A$)$ Starting with the $1$st Law, show that

T T

U S T P

V V

$.($Don$$t overthink this one!$)$

B$)$ Using an appropriate Maxwell relation, show that

T V

U P T P

V T

$.$

C$)$ The van der Waals equation for a non$-$ideal gas is $2$

$2$

n P a V nb nRT

V

$$

$$

$,$ where a and b

are constants. Find the change in internal energy when a non$-$ideal gas expands isothermally from initial

volume Vi to final volume Vf$.$

D$)$ Based on your answer in $($C$),$ provide a physical meaning for the constant a$.$A$)$ Starting with the $1^{st}$ Law, show that $(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{V}}{)}_T-P.($Don$\text{'}$t overthink this one!$)$

B$)$ Using an appropriate Maxwell relation, show that $(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V-P.$

C$)$ The van der Waals equation for a non$-$ideal gas is $(P+a\frac{n^2}{V^2})(V-nb)=nRT,$ where a and $b$

are constants. Find the change in internal energy when a non$-$ideal gas expands isothermally from initial

volume $V_i$ to final volume $V_f.$

D$)$ Based on your answer in $($C$),$ provide a physical meaning for the constant $a.$