Additional Problems:

$($a$)$ Use the relationship between $U$ and $A$ as well as the expression for $dU,$

where $dU=TdS-PdV,$ to derive the expression for $dA.($b$)$ Use the result in $($a$)$ to

derive the expression for the corresponding Maxwell relation.

Use the expression $S={\displaystyle \underset{V_i}{\overset{V_f}{}}}(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_{V}dV$ to find the change in entropy during an

isothermal expansion for $($a$)$ an ideal gas, $($b$)$ a gas that obeys van der Waals equation, and

$($c$)$ a gas that obeys the Redlich$-$Kwong equation.

Use the van der Waals equation and the constants for $C{O}_2$ in Table $16\mathrm{.}3$ to

calculate $($a$)$ the pressure $P,($b$)$ the compressibility factor $Z,$ and $($c$)$ the fugacity coefficient

$$ for one mole of $C{O}_2$ gas in a $200mL$ container at $300K.($d$)$ How do these values compare

to an ideal gas? Use the equations in Section $16-2$ and from Problem $22-33.$

The molar enthalpy of vaporization of water is $40\mathrm{.}65k\frac{J}{m}ol$ at its normal

boiling point. Use the Clausius$-$Clapeyron equation to calculate the vapor pressure of

water at $90C.$

Show a general Clausius$-$Clapeyron plot and explain how it can be used to

determine the molar enthalpy change upon vaporization.