net magnetic moment of Fe3+ ion. Hence solutions containing ferric ions such as iron(III) chloride have positive magnetic susceptibility and thus paramagnetic.
The objectives of the experiment are to measure the magnetic susceptibility of an iron(III) chloride solution of known concentration, and hence to deduce the magnetic moment of Fe3+ ion.
The magnetic susceptibility χ can be deduced by measuring the force on a column of the solution contained in a capillary tube which extends from the center of the gap of an electromagnet to a place where the magnetic induction is zero. Consider a small volume dV of the solution in a capillary tube at the position with magnetic field H(x) as shown in Figure 1. The small volume of the solution experiences a force Fx, which arises from the inhomogenity of the magnetic field along the column
Integrating equation (1) over the column from the bottom of the solution where H1 = 0 to the the centre of the field with H2, the total force acting the column is given by
where
A is the cross-sectional area of the column. For a paramagnetic solution in
which χ is positive, Fx is positive and the column experiences an
upward thrust.
The
force Fx can be found by a hydrostatic method, using a uniform-bore
U-tube with one arm placed in the magnetic field and the other one well outside.
The meniscus of the solution in the arm inside the magnetic field will be raised
due to Fx, while the level outside arm will fall. Let ρ be the
density of the solution and Δh be the difference between the two liquid levels,
then
Fx = Δh · A · ρ · g
(3)
Equating equations (2) and (3), and neglecting χair, magnetic susceptibility is given by
where
B is the magnetic induction at the centre of the gap between the poles.
The
magnetic moment μ of each Fe3+ ion can be found from the Curie’s
Law
where
μo is the permeability of free space, N is the number of ions per
unit volume of the solution, and k is the Boltzmann constant.
The effective number of Bohr magneton per ion P is given by
where
μB is the Bohr magneton with value 9.27 x 10-24 Am2.
For Fe3+, electron spin is
and the theoretical value of P is √(35)
Experimental
Apparatus and Procedures
An electromagnet was setup and then connected in series. A multimeter was connected to measure the current of the circuit. A teslameter and hall probe were used to measure the magnetic field strength produced by the electromagnet.
Iron (III) chloride solution of known concentration and density was introduced into the U-tube using syringe. The U-tube was then mounted on the clamp and stand so that one arm lies between the poles with the meniscus in the middle of the gap. The other arm and the bottom of the U-tube were well outside the magnetic field.
The change in level of the position of the meniscus was measured using the traveling microscope. The magnetic field in the middle of the gap was measured by teslameter and hall probe. The measurement was repeated several times by varying the magnetic field, the concentration of the solution and the diameters d of the capillary tube.
Data
Analysis and Results
For each set of data obtained, a graph of change of height against square of magnetic field strength was plotted, and a linear relationship was found. The value of magnetic susceptibility χ of FeCl3 solution was then determined from the slope using equation (4). The values of magnetic moment μ and effective number of Bohr magneton per ion P of Fe3+ ions were calculated by equations (5) and (6) respectively.
When ρ is 1.09 gcm-3 and d is 0.161cm, the graph is shown in Figure 2(a), with slope (1.2 + 0.2) x 10-3 mT-2. The value of χ is (6.9 + 0.9) x 10-5. The values of μ and P are (4.1 + 0.3) x 10-23 and 4.39 + 0.07 respectively.
When ρ is 1.35 gcm-3 and d is 0.161cm, the graph is shown in Figure 2(b), with slope (3.1 + 0.2) x 10-3 mT-2. The value of χ is (2.0 + 0.1) x 10-4. The values of μ and P are (3.37 + 0.08) x 10-23 and 3.63 + 0.09 respectively.
When ρ is 1.09 gcm-3 and d is 0.091cm, the graph is shown in Figure 2(c), with slope (1.14 + 0.09) x 10-3 mT-2. The value of χ is (6.2 + 0.5) x 10-5. The values of μ and P are (3.9 + 0.2) x 10-23 and 4.2 + 0.2 respectively.
