The State of the System

The State of the System

To do any useful calculation in thermodynamics, you must first describe the system. This means specifying its measurable properties quantitatively — things like pressure ($p$), volume ($V$), temperature ($T$), and the amount of substance ($n$). You need to know these values both before and after a change occurs.

In your physics course, you learned that the state of a mechanical system is completely fixed by the position and velocity of every particle at a given instant. Thermodynamics takes a much simpler approach. Because we deal with the average behaviour of enormous numbers of particles, we don't need to track each one. Instead, we describe the system using its macroscopic, bulk properties — the ones you can measure directly.

These measurable properties — $p$, $V$, $T$, $n$ — are called state variables or state functions. Their defining feature is this: their value depends only on the current state of the system, not on the path taken to reach that state.

For example, if a gas is at $300\ \text{K}$ and $1\ \text{atm}$, its temperature is $300\ \text{K}$ regardless of whether it was heated slowly, compressed suddenly, or cooled from a higher temperature. The temperature is a state function; the history is irrelevant.

How Many Variables Do We Need?

You do not need to specify every property of the system to define its state completely. Only a certain minimum number of variables can be changed independently. Once you fix those, all other properties automatically take on definite values.

The number of independent variables depends on the nature of the system. For a fixed amount of a pure gas (say, $n$ moles), you only need to specify two of the three variables $p$, $V$, and $T$. The third is then determined by the ideal gas equation:

So if you know $p$ and $T$, $V$ is fixed. If you know $V$ and $T$, $p$ is fixed. The state of the gas is completely defined by any two of these three variables. …