It will be very useful to represent ideal-gas processes on a graph called a $\mathit{p}\mathit{V}$ diagram. This is nothing more than a graph of pressure versus volume. The important idea behind the $pV$ diagram is that each point on the graph represents a single, unique state of the gas. That seems surprising at first, because a point on the graph only directly specifies the values of $p$ and $V$. But knowing $p$ and $V$, and assuming that $n$ is known for a sealed container, we can find the temperature by using the ideal-gas law. Thus each point actually represents a triplet of values $\left(p\text{,}V\text{,}T\right)$ specifying the state of the gas.

For example, FIGURE 18.9 is a $pV$ diagram showing three states of a system consisting of 1 mol of gas. The values of $p$ and $V$ can be read from the axes for each point, then the temperature at that point determined from the ideal-gas law.

FIGURE 18.9 The state of the gas and ideal-gas processes can be shown on a $pV$ diagram.

An ideal-gas process is a “trajectory” in the $pV$ diagram showing all the intermediate states through which the gas passes. Figure 18.9 shows two different processes by which the gas can be changed from state 1 to state 3.

There are infinitely many ways to change the gas from state 1 to state 3. Although the initial and final states are the same for each of them, the particular process by which the gas changes—that is, the particular trajectory—will turn out to have very real consequences. The $pV$ diagram is an important graphical representation of the process.