Chapter 6.6, Problem 15P

A)

Interpretation Introduction

Interpretation:

• $Q,W,\text{ and }\Delta U$ for the gas in each of the two steps of this process has to be determined.
• Temperature at stage 2 and pressure at stage 3 has to be determined.

Concept introduction:

Use the energy balance and ideal gas equation obtain the unknown temperature of the state, then use the general expression to obtain the remaining properties of the both the steps.

Write the expression for the energy balance for a closed system.

$dU=dQ+d{W}_{EC}$

Here, change in internal energy is $dU$, change in heat is $dQ$, and change in compression work is $W_{EC}$.

Write the energy balance for the ideal gas by considering the compression is adiabatic.

$\begin{array}{l}dU=d{W}_{EC}\\ Nd\underset{\_}{U}=-PdV\end{array}$        (1)

Here, number of moles is $N$, change in molar internal energy is $d\underset{\_}{U}$, and pressure is $P$.

Write the expression for the change in internal energy $\left(\Delta U\right)$.

$\begin{array}{c}\Delta \underset{\_}{U}=W\\ =N{C}_V^{\ast }\left(T_2-T_1\right)\\ =N\left(C_P^{\ast }-R\right)\left(T_2-T_1\right)\end{array}$

Here, work is $W$.

B)

Interpretation Introduction

Interpretation:

The change in internal energy $\left(d{U}_{1-2}\right)$ with the transition from state 1 - 2 and state 3-1 has to be determined.

Concept introduction:

Calculate the change in internal energy $\left(d{\underset{\_}{U}}_{1-2}\right)$ with the transition from state 1 to state 2.

$\begin{array}{c}d{\underset{\_}{U}}_{1-2}=\left({\underset{\_}{U}}_2-{\underset{\_}{U}}_2^{ig}\right)+\left({\underset{\_}{U}}_2^{ig}-{\underset{\_}{U}}_1^{ig}\right)-\left({\underset{\_}{U}}_1-{\underset{\_}{U}}_1^{ig}\right)\\ =\left(-a{T}_2{P}_2^3\right)+C_V^{\ast }\left(T_2-T_1\right)-\left(-a{T}_1{P}_1^3\right)\\ =\left(-a{T}_2{P}_2^3\right)+\left(C_P^{\ast }-R\right)\left(T_2-T_1\right)-\left(-a{T}_1{P}_1^3\right)\end{array}$        (7)

Here, molar internal energy at state 2 is ${\underset{\_}{U}}_2$, molar internal energy at state 1 is ${\underset{\_}{U}}_1$, molar internal energy for an ideal gas state at state 2 and 1 is ${\underset{\_}{U}}_2^{ig},\text{and}{\underset{\_}{U}}_1^{ig}$ respectively, and constant is $a$.

Calculate the change in internal energy $\left(d{\underset{\_}{U}}_{3-2}\right)$ with the transition from state 2 to state 3.

$\begin{array}{c}d{\underset{\_}{U}}_{3-2}=\left({\underset{\_}{U}}_3-{\underset{\_}{U}}_3^{ig}\right)+\left({\underset{\_}{U}}_3^{ig}-{\underset{\_}{U}}_2^{ig}\right)-\left({\underset{\_}{U}}_2-{\underset{\_}{U}}_2^{ig}\right)\\ =\left(-a{T}_3{P}_3^3\right)+C_V^{\ast }\left(T_3-T_2\right)-\left(-a{T}_2{P}_2^3\right)\\ =\left(-a{T}_3{P}_3^3\right)+\left(C_P^{\ast }-R\right)\left(T_3-T_2\right)-\left(-a{T}_2{P}_2^3\right)\end{array}$        (8)

Here, molar internal energy at state 3 is ${\underset{\_}{U}}_3$, molar internal energy for an ideal gas state at state 3 is ${\underset{\_}{U}}_3^{ig}$, constant is $a$, temperature at state 3 is $T_3$, and pressure at state 3 is $P_3$.