Physics Problems & Solutions: Partition Function

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Thermal Physics - Maxwell-Boltzmann Statistics

In a Maxwell-Boltzmann system with two states of energies ε and 2ε respectively, and a degeneracy of 2 for each state, the partition function is

A. e ^{−ε / kT}
B. 2e^{−2ε / kT}
C. 2e^{−3ε / kT}
D. e^{−ε / kT} + e^{−2ε / kT}
E. 2(e^{−ε / kT}+ e^{−2ε / kT})

(GR0177 #49)

Solution:

Canonical partition function: Z = Σ_i e^−βE_i
β = 1/kT

For Maxwell-Boltzmann System: Z = Σ_i g_i e^−βE_i
g_i = degeneracy

Thus, the partition function is
Z = 2e⁻^βε + 2e⁻²^β^ε = 2(e^{−ε / kT}+ e⁻²^{ε / kT})

Answer: E

Thermal Physics - Partition Function

A system consists of N weakly interacting subsystems, each with two internal quantum states with energies 0 and $\epsilon$ . The internal energy for this system at absolute temperature T is equal to

A. $N\epsilon$

B. $\frac{3}{2}NkT$

C. $N\epsilon e^{-\epsilon/kT}$

D. $\frac{N\epsilon}{e^{\epsilon/kT} + 1}$

E. $\frac{N\epsilon}{1+ e^{-\epsilon/kT} }$

(GR9677 #94)

Solution:

A. FALSE. It’s not an exponential function

B. FALSE. It’s not an exponential function

C. FALSE. $E\rightarrow 0$ as $T\rightarrow 0$    but   $E\rightarrow N\epsilon$    as    $T\rightarrow \infty$

D. TRUE. $E\rightarrow 0$ as   $T\rightarrow 0$ and   $E\rightarrow N\epsilon/2$ as   $T\rightarrow \infty$
(fits the equipartition theorem for high temperature/classical region/Maxwell-Boltzmann Statistics)

E. FALSE. $E\rightarrow N\epsilon$ as   $T\rightarrow 0$ and   $E\rightarrow \infty$ as   $T\rightarrow \infty$

Alternative Solution #1:

Partition function

$Z=\sum_i e^{-E_i/kT}$

with energies 0 and $\epsilon$ ,

$Z=e^{0/kT}+e^{-\epsilon/kT} = 1+e^{-\epsilon/kT}$

Energy

$E=\frac{\displaystyle \sum_i E_i e^{-E_i/kT}}{Z}$

$E=\frac{0+\epsilon e^{-\epsilon/kT}}{1+e^{-\epsilon/kT}}\times\frac{e^{\epsilon/kT}}{e^{\epsilon/kT}}=\frac{\epsilon }{e^{\epsilon/kT}+1}$

For N subsystems,

$E=\frac{N\epsilon }{e^{\epsilon/kT}+1}$

Answer: D

Alternative Solution #2:

We can also use

$E=-\frac{\partial \ln Z}{\partial \beta}$

with $\beta=1/kT$ , as

$E=\frac{\displaystyle \sum_i E_i e^{-E_i\beta}}{Z}=-\frac{\partial \ln Z}{\partial \beta}$

$\frac{\partial \ln Z}{\partial \beta}=\frac{1}{Z}\frac{\partial Z}{\partial \beta} = \frac{-\epsilon e^{-\epsilon \beta}}{1+e^{-\epsilon \beta}} \times \frac{e^{\epsilon \beta}}{e^{\epsilon \beta}} = \frac{-\epsilon}{e^{\epsilon \beta}+1}$

$E=-\frac{\partial \ln Z}{\partial \beta}= \frac{\epsilon}{e^{\epsilon \beta}+1}$

For N subsystems,

$E=\frac{N\epsilon }{e^{\epsilon/kT}+1}$

Thermal Physics - Thermal Equilibrium

Suppose that a system in a quantum state i has energy E_i. In thermal equilibrium, the expression

represents which of the following?

A. The average energy of the system
B. The partition function
C. Unity
D. The probability to find system with energy E_i
E. The entropy of the system

(GR0177 #98)

Solution:

The units of the given expression are the units of E_i which as given is energy.

The average energy is given by

<E> = ^{∑_i E_ie^−E_i/kT}/ _Z

where

Z = ∑_i e^−E_i/kT

is the partition function.

Answer: A