Physics 4311: Thermal Physics – HW Solution 8

due date: Wednesday, Apr 1, 2026

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Problem 1: Entropy of the ideal gas (12 points)

a)

𝑑𝑈 = T 𝑑𝑆 - p 𝑑𝑉 . 𝑑𝑆 = \frac{1}{T} 𝑑𝑈 + \frac{p}{T} 𝑑𝑉

use $𝑝𝑉 = N k_{B} T$ and $U = (3 ∕ 2) N k_{B} T$

𝑑𝑆 = \frac{3 N k_{B}}{2 T} 𝑑𝑇 + \frac{N k_{B}}{V} 𝑑𝑉

S = \frac{3}{2} N k_{B} 𝑙𝑛 (\frac{T}{T_{0}}) + N k_{B} ln (\frac{V}{V_{0}})

b)

for $T \to 0$ , entropy $S \to - \infty$ , this contradicts condition $S \geq 0$ which follows from $S = k_{B} 𝑙𝑛 Ω$ .
$\Rightarrow$ ideal gas laws cannot hold down to $T = 0$

Problem 2: Maxima of entropy (16 points)

a)

define

f = - k_{B} \sum_{i} p_{i} ln p_{i} - λ (\sum_{i} p_{i} - 1)

\frac{∂𝑓}{\partial p_{j}} = - k_{B} (𝑙𝑛 p_{j} + 1) - λ = 0

p_{j} = e^{= - 1 - λ ∕ k_{B}} independent of 𝑗

\sum_{j} p_{j} = 1 \Rightarrow p_{j} = \frac{1}{N}

b)

f = - k_{B} \sum_{i} p_{i} ln p_{i} - λ (\sum_{i} p_{i} - 1) - μ (\sum_{i} p_{i} E_{i} - ⟨ E ⟩)

\frac{∂𝑓}{\partial p_{j}} = - k_{B} (𝑙𝑛 p_{j} + 1) - λ - μ E_{j} = 0

p_{j} = e^{= - 1 - λ ∕ k_{B} - μ E_{j} ∕ k_{B}} structure of Boltzmann factor

now find $λ$ and $μ$ from $\sum_{i} p_{i} = 1$ and $\sum_{i} p_{i} E_{i} = ⟨ E ⟩$

Problem 3: Additivity of the entropy (12 points)

S = - k_{B} \sum_{i = 1}^{n} ∑_{j = 1}^{m} p (i, j) ln p (i, j)

if subsystems are statistically independent, $p (i, j) = p_{A} (i) p_{B} (j)$

S = - k_{B} \sum_{i = 1}^{n} ∑_{j = 1}^{m} p_{A} (i) p_{B} (j) ln [p_{A} (i) p_{B} (j)]

S = - k_{B} \sum_{i = 1}^{n} ∑_{j = 1}^{m} p_{A} (i) p_{B} (j) [ln p_{A} (i) + ln p_{B} (j)]

S = - k_{B} \sum_{i = 1}^{n} p_{A} (i) ln p_{A} (i) \sum_{j = 1}^{m} p_{B} (j) - k_{B} \sum_{j = 1}^{m} p_{B} (j) ln p_{B} (j) \sum_{i = 1}^{n} p_{A} (i)

S = S_{A} + S_{B}