Physics 4311: Thermal Physics – HW Solution 9

Problem 1: Thermodynamic potentials of a surface (10 points)

a)

$H=U-\mathit{𝜎𝐴},\mathit{𝑑𝐻}=\mathit{𝑇𝑑𝑆}-\mathit{𝐴𝑑𝜎}$

b)

$F=U-\mathit{𝑇𝑆},\mathit{𝑑𝐹}=-\mathit{𝑆𝑑𝑇}+\mathit{𝜎𝑑𝐴}$

c)

$G=U-\mathit{𝑇𝑆}-\mathit{𝜎𝐴},\mathit{𝑑𝐺}=-\mathit{𝑆𝑑𝑇}-\mathit{𝐴𝑑𝜎}$

Problem 2: Maxwell relations for a surface (10 points)

a)

from $U$: ${\left(\frac{\mathit{\partial 𝜎}}{\mathit{\partial 𝑆}}\right)}_A={\left(\frac{\mathit{\partial 𝑇}}{\mathit{\partial 𝐴}}\right)}_S$

b)

from $H$: ${\left(\frac{\mathit{\partial 𝐴}}{\mathit{\partial 𝑆}}\right)}_{\sigma }=-{\left(\frac{\mathit{\partial 𝑇}}{\mathit{\partial 𝜎}}\right)}_S$

c)

from $F$: ${\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝐴}}\right)}_T=-{\left(\frac{\mathit{\partial 𝜎}}{\mathit{\partial 𝑇}}\right)}_A$

d)

from $G$: ${\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝜎}}\right)}_T={\left(\frac{\mathit{\partial 𝐴}}{\mathit{\partial 𝑇}}\right)}_{\sigma }$

Problem 3: Response functions of the ideal gas (20 points)

a)

$C_V={\left(\frac{\mathit{𝛿𝑄}}{\mathit{\partial 𝑇}}\right)}_V={\left(\frac{\mathit{\partial 𝑈}}{\mathit{\partial 𝑇}}\right)}_V=\frac{3}{2}N{k}_B$

b)

$C_p={\left(\frac{\mathit{𝛿𝑄}}{\mathit{\partial 𝑇}}\right)}_p={\left(\frac{\mathit{\partial 𝑈}}{\mathit{\partial 𝑇}}\right)}_p+p{\left(\frac{\mathit{\partial 𝑉}}{\mathit{\partial 𝑇}}\right)}_p=\frac{3}{2}N{k}_B+N{k}_B=\frac{5}{2}N{k}_B$

c)

${\kappa }_T=-\frac{1}{V}{\left(\frac{\mathit{\partial 𝑉}}{\mathit{\partial 𝑝}}\right)}_T=-\frac{1}{V}{\left(\frac{\partial (N{k}_{B}T∕p)}{\mathit{\partial 𝑝}}\right)}_T=\frac{1}{p}$

d)

use equation of adaibatic curve $p{V}^{5∕3}=\mathrm{\text{const}}$ or $p^{3∕5}V=A=\mathrm{\text{const}}$
${\kappa }_S=-\frac{1}{V}{\left(\frac{\mathit{\partial 𝑉}}{\mathit{\partial 𝑝}}\right)}_S=-\frac{1}{V}{\left(\frac{\partial (A∕p^{3∕5})}{\mathit{\partial 𝑝}}\right)}_S=\frac{3}{5p}$

e)

$\alpha =\frac{1}{V}{\left(\frac{\mathit{\partial 𝑉}}{\mathit{\partial 𝑇}}\right)}_p=\frac{1}{V}{\left(\frac{\partial (N{k}_{B}T∕p)}{\mathit{\partial 𝑇}}\right)}_p=\frac{1}{T}$