Chapter 17, Problem 32P

In Figure P17.32, the change in internal energy of a gas that is taken from A to C along the blue path is +800 J. The work done on the gas along the red path ABC is −500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D?

Figure P17.32

To determine

The energy added to the system by heat as it goes from $A$ through $B$ to $C$.

Answer to Problem 32P

The energy added to the system by heat as it goes from $A$ through $B$ to $C$ is $\underset{\_}{1300\text{J}}$.

Explanation of Solution

In this case, the change in the internal energy from $A$ to $C$  is equal to the change in the internal energy from $A$ through $B$ to $C$.

    $\Delta E_{\mathrm{int},AC}=\Delta E_{\mathrm{int},ABC}$        (I)

Here, $\Delta E_{\mathrm{int},AC}$ is the change in the internal energy from $A$ to $C$, $\Delta E_{\mathrm{int},ABC}$ is the change in the internal energy from $A$ through $B$ to $C$.

Write the expression for the energy added to the system by heat as it goes from $A$ through $B$ to $C$,

  $\Delta E_{\mathrm{int},ABC}=Q_{ABC}+W_{ABC}$        (II)

Here, $W_{ABC}$ is the work done on the gas and $Q_{ABC}$ is the energy added to the system by heat as it goes from $A$ through $B$ to $C$.

Rewrite the above expression,

  $Q_{ABC}=\Delta E_{\mathrm{int},ABC}-W_{ABC}$        (III)

Conclusion:

Substitute $800\text{J}$ for $\Delta E_{\mathrm{int},ABC}$ and $-500\text{J}$ for $W_{ABC}$ in (III) to find $Q_{ABC}$,

    $\begin{array}{c}{Q}_{ABC}=800\text{J}-\left(-500\text{J}\right)\\ =800\text{J}+500\text{J}\\ =1300\text{J}\end{array}$

Therefore, the energy added to the system by heat as it goes from $A$ through $B$ to $C$ is $\underset{\_}{1300\text{J}}$.

To determine

The work done on the system from $C$ to $D$ when the pressure at point $A$ is five times that of point $C$.

Answer to Problem 32P

The work done on the system from $C$ to $D$ when the pressure at point $A$ is five times that of point $C$ is $\underset{\_}{100\text{J}}$.

Explanation of Solution

Write the expression for the work done the system from $C$ to $D$,

  $W_{CD}=-P_C\Delta V_{CD}$        (IV)

Here, $W_{CD}$ is the work done the system from $C$ to $D$, $P_C$ is the pressure at point $C$ and $\Delta V_{CD}$ is the change in the volume at $CD$.

Substitute $-\Delta V_{AB}$ for $\Delta V_{CD}$, $\frac{P_A}{5}$ for $P_C$ and . $W_{AB}$ for $-P_A\Delta V_{AB}$ in the above equation and rewrite it,

    $\begin{array}{c}{W}_{CD}=\frac{1}{5}{P}_A\Delta V_{AB}\\ =-\frac{1}{5}{W}_{AB}\end{array}$        (V)

Conclusion:

Substitute $-500\text{J}$ for $W_{AB}$ in (V) to find $W_{CD}$ ,

    $\begin{array}{c}{W}_{CD}=-\frac{1}{5}\left(-500\text{J}\right)\\ =100\text{J}\end{array}$

Therefore, the work done on the system from $C$ to $D$ when the pressure at point $A$ is five times that of point $C$ is $\underset{\_}{100\text{J}}$.

To determine

The energy exchanged with the surroundings by heat from $C$ to $A$ along the green path.

Answer to Problem 32P

The energy exchanged with the surroundings by heat from $C$ to $A$ along the green path is $\underset{\_}{-900\text{J}}$ .

Explanation of Solution

Write the expression for the work done the system from $C$ through $D$ to $A$,

  $W_{CDA}=W_{CD}$        (VI)

Here, $W_{CD}$ is the work done the system from $C$ to $D$ and $W_{CDA}$ is the work done the system from $C$ through $D$ to $A$.

Write the expression for the energy exchanged with the surroundings by heat from $C$ to $A$ along the green path,

    $Q_{CA}=\Delta E_{\mathrm{int},CA}-W_{CDA}$        (VII)

$Q_{CA}$ is the energy exchanged with the surroundings by heat from $C$ to $A$ and $\Delta E_{\mathrm{int},CA}$ is the energy exchanged from $C$ to $A$ along green path.

Conclusion:

Substitute $100\text{J}$ for $W_{CDA}$ and $-800\text{J}$ for $\Delta E_{\mathrm{int},CA}$ in (VII) to find $Q_{CA}$ ,

    $\begin{array}{c}{Q}_{CA}=-800\text{J}-100\text{J}\\ =-900\text{J}\end{array}$

Therefore, the energy exchanged with the surroundings by heat from $C$ to $A$ along the green path is $\underset{\_}{-900\text{J}}$.

To determine

The energy added to the system by heat from point $C$ to point $D$.

Answer to Problem 32P

The energy added to the system by heat from point $C$ to point $D$ is $\underset{\_}{-1400\text{J}}$.

Explanation of Solution

Write the expression for the energy added to the system by heat as it goes from $C$ to $D$,

  $\Delta E_{\mathrm{int},CD}=\Delta E_{\mathrm{int},CDA}-\Delta E_{\mathrm{int},DA}$        (VIII)

Here, $\Delta E_{\mathrm{int},CD}$ is the energy exchanged to the system by heat as it goes from $C$ to $D$,. $\Delta E_{\mathrm{int},CDA}$ is the energy exchanged to the system by heat as it goes from $C$ through $D$ to $A$ and $\Delta E_{\mathrm{int},DA}$ is the energy exchanged to the system by heat as it goes from $D$ to $A$

Write the expression for the energy added to the system by heat from point $C$ to point $D$,

  $Q_{CD}=\Delta E_{\mathrm{int},CD}-W_{CD}$        (IX)

Here, $Q_{CD}$ is the energy added to the system by heat from point $C$ to point $D$.

Conclusion:

Substitute $-800\text{J}$ for $\Delta E_{\mathrm{int},CDA}$ and $-800\text{J}$ for $\Delta E_{\mathrm{int},DA}$ in (VIII) to find $\Delta E_{\mathrm{int},CD}$,

    $\begin{array}{c}\Delta E_{\mathrm{int},CD}=-800\text{J}-500\text{J}\\ \text{=}-\text{1300}\text{J}\end{array}$

Substitute $-1300\text{J}$ for $\Delta E_{\mathrm{int},CD}$ and $100\text{J}$ for $W_{CD}$ in (IX) to find $Q_{CD}$ ,

    $\begin{array}{c}{Q}_{CD}=-1300\text{J}-100\text{J}\\ =-1400\text{J}\end{array}$

Therefore, the energy added to the system by heat from point $C$ to point $D$ is $\underset{\_}{-1400\text{J}}$.