When ρ is 1.09 gcm-3 and d is 0.088cm, the graph is shown in Figure 2(d), with slope (1.7 + 0.2) x 10-3 mT-2. The value of χ is (9.1 + 0.1) x 10-5. The values of μ and P are (4.67 + 0.03) x 10-23 and 5.04 + 0.03 respectively.
When ρ is 1.16 gcm-3 and d is 0.088cm, the graph is shown in Figure 2(e), with slope (1.7 + 0.2) x 10-3 mT-2. The value of χ is (1.0 + 0.1) x 10-4. The values of μ and P are (3.9 + 0.2) x 10-23 and 4.2 + 0.2 respectively.
From the results, it can be see that the denser (and hence the higher concentration) of the solution, the larger the value of χ. For the same density of the solution but different diameters of the capillary tubes, the difference of the values of χ is small. Hence the value of χ is independent of the diameter of the capillary tube. The values of μ and P are similar for each set of data.
Discussion
In deriving the expression for χ, an assumption made is that the bottom of the solution lies far enough away from the magnetic field such that H = 0. However, this assumption may not be valid because magnetic field induced by connecting wires accounts for uncertainty to the accuracy, but calibration cannot eliminate the effect. Therefore, zero magnetic field at the bottom is not a good assumption.
The magnetic field of the Earth, however, does not introduce significant error on the accuracy. This is because the magnitude of the field is about 50 μT, which is 103 times compared with the magnetic field used in the experiment. [1] Hence the magnetic field of the Earth is small enough to be neglected.
Sometimes, when measuring the change in height of the liquid levels, abnormal rises were observed. For small magnetic field, the change in height is not apparent. However, there is a sudden rise for further increase in the magnetic field. The reason accounting for such sudden rise is that the force arising from increasing magnetic field, which acts on the meniscus of the solution, is smaller than the static friction force, until when the force overcome the limiting frictional force. [2]
The value of the magnetic susceptibility of the FeCl3 solution can be considered as arising from the Fe3+ ions alone. The electronic configuration of chloride ion is 1s22s22p63s23p6, and all the electrons in the 3p subshell of chloride ion are paired. There is almost no magnetic moment of chloride and the susceptibility of chloride ion can then be neglected. On the other hand, the proportion of water molecules that ionize into H3O+(aq) and OH-(aq) is very small and insignificant. [3] Hence the magnetic susceptibility of water molecules can also be neglected.
Some sources of uncertainty in the experiment include the unknown frictional force between the inner wall of the capillary tube and the solution, the magnetic field induced by the connecting wires, the non-uniformity of the concentration of the solution over the tube, the presence of unknown substances and impurities, and the judgement of the change of height using traveling microscope.
Conclusion
To summarize, the magnetic susceptibility χ of FeCl3 solutions is dependent to the density ρ of the solutions, but is independent from the diameter of the diameter of the capillary tube. Higher concentration of the FeCl3 solution gives larger value of χ. The magnetic moment of Fe3+ is of the order of 10-23. The value of effective number of Bohr magneton per ion is close to the theoretical one.
References
[1] Duncan T., Advanced Physics for Hong Kong, Volume 1, 1999, London: John Murray, pp.178
[2] Duncan T., Advanced Physics for Hong Kong, Volume 1, 1999, London: John Murray, pp.22
[3] Wong Y.C. and Wong C.T., New Way Chemistry for Hong Kong A-Level, Volume 2, Hong Kong: Manhattan, 1998, pp.130
Figure
Captions
Figure 1: Setup of the experiment. A capillary tube containing FeCl3 solution is placed in the magnetic field produced by an electromagnet.
Figure 2: Graphs of change of height against square of magnetic field strength for different sets of data with (a) ρ is 1.09 gcm-3 and d is 0.161cm; (b) ρ is 1.35 gcm-3 and d is 0.161cm; (c) ρ is 1.09 gcm-3 and d is 0.091cm; (d) ρ is 1.09 gcm-3 and d is 0.088cm; (e) ρ is 1.16 gcm-3 and d is 0.088cm.