BioPHYSICAL chemistry

166 PARTI THERMODYNAMICS AND KINETICS

Energy

Figure 8.3
The number of
configurations at
each energy with
weighting by the
Boltzmann factor.

Thus, the relative population of the higher-energy state will exponentially

decrease according to the difference in energies. Consider the probability

of an electron occupying different electron orbitals (which will be dis-

cussed in detail in Chapter 12). The difference in energy for the two lowest

orbitals of the hydrogen atom, the 1s and 2s orbitals, is 1.64 × 10 −^18 J, so

the population ratio is:

e−(E^2 −E^1 )/kBT=e−(1.64×^10

− (^18) J)/[(1.38× 10 − (^23) J/K)(298 K)]
=e−^397 ≈ 0 (8.10)
Essentially all of the electrons will occupy the lowest-energy states of
the hydrogen atom model. When an unpaired electron is in the presence
of a magnetic field, the energy levels will split for the two spin states. In
this case the difference in energy is much smaller, at about 6.7 × 10 −^25 J,
yielding a probability of just less than one:
e−(E^2 −E^1 )/kBT=e−(6.7×^10
− (^25) J)/[(1.38× 10 − (^23) J/K)(298 K)]
=e−0.0016=0.9984 (8.11)
Note that the lower-energy state will be only slightly more populated.
For this reason, spectroscopic measurements of the electron spins using
magnetic fields (Chapter 16) often make use of low-temperature measure-
ments that have an increase in the population difference compared to
room temperature. For example, lowering the temperature from 298 K
to 4 K results in the population ratio becoming measurably smaller:
e−(E^2 −E^1 )/kBT=e−(6.7×^10
− (^25) J)/[(1.38× 10 − (^23) J/K)(4 K)]
=e−0.12=0.887 (8.12)

Partition function

One convention in problems with states possessing different energies is

that only the relative energies are considered, with the zero energy usu-

ally defined to be the ground state. The partition factor, q, is the sum of all

terms that describe the probability associated with the variable of interest.

For the Boltzmann factor, the partition function, q, is given by summation

of the exponential terms at the energies Ei:

(8.13)

The probability Piof occupying the energy le 9 el Eican be written in terms

of the partition function by inserting eqn 8.13 into eqn 8.8:

qeEkT

i

=∑−iB/

e

e

eee

a

b

==abab−−