Chapter 6, Problem 6.91P

(a)

Interpretation Introduction

Interpretation:

Derive $Z$, $U$ and $C_V$ in relation to $G$, $T$, $P$ and/or derivatives of $G$ with respect to $T$ and $P$ or ${\left(\frac{\partial G}{\partial T}\right)}_P$

  ${\left(\frac{\partial G}{\partial P}\right)}_T$

Concept Introduction:

The general correlation which define compressibility factor $Z$ is

  $Z=\frac{PV}{RT}$

The fundamental property relation for internal energy $U$ is defined as:

  $\begin{array}{l}G=H-TS\\ G=U+PV-TS\\ U=G-PV+TS\end{array}$

And

The fundamental property relation for heat capacity at constant volume $C_V$ is defined as:

  $C_V={\left(\frac{\partial U}{\partial T}\right)}_V$

And the chain rule of partial derivatives or triple product rule is

  ${\left(\frac{\partial x}{\partial y}\right)}_Z{\left(\frac{\partial y}{\partial z}\right)}_x{\left(\frac{\partial z}{\partial x}\right)}_y=-1$

And one useful equation derived from exact differentials in two dimensions, considering four state functions $z,x,y\text{and }v$ and state space is two dimension and any of these four quantities are exact differential

  ${\left(\frac{\partial z}{\partial y}\right)}_v={\left(\frac{\partial z}{\partial y}\right)}_x+{\left(\frac{\partial z}{\partial x}\right)}_y{\left(\frac{\partial x}{\partial y}\right)}_v$

(b)

Interpretation Introduction

Interpretation:

Derive $Z$, $H$ and $C_P$ in relation to $A$, $T$, $V$ and/or derivatives of $A$ with respect to $T$ and $V$ or ${\left(\frac{\partial A}{\partial T}\right)}_V$

  ${\left(\frac{\partial A}{\partial V}\right)}_T$

Concept Introduction:

The general correlation which define compressibility factor $Z$ is

  $Z=\frac{PV}{RT}$

The fundamental property relation for enthalpy $H$ is defined as:

  $\begin{array}{l}A=U-TS\\ A=H-PV-TS\\ H=A+PV+TS\end{array}$

And

The fundamental property relation for heat capacity at constant pressure $C_P$ is defined as:

  $C_P={\left(\frac{\partial H}{\partial T}\right)}_P$

And the chain rule of partial derivatives or triple product rule is

  ${\left(\frac{\partial x}{\partial y}\right)}_Z{\left(\frac{\partial y}{\partial z}\right)}_x{\left(\frac{\partial z}{\partial x}\right)}_y=-1$

And one useful equation derived from exact differentials in two dimensions, considering four state functions $z,x,y\text{and }v$ and state space is two dimension and any of these four quantities are exact differential

  ${\left(\frac{\partial z}{\partial y}\right)}_v={\left(\frac{\partial z}{\partial y}\right)}_x+{\left(\frac{\partial z}{\partial x}\right)}_y{\left(\frac{\partial x}{\partial y}\right)}_v$