The Unexpected State Function: Entropy

We determined that for an ideal gas:

$$\input{equations/81D}$$ (12-1)

Notice that we have separated the equation into something that is integrable over segments of $dT$ (holding $V$ constant) and $dV$ (holding $T$ constant).

Therefore, over any curve that we can divide up into small orthogonal segments:

Figure 12-1: Any continuous curve can be composed of infinitesimal segments.

Therefore, for an ideal gas, $\frac{dq_{rev}}{T}$ is a ``perfect differential.''

The integral of $\frac{dq_{rev}}{T}$ depends only on its endpoints; therefore it has the properties of a state function.

Let's call the perfect differential in Equation 12-1 $dS$ and $S$ is defined as the new state function entropy.

$$\input{equations/Sdef}$$ (12-2)

The numerical value of entropy depends only on the state itself relative to some reference state.

Question: According to the definition in Equation 12-2, is entropy an intensive or extensive quantity?

Therefore, for an ideal gas at least,

$$\input{equations/du=tds-pdv}$$ (12-3)

Notice that:

$T$ is conjugate to $S$ in the same way as $-P$ is conjugate to $V$

This gives a new interpretation of $T$: