Abstract:
    The change in weight induced by a magnetic field for three solutions of complexes was recorded. The change in weight of a calibrating solution of 29.97% (W/W) of NiCl2 was recorded to calculate the apparatus constant as 5.7538. cv and cm for each solution was determined in order to calculate the number of unpaired electrons for each paramagnetic complex. Fe(NH4)2(SO4)2€6(H20) had 4 unpaired electrons, KMnO4 had zero unpaired electrons, and K3[Fe(CN)6] had 1 unpaired electron. The apparent 1 unpaired electron in K3[Fe(CN)6] when there should be five according to atomic orbital calculations arises from a strong ligand field produced by CN-.

Introduction:
    The magnetic susceptibility is a phenomena that arises when a magnetic moment is induced in an object. This magnetic moment is induced by the presence of an external magnetic field. This induced magnetic moment translates to a change in the weight of the object when placed in the presence of an external magnetic field. This induced moment may have two orientations: parallel to the external magnetic field of or perpendicular to the external magnetic field. The former is known as paramagnetism and the later is known as diamagnetism. The physical effect of paramagnetism is an attraction to the source of magnetism (increase in weight when measured by a Guoy balance) and the physical effect of diamagnetism is a repulsion from the source of magnetic field (decrease in weight when measured by a Guoy balance).
    The observed magnetic moment is derived by the change in weight. This observed magnetic moment arises from a combination of the orbital and spin moments of the electrons in the sample with the spin component being the most important source of the magnetic moment. This magnetic moment is caused by the spinning of an electron around an axis acting like a tiny magnet. This spinning of the �magnet� results in the magnetic moment.
    Paramagnetism results from the permanent magnetic moment of the atom. These permanent magnetic moments arise from the presence of unpaired electrons. These unpaired electrons result in unequal number of electrons in the two possible spin states (+1/2. -1/2). When in the absence of an external magnetic field, these spins tend to orient themselves randomly accordingly to statistics. When they are placed in the presence of an external magnetic field, the moments tend to align in directions anti parallel and parallel to the magnetic field. According to statistics, more electrons will occupy the lower energy state then the higher energy state. In the presence of a magnetic field, the lower energy state is the state when the magnetic moments are aligned parallel to the external field. This imbalance in the orientation favoring the parallel orientation results in attraction to the source of the external magnetic field.
    Diamagnetism is a property of substances that contain no unpaired electrons and lack a permanent dipole moment. The magnetic moment induced by one electron is canceled by the magnetic moment of an electron having the opposite spin state. The force of diamagnetism results from the effect of the external magnetic field on the orbital motion of the paired electrons. The susceptibility is correlated to the radii of the electronic orbits and the precession of the electronic orbits. The complex mathematical system describing this system is beyond the scope of the experiment. It must be included that paramagnetic substances do have a diamagnetic component to them but it is much smaller than the paramagnetic component and therefore can be ignored.
Calculation. cm (the mass susceptibility)is found for a calibrating solution of NiCl2 using the equation
                    (1)
where p is the mass fraction (w/w) of NiCl2 of the solution and T is the absolute temperature.
cv (the volume susceptibility)is determined using equation
                                            (2)
where r is the density of the solution.
The apparatus constant moH2A/2 is evaluated using equation
                                        (3)
With the apparatus constant known and W (mass(kg) x 9.8 m s-2) known, it is possible to determine cv for each solution using the equation
                                    (4)
cM (molar susceptibility) is calculated (in SI units) using the equation
                        (5)
With cM determined, the Curie Constant C is calculated by the equation:
                                        (6)
The small diamagnetic term can be neglected for paramagnetic compounds and the equation becomes:
                                            (7)
The atomic moment � can then be calculated using the equation:
                                (8)
The number of unpaired electrons can be found approximately by the equation:
                                (9)
where n is the number of unpaired electrons.

Experimental Method:
    The method described in Experiments in Physical Chemistry was followed. The density of all solutions were measured using a pycnometer.
    A solution of NiCl2 was made with the following parameters (table one):
Table One: Parameters of NiCl2
Solution    Concentration (M)    Weight Fraction     Density (kg/m3)
NiCl2     2.308 .016    0.2997    1.3552 .003x103
Three test solutions were prepared as follows (table two):
Table Two: Parameters of Solutions
Solution    Concentration (M)    Density (kg/m3)
Fe(NH4)2(SO4)2€
6(H20)    0.705 .016    1.1148 .003x103
KMnO4    0.377 .016    1.0201 .003x103
K3[Fe(CN)6]    0.498 .016    1.0834 .003x103

    Measurements in the presence and absence of magnetic fields were made using a Guoy balance as described in Experiments in Physical Chemistry and were made in triplicate.

