Paths on a Pressure-Volume Plot

The points on a pressure-volume plot represent states of the system. When the pressure or volume is changed the system moves from one point in P-V space to another. The system does not usually move in a straight line, but rather follows a path. Ideally, there are three paths that we discuss

Constant temperature – isothermal

Constant pressure – isobaric

Constant volume – isochoric

Here we demonstrate this for both an expansion and a compression. First consider the expansion shown below.

The blue-violet curve is a constant temperature (isothermal) path between the initial state and the final state

Here P_i = 100 atm, V_i = 0.246 L, and T_i = 300 K.

And P_f = 10 atm, V_f = 2.46 L, and T_f = 300 K.

Note that T_i = T_f. In other words we are following an isotherm.

For the isothermal path the work is:

In this specific example the numerical result requires us to solve for the number of moles of gas. Using the ideal gas equation of state we obtain

Thus, the work is:

w = -5740 J.

If we choose a path consisting of a constant pressure expansion (the red-violet line) then we must also follow the constant volume decompression shown as the red line. The path follows the two lines and makes a sharp right angle turn. Note that after the P step

(decompression step) the final pressure has been reached and the temperature has decreased accordingly. In fact the temperature has increased according to T_e/T_i = P_f/P_i, in this case a factor of 10!

The work extracted from the system following the constant pressure path is:

w = -P_f(V_f - V_i) = -10 atm(2.46 L – 0.246 L) = -22.14 L-atm. Notice that the units are not Joules. We can convert to Joules by recalling that R = 8.31 J/mol-K and R = 0.082 L-atm/mol-K so that the ratio is 8.31/0.082 J/L-atm = 101.34 J/L-atm. This the work for the constant pressure path is:

w = -2244 J.

The energy extracted from the system by isothermal (reversible) path is greater than that extracted by the constant pressure (irreversible) path. Work is a path function and so we expect the amount of work to be path dependent.

A compression has the following shown in Figure below. Here we must apply an external pressure at least as great as the final pressure we wish to achieve in the sample volume. We follow the red line to indicate that the pressure is applied at constant volume. Then in a subsequent constant pressure step the sample volume is decreased from 2.46 L to 0.246 L. Note that the temperature at P step is much higher than it is in either the initial or final states. The work of compression is equal to the area under the violet line across the top. The reversible work of compression is equal to the area under the curve representing an isothermal reversible compression.