Question 20.5: An Isothermal Expansion A 1.0-mol sample of an ideal gas is ......
An Isothermal Expansion
A 1.0-mol sample of an ideal gas is kept at 0.0^{\circ} \mathrm{C} during an expansion from 3.0 \mathrm{~L} to 10.0 \mathrm{~L}.
(A) How much work is done on the gas during the expansion?
(B) How much energy transfer by heat occurs between the gas and its surroundings in this process?
(C) If the gas is returned to the original volume by means of an isobaric process, how much work is done on the gas?
(A) Conceptualize Run the process in your mind: the cylinder in Active Figure 20.8 is immersed in an ice-water bath, and the piston moves outward so that the volume of the gas increases. You can also use the graphical representation in Figure 20.9 to conceptualize the process.
Categorize We will evaluate parameters using equations developed in the preceding sections, so we categorize this example as a substitution problem. Because the temperature of the gas is fixed, the process is isothermal.
Substitute the given values into Equation 20.14:
\begin{aligned} W & =n R T \ln \left(\frac{V_{i}}{V_{f}}\right) \\ & =(1.0 \mathrm{~mol})(8.31 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K})(273 \mathrm{~K}) \ln \left(\frac{3.0 \mathrm{~L}}{10.0 \mathrm{~L}}\right) \\ & =-2.7 \times 10^{3} \mathrm{~J} \end{aligned}
(B) Find the heat from the first law:
\begin{aligned} & \Delta E_{\text {int }}=Q+W \\ & 0=Q+W \\ & Q=-W=2.7 \times 10^{3} \mathrm{~J} \end{aligned}
(C) Use Equation 20.12. The pressure is not given, so incorporate the ideal gas law:
\begin{aligned} W & =-P\left(V_{f}-V_{i}\right)=-\frac{n R T_{i}}{V_{i}}\left(V_{f}-V_{i}\right) \\ & =-\frac{(1.0 \mathrm{~mol})(8.31 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K})(273 \mathrm{~K})}{10.0 \times 10^{-3} \mathrm{~m}^{3}}\left(3.0 \times 10^{-3} \mathrm{~m}^{3}-10.0 \times 10^{-3} \mathrm{~m}^{3}\right) \\ & =1.6 \times 10^{3} \mathrm{~J} \end{aligned}
We used the initial temperature and volume to calculate the work done because the final temperature was unknown. The work done on the gas is positive because the gas is being compressed.
