19.2.3 Isothermal Process

FIGURE 19.8 shows an isothermal process. Here we need to know the pressure as a function of volume before we can integrate Equation 19.8. From the ideal-gas law, $p = n R T / V$ . Thus the work on the gas as the volume changes from $V_{i}$ to $V_{f}$ is

W = - \int_{V_{i}}^{V_{f}} p d V = - \int_{V_{i}}^{V_{f}} \frac{n R T}{V} d V = - n R T \int_{V_{i}}^{V_{f}} \frac{d V}{V}

(19.11)

where we could take the $T$ outside the integral because temperature is constant during an isothermal process. This is a straightforward integration, giving

\begin{array}{l} W & = - n R T \int_{V_{i}}^{V_{f}} \frac{d V}{V} = - n R T ln V |_{V_{i}}^{V_{f}} \\ = - n R T (ln V_{f} - ln V_{i}) = - n R T ln (\frac{V_{f}}{V_{i}}) \end{array}

(19.12)

Because $n R T = p_{i} V_{i} = p_{f} V_{f}$ during an isothermal process, the work is:

\begin{matrix} W = - n R T ln (\frac{V_{f}}{V_{i}}) = - p_{i} V_{i} ln (\frac{V_{f}}{V_{i}}) = - p_{f} V_{f} ln (\frac{V_{f}}{V_{i}}) \\ (isothermal process) \end{matrix}

(19.13)

Which version of Equation 19.13 is easiest to use will depend on the information you’re given. The pressure, volume, and temperature must be in SI units.

The figure illustrates a pressure versus volume graph.

FIGURE 19.8 Full Alternative Text

EXAMPLE 19.2 The work of an isothermal compression

A cylinder contains 7.0 g of nitrogen gas. How much work must be done to compress the gas at a constant temperature of $80 ° C$ until the volume is halved?

MODEL This is an isothermal ideal-gas process.

SOLVE Nitrogen gas is $N_{2}$ , with molar mass $M_{mol} = 0.028 kg/mol = 28 g/mol$ , so 7.0 g is 0.25 mol of gas. The temperature is $T = 353 K$ . Although we don’t know the actual volume, we do know that $V_{f} = \frac{1}{2} V_{i}$ . The volume ratio is all we need to calculate the work:

\begin{array}{l} W & = - n R T ln (\frac{V_{f}}{V_{i}}) \\ = - (0.25 mol) (8.31 J/mol K) (353 K) ln (1 / 2) = 508 J \end{array}

REVIEW The work is positive because a force from the environment pushes the piston inward to compress the gas.