Results:
All measurements were performed at 293K.
Table Three: Mass (field on -field off)
Solution            Mass (g)    
    Run One    Run Two    Run 3     Average
NiCl2     0.09349 0.0001    0.09381 0.0001     0.10427 0.0001    0.09719 0.0001
Fe(NH4)2(SO4)2€
6(H20)    0.03548 0.0001    0.03665 0.0001     0.04785 0.0001    0.03999 0.0001
KMnO4    -0.00406 0.0001    -.00404 0.0001     -0.00399 0.0001    -0.00403 0.0001
K3[Fe(CN)6]    0.00252 0.0001    0.00258 0.0001     0.00386 0.0001    0.00299 0.0001

Table Four: Weight for Solutions
Solution     Weight (N)
NiCl2     9.5246 .0098x10-4
Fe(NH4)2(SO4)2€6(H20)    3.9190 .0098x10-4
KMnO4    -3.948 .098x10-5
K3[Fe(CN)6]    2.930 .098x10-5


The following parameters of NiCl2 were determined (table five) using equations 1 and 2:
Table Five: Parameters of NiCl2
cm    1.22 .04x10-7 m3kg-1.
cv     1.66 .06x10-4

The apparatus constant moH2A/2 was evaluated using equation 3 as 5.73 .02.

cv was calculated for each solution (table six) using equation 4.
Table Six
Solution     Weight (N)    cv
Fe(NH4)2(SO4)2€6(H20)    3.9190 .0098x10-4    6.831 .003x10-5
KMnO4    -3.948 .098x10-5    -6.88 .02x10-6
K3[Fe(CN)6]    2.930 .098x10-5    5.10 .02x10-6

cM is calculated (in SI units) using equation 5 (table seven):
Table Seven
Solution    cv     cM
Fe(NH4)2(SO4)2€6(H20)    6.831 .003x10-5    1.087 .007x10-7
KMnO4    -6.88 .02x10-6    4.79 .03x10-9
K3[Fe(CN)6]    5.10 .02x10-6    2.69 .02x10-8

With cM determined, the Curie Constant C is calculated by equation 7 (table eight):
Table Eight
Solution    C
Fe(NH4)2(SO4)2€6(H20)    3.18 .02x10-5
KMnO4    1.406 .008x10-6
K3[Fe(CN)6]    7.90 .06x10-6

The atomic moment was then be calculated using equation 8 (table nine):
Table Nine
Solution    � (Bohr Magneton)
Fe(NH4)2(SO4)2€6(H20)    4.5044
KMnO4    0.9462
K3[Fe(CN)6]    2.2428

The number of unpaired electrons was found approximately by the equation 9 (table ten):
Table Ten
Solution    n     # unpaired electrons
Fe(NH4)2(SO4)2€6(H20)    3.6141    4
KMnO4    0.3767    0
K3[Fe(CN)6]    1.4557    1

Discussion:
    The number of unpaired electrons determined experimentally is correct as compared to atomic orbital calculations except for K3[Fe(CN)6](table eleven):
Solution    Experimental Determined    A.O. Calculations
Fe(NH4)2(SO4)2€6(H20)    4    4
KMnO4    0    0
K3[Fe(CN)6]    1    5

    The cause of the discrepancy of the K3[Fe(CN)6] complex is not experimental error but is from the physical properties of transition metal complexes such as K3[Fe(CN)6]. These properties are characterized by ligand field theory.
    The compound K3[Fe(CN)6] is characterized as a low spin case. A low spin case causes the measured numbers of unpaired electrons to be considerably less than that calculated theoretically. This is caused by splitting of the five degenerate d- level electronic orbitals into two or more levels of different energies by the fields put out by the ligand.
    In the case of K3[Fe(CN)6], CN- exerts a strong ligand field. This strong splitting field results in a greater energy difference between the bonding and antibonding orbitals. (see picture one) making it more probable that all 5 e- will occupy the lower energy bonding orbital.
   
Picture One: A diagram of the weak field and strong field effect on electron arrangement in Fe+3

The strong ligand field produced by CN- results in spin moment cancellation of four out of the five unpaired electrons. This results to the apparent 1 free electron determine by the experiment.
    The sources of error in this experiment are the solutions, density and mass. All confidence limits were determined using the method of partial fractions just how all lab reports are done. Although we initially had trouble with the scale, these problems were resolved prior to taking measurements.
    Although more accurate results are not needed, a possible way to increase accuracy is to use more volume of solutions. When we performed this experiment, we had to cut the volume used in other years by 50% because the weight exceeded the capacity of the balance. Using more solution would decrease the significance of the error in mass.
   
References:
1. Shoemaker, Garland, and Nibler, Experiments in Physical Chemistry, Fifth Edition, McGraw-Hill Company, New York, 1989.

2. Mulay, L.N., Magnetic Susceptibility,. Intersceince Publishers, New York, 1963

3. Adamson, Arthur W., A Textbook of Physical Chemistry,. Tjird Edition, Academic Press College Division, Orlando, Flrida, 1986.

4. Barrow, Gordon M., Physical Chemistry,. Third Edition, McGraw-Hill Company, New York, 1973.










Experiment 33
Magnetic Susceptibility

Michael J. Horan II




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