Magnetic Susceptibility

Magnetic susceptibility Introduction A magnetic field causes unpaired electrons to tend to align with the field. The resulting magnetization M (magnetic dipole moment per unit volume) induced in a substance is proportional to applied field H: M = χ H.

(1.)

The constant of proportionality χ is the volume susceptibility, which is unit less. From the chemist’s point of view, this quantity gives insight into the number of unpaired electrons present, although other effects (orbital magnetic moment, diamagnetic ligands and ions) complicate the picture. Substances having unpaired electrons are called paramagnetic and have values of χ around +10−4. Diamagnetic substances, those lacking unpaired electrons, have χ around –10-6. mks and cgs units for quantities related to magnetic susceptibility are summarized in Table 2. We will use cgs units here because of the convenience of working with cm3 and g. Because heat affects the alignment of electrons, paramagnetic substances follow the Curie law approximately:1 N µ χ M = A eff . 3kT 2

(2.)

χM is the molar susceptibility, defined below (equation 5). It is the quantity that will be determined in this experiment. µeff is the effective magnetic moment. From equation (2) (ref. 1), µeff /β = 2.83 mol1/2 cm−3/2 K−1/2 χ M T .

(3.)

β is the Bohr magneton, a convenient unit of magnetic moment whose value is given in Table 2 below. In the first transition series, orbital contribution to magnetic moment “is almost completely quenched by the ligand fields.” 1 In this case, µeff is related to spin only and µeff /β ≈ 2 [S (S + 1)]1/2 = [n (n+2)]1/2 , and n(n+2) ≈ 8.01 mol cm−3 K−1 χM T,

(4.)

in which S is total spin and n is the number of unpaired electrons of the transition metal ion. Gram and molar susceptibilities The gram susceptibility χg (units cm3 g−1) and the molar susceptibility (units cm3 mol−1) are χg = χ/ρ,

and

χM = M × χ/ρ,

(5.)

in which χ is the volume susceptibility, M is the molar mass and ρ is the bulk density. Example: χM of CuSO4.5H2O = 1460×10-6 cm3 mol−1, so

χ g = 1460 × 10 −6 cm 3 mol −1 × mol = 5.847 × 10 −6 cm 3 g −1 . 249.7g Dr Amit Kumar, Centro Química Estrutural Instituto Superior Técnico

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In order to find χ, we would need the bulk density, which would depend on the particle size, packing etc. in the case of a solid. NMR method2 NMR can be used to measure magnetic susceptibility. The presence of a paramagnetic ion in solution causes a change in local magnetic field and therefore a shift in proton resonance lines for a reference molecule such as t-butanol. Since the paramagnetic ion strengthens the local magnetic field, the shift will be upfield (to the left). In a typical experiment2,3 the paramagnetic substance is dissolved in a 10% solution of t-butanol in D2O and the solution is placed in an NMR tube. The mass fraction of solute must be accurately known. In addition, some butanol/D2O solution is sealed in a capillary tube which is placed in the NMR tube as a reference. The signal will consist of the unshifted methyl proton resonance at ~1 ppm (from the capillary tube) plus a second line shifted by the paramagnetic ion. A pair of lines at ~5 ppm from H2O will probably also be present. The shift, in ppm, caused by the paramagnetic ion is ∆H 2π = ( χ solution − χ solvent ) . H 3

(6.)

∆H/H × 106 is the shift in ppm. The volume susceptibility of the solution is χ(solution) = [χg(solute)×p + χg(solvent)×(1 − p)] × ρ ,

(7.)

in which p is the mass fraction of solute in the solution and ρ is the density of the solution. Example For a 10% solution of t-butanol in D2O (see Table 2), χ(solution) = [−0.77×0.10 + (−0.64)×0.9] 10−6cm3g−1 × 1.07g cm−3 = −0.70×10−6. If CuSO4.5H2O is dissolved at 5% by mass in D2O, then −6 3 −6 3   1.10g χ(solution ) =  4.7 ×10 cm × 0.05 + − 0.64 × 10 cm × 0.95 × 3 g g   cm = −0.41×10 − 6 .

The expected shift in magnetic resonance would be ∆H = 2π (− 0.41 × 10 −6 − ( −0.70 × 10 −6 ) ) H 3 = 0.61 × 10 −6 = 0.61ppm.

Guoy balance method In the Gouy method, a sample is weighed with and without applied magnetic field (references 1,2,3). Paramagnetic materials are drawn into the field; diamagnetic materials are repelled. The resulting force changes the apparent mass of the sample. Force = [ mass with field − mass no field ] × g 2 = 12 Aχ( H 2 − H 0 ),

(8.) Dr Amit Kumar, Centro Química Estrutural Instituto Superior Técnico

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in which g is the gravitational acceleration, A is sample cross-sectional area, H and H0 are the magnetic fields inside and outside of the magnet and χ is the sample’s volume susceptibility.* Mass changes are typically quite small. For careful work, air displacement and sample tube paramagnetism must be allowed for. Experiment A good choice of compounds for the experiment would illustrate the effect of different ligand environments and oxidation states on the number of unpaired electrons in a transition metal compound. Examples: K3Fe(CN)6, K4Fe(CN)6 ⋅3H2O, (NH4)2 Fe(SO4)2 ⋅6H2O; KMnO4 , MnSO4⋅H2O; Zn+2, Cu+2, Ni+2 etc. Determine the value of χM for each compound and compare to literature values. Use χM to estimate the apparent number of unpaired electrons. Discuss result taking into account Hund’s rule,5 crystal field effects5,6 and the Boltzmann distribution. For the NMR method, it is crucial to calculate the appropriate concentration of each paramagnetic sample to produce a shift of about -1 ppm. If an arbitrary concentration is used, the shift may be too small or too large to see in the spectrum. Note: in the past, the NMR has confused the shifted CH3 peak with TMS and assigned it a shift of 0.000. Peak positions are incorrect, but shifts are not affected. References 1.

W.L.Jolly, The Synthesis and Characterization of Inorg anic Compounds, Prentice Hall, Englewood Cliffs (1970)

2.

D.P.Shoemaker et al., Experiments in Physical Chemistry, 5th Edition, McGraw Hill, New York (1989)

3.

C.F.Bell and K.A.K.Lott, Un Esquema Moderno de la Química Inorgánica, Alhambra, Madrid (1969)

4.

Handbook of Chemistry and Physics

5.

R. Petrucci and W. Harwood, General Chemistry, 6th Ed., Macmilla n, New York (1993), Chapter 25

6.

F. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, 2nd E d. Interscience, New York (1966), Chapter 26

7.

W.L.Jolly, Modern Inorganic Chemistry, McGraw-Hill, New York (1984)

8.

R. Wangsness, Electromagnetic Fields, J. Wiley, New York (1979)

9.

E. Purcell, Electricity and Magnetism,

In mks units, force = ½Aχµ0(H2−H02), with χ(mks) = 4πχ(cgs). The quantity A(H2−H02)/2g is sometimes called the instrument constant. *

Dr Amit Kumar, Centro Química Estrutural Instituto Superior Técnico

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Appendix: Units Susceptibilities are often given without units, but only the volume susceptibility is unitless. The mks volume susceptibility is obtained from cgs by multiplying by 4π. Table 1: Symbols and units related to magnetic susceptibility (refs 8,9) equation

cgs units

mks units

applied magnetic field magnetization

symbol H M

M=χH

Oersted = Gauss= erg G–1 cm–3

A m−1 = J T–1 m–3

volume susceptibility mass susceptibility molar susceptibility mks vacuum permeability magnetic dipole moment

χ χg χM µ0 µ

4πχ(cgs) = χ(mks) χg = χ/ρ χM = χg × M µ0 = 4π × 10–7 T m A–1 ∆E = µ B

unitless

unitless

Bohr magneton

β

magnetic field strength

B

cm g cm3 mol−1 —

m3 kg−1 m3 mol−1 T m A–1

β = eh / 2 m e

erg G−1 9.27 × 10−21 erg G−1

J T−1 = A m2 9.27 × 10−24 J T−1

B = (1+4πχ)H (cgs) B = µ0(1+χ)H (mks)

Gauss 1T = 104 G

Tesla T = kg A−1 s−2

3

−1

Table 2: Magnetic susceptibilities of selected compounds at 200C (cgs) compound (ref) comments χ, χg or χM (cgs) -6 oxygen is paramagnetic air (3,4) 0.029×10 -6 3 −1 diamagnetic water (1,4) −0.720×10 cm g -6 3 −1 " D2O (4) −0.638×10 cm g -6 3 −1 " t-butanol (4) −0.77×10 cm g -6 3 −1 recommended standard for solid samples HgCo(SCN)4 (1) 16.44×10 cm g -6 3 −1 recommended standard for solution samples NiCl2 (1) 34.2×10 cm g -6 3 −1 CuSO4.5H2O (7) 1460×10 cm mol ZnSO4.7H2O (7) −143×10-6 cm3 mol−1 water (4,7) −13.0×10-6 cm3 mol−1

Dr Amit Kumar, Centro Química Estrutural Instituto Superior Técnico

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