Physical Chemistry Third Edition

24.2 Electronic and Nuclear Magnetic Dipoles 1009

Solution

∆EmaggpβNBz(5.58569)(5. 050787 × 10 −^27 JT−^1 )(0.500 T)

 1. 41 × 10 −^26 J

∆Eproton

∆Eelectron



1. 41 × 10 −^26 J

9. 28 × 10 −^24 J

 1. 52 × 10 −^3 

1

658

Exercise 24.5

Find the ratio of the populations of the two energy proton levels in the previous example at

298.15 K.

Many other nuclei besides the proton have nonzero spin angular momenta. Chemists

ordinarily encounter nuclei only in their ground states, so we take each nucleus to have

a fixed magnitude of its spin angular momentum:

|I|h ̄

√

I(I+1) (24.2-13)

whereIis a fixed quantum number for a given nucleus:I1 for^2 H,I 1 /2 for^13 C,

I0 for^12 C and^16 O, and so on. Each nucleus has a characteristic magnetic dipole

moment:

|μ||gN|βN

√

I(I+1)hγ ̄

√

I(I+1) (24.2-14)

wheregNis a characteristic factor for the given nucleus, called thenucleargfactor.

The quantityγis called themagnetogyric ratio. It has a different value for each nucleus.

The value of the magnetogyric ratio for the proton is

γp

e

2 mp

gp

gpβN

h ̄

 2. 67522 × 108 s−^1 T−^1  2. 67522 × 108 Hz T−^1 (24.2-15)

where we use the hertz (Hz, same as s−^1 ) as the unit of frequency. For another nucleus,

the magnetogyric ratio must be calculated from thegNvalue:

γN

e

2 mp

gN(4. 78941 × 107 JT−^1 )gN (24.2-16)

The nuclear magnetonβNof Eq. (24.2-12) contains the mass and charge of the proton

but is used for all nuclei. The necessary correction for different masses and charges is

incorporated into the nucleargfactorgNof the specific nucleus. Table A.24 of Appendix

A lists the nucleargfactors and spin quantum numbers of several common nuclides.

Surprisingly, some nuclides have negative values ofgN. In these cases the magnetic

dipole of the nucleus has the direction that would be expected for a negative particle.

It is as though such a nucleus contained both positive and negative charges, with the

negative charges concentrated near the exterior of the nucleus.