Find $A_2,$ the area of the rectangle.

$,A_2=b{h}_2=(0\mathrm{.}200m^3)(1\mathrm{.}00{10}^5(Pa))=2\mathrm{.}00{10}^{4}J$

$W=-(A_1+A_2)=-3\mathrm{.}50{10}^{4}J$

$($b$)$ Find the change in the internal energy during the process.

Compute the temperature at points A and $B$ with the ideal gas law.

$T_A=\frac{P_A{V}_A}{nR}=\frac{(1\mathrm{.}00{10}^5(Pa))(0\mathrm{.}100m^3)}{(4\mathrm{.}00(mol))(8\mathrm{.}31\frac{J}{K}*(mol))}=301K$

$T_B=\frac{P_B{V}_B}{nR}=\frac{(2\mathrm{.}50{10}^5(Pa))(0\mathrm{.}300m^3)}{(4\mathrm{.}00(mol))(8\mathrm{.}31\frac{J}{K}*(mol))}=2\mathrm{.}26{10}^{3}K$

Compute the change in internal energy.

$U=\frac{3}{2}nRT$

$=\frac{3}{2}(4\mathrm{.}00$mol$)(8\mathrm{.}31\frac{J}{K}*mol)(2\mathrm{.}26{10}^3(K)-301(K))$

$=9\mathrm{.}77{10}^{4}J$

$($c$)$ Find the thermal energy transferred to the gas.

$Q=U-W=9\mathrm{.}77{10}^{4}J-(-3\mathrm{.}50{10}^4(J))$

$=1\mathrm{.}33{10}^{5}J$

Remarks

As long as it's possible to compute the work, cycles involving these more exotic processes can be completely analyzed. Usually, however, it's necessary to use calculus. Note that the solution to part $($b$)$ could have been facilitated by yet another application of $PV=nRT.$

$U=\frac{3}{2}nRT=\frac{3}{2}(PV)=\frac{3}{2}(P_B{V}_B-P_A{V}_A)$

This result means that even in the absence of information about the number of moles or the temperatures, the problem could be solved knowing the initial and final pressures and volumes.

Question

If the curve in the figure is deformed so that it involves lower pressures between the same endpoints, how would that affect the calculation of the thermal energy transferred? $($Select all that apply.$)$

The thermal energy transferred would remain the same.

The magnitude of the work done would remain the same.

The thermal energy transferred would become smaller.

The magnitude of the work done would become smaller.

The magnitude of the work done would become larger.

The thermal energy transferred would become larger.

Exercise

The figure below represents a process involving $1\mathrm{.}97$ moles of a monatomic ideal gas expanding from $0\mathrm{.}100m^3$ to $0\mathrm{.}200m^3.$

$($i$)$

$($a$)$ Find the work done on the system.

$w=$

Is the system expanding or being compressed? Are you asked to find the work done on the system or the work done by the system? J

$($b$)$ Find the change in the internal energy of the system.

$U=q,$

Follow the example. J

$($c$)$ Find the thermal energy transferred in the process.

$Q=$

Check your algebra. What does $W$ represent, the work done by the system or the work done by the environment? J