We’ve seen that state variables, such as p and T, undergo changes that we denote as $\text{Δ}p$ and $\text{Δ}T$. Thus the quantity $\text{ln}\text{Ω}$, which changes by $\text{Δ}(\text{ln}\text{Ω})$ during the infinitesimal isothermal expansion, appears to be a state variable. Specifically, $\text{ln}\text{Ω}$ seems to be a state variable that measures the dispersal of energy in a system—exactly what we are looking for.

With that in mind, we define the entropy S of a macrostate that has multiplicity $\text{Ω}$ as

That is, entropy is a state variable that measures the number of ways a macrostate can differ microscopically. The Boltzmann constant is included in the definition for historical reasons. As a result, the units of entropy are the same as the units of the Boltzmann constant: $\text{J}\text{/}\text{K}$.

Equation 20.50 defines entropy in terms of the multiplicity of the macrostate, but in practice we rarely know the multiplicity $\text{Ω}$. On the other hand, we can easily measure heat transfer. For an infinitesimal heating, such as we’ve been considering, the quantity $\text{Δ}(\text{ln}\text{Ω})=\text{Δ}S/k_{\text{B}}$ on the left-hand side of Equation 20.49 is better written in terms of an infinitesimal change in entropy as $dS/k_{\text{B}}$. Thus Equation 20.49 is

Equation 20.51 is equivalent to Equation 20.50 but will turn out to be much more useful because it directly relates entropy to heat. Although we used the slow heating of a gas to arrive at Equation 20.51, the equation turns out to be true in general for any slow quasi-static process.