Chapter 6: Boltzmann statistics

Question 1

Boltzmann factor

Answer 1

$e^{-E(s)/kT}$

Question 2

P(s)

Answer 2

$P(s) = \frac{1}{Z}e^{-E(s)/kT}$

The prob that system is in state S (one microstate at equilat given temp T)

Question 3

P(s) ranges with T

Answer 3

^^T => $e^{E(s)/kT} = e^{-E(s)/big} \approx e^0 \approx 1$ linked to more probable state

low T => $e^{E(s)/kT} = e^{-E(s)/small} \approx e^-\infty \approx 0$ linked to LESS probable state

Question 4

P(s) range with E

Answer 4

Plot P(s) vs E(s) and see tht high E is less probable 
Ground state most likely with \$\$P(s) = \frac{1}{Z}\$\$

Question 5

Partition function

Answer 5

sum over all the boltzmann factors

$Z = \sum_s e^{E(s)/kT}$

Question 6

Comments on partition func dep

Answer 6

Independent of state (s) - constant for a system
Dependent on Temperature
“counts” how many states accessible to an atom weigting each in proportion to its probability

Question 7

Z rel to temp

Answer 7

^T => $Z = \sum_s e^{-E(s)/kT}$ =>
$Z = \sum_s e^{-E(s)/big}$ => $Z \approx \sum_s 1$ => Z&raquo_space; 1

low temp => $Z = \sum_s e^{-E(s)/kT}$ => $Z = \sum_s e^{-E(s)/small}$ => $Z = \sum_s e^{-big} \approx 0$ so summing all up we get that
low T => $Z \approx 1$

Question 8

Average energy

Answer 8

$E_{avg}(s) = \frac{1}{Z} \sum_s E(s) e^{-E(s)/kT}$

Question 9

Average quantity X(s)

Answer 9

$X_{avg}(s) = \frac{1}{Z} \sum_s X(s) e^{-E(s)/kT}$

Question 10

Additivity of avgs

Answer 10

$$U_{total} = NE$$
where N is # particles
E is avg energy of 1 particle

Question 11

Magical formula eqn 6.25

Answer 11

$\bar E = - \frac{1}{Z} \frac{\partial Z}{\partial \beta}$

Question 12

Helmholtz free energy

Answer 12

$$F = U - TS$$

Question 13

Related F to entropy

Answer 13

$$(\frac{\partial F}{\partial T})_{V,N} = -S$$
derive from taking partial of
F = U - TS

Question 14

Relate paritiaion func and free energy

Answer 14

$$F = -kT\ln{Z}$$

i.e. $$Z = e^{-\frac{F}{kT}}$$

Question 15

Relate partition functions for distinguishable non interacting sytems

Answer 15

$Z_{tot} = Z_1 Z_2 Z_3 … Z_N$

Question 16

Relate partition functions for indistinguishable non interacting sytems

Answer 16

$Z_{tot} = \frac{1}{N!} Z_1^N$

where $Z_1$ is the partition func for any single indiv particle

Question 17

Helholtx energy describd as competition

Answer 17

F wants to decrease (min @ equil)

F = U - TS
U total energy wants to decrease and total entorpy wants to increase (S)
SO competition between energy and entrooy which wants to decrease one increase temp facilitates