11.3.6: Reversible Energy Flow

We saw in section 11.3.4 that when a system is allowed to interact with its environment, total entropy generally increases. In this case it is not possible to restore the system and the environment to their prior states by further mixing, because such a restoration would require a lower total entropy. Thus mixing in general is irreversible.

The limiting case where the total entropy stays constant is one where, if the system has changed, it can be restored to its prior state. It is easy to derive the conditions under which such changes are, in this sense, reversible.

From the formulas given earlier, specifically Equation 11.23, the change in system entropy is proportional to the part of the change in energy due to heat. Thus

\(\begin{align*}
d S_{s} &=k_{B} \beta_{s} d E_{s}-k_{B} \beta_{s} \sum_{i} p_{s, i} d E_{s, i}(H) \tag{11.53}\\
&=k_{B} \beta_{s}\left[d E_{s}-\sum_{i} p_{s, i} d E_{s, i}(H)\right] \tag{11.54} \\
&=k_{B} \beta_{s} d q_{s} \tag {11.55}
\end{align*}\)

where \(dq_s\) stands for the heat that comes into the system due to the interaction mechanism. This formula applies to the system and a similar formula applies to the environment:

\(dS_e = kB\beta_e dq_e \tag{11.56}\)

The two heats are the same except for sign

\(dq_s = −dq_e \tag{11.57}\)

and it therefore follows that the total entropy \(S_s + S_e\) is unchanged (i.e., \(dS_s = −dS_e\)) if and only if the two values of \(\beta\) for the system and environment are the same:

\(\beta_s = \beta_e \tag{11.58}\)

Reversible changes (with no change in total entropy) can involve work and heat and therefore changes in energy and entropy for the system, but the system and the environment must have the same value of \(\beta\). Otherwise, the changes are irreversible. Also, we noted in Section 11.3.4 that interactions between the system and the environment result in a new value of \(\beta\) intermediate between the two starting values of \(\beta_s\) and \(\beta_e\), so reversible changes result in no change to \(\beta\).

The first-order change formulas given earlier can be written to account for reversible interactions with the environment by simply setting \(d\beta = 0\)

\(\begin{align*}
0 &=\sum_{i} d p_{i} \tag{11.59} \\
d E &=\sum_{i} E_{i}(H) d p_{i}+\sum_{i} p_{i} d E_{i}(H) \tag{11.60} \\
d S &=k_{B} \beta d E-k_{B} \beta \sum_{i} p_{i} d E_{i}(H) \tag{11.61} \\
d \alpha &=-\beta \sum_{i} p_{i} d E_{i}(H) \tag{11.62} \\
d p_{i} &=-p_{i} \beta d E_{i}(H)+p_{i} \beta \sum_{j} p_{j} d E_{j}(H) \tag{11.63} \\
d E &=\sum_{i} p_{i}\left(1-\beta\left(E_{i}(H)-E\right)\right) d E_{i}(H) \tag{11.64} \\
d S &=-k_{B} \beta^{2} \sum_{i} p_{i}\left(E_{i}(H)-E\right) d E_{i}(H) \tag{11.65}
\end{align*}\)

As before, these formulas can be further simplified in the case where the energies of the individual states is proportional to \(H\)

\(\begin{align*}
0 &=\sum_{i} d p_{i} \tag{11.66} \\
d E &=\sum_{i} E_{i}(H) d p_{i}+\left(\frac{E}{H}\right) d H \tag{11.67} \\
d S &=k_{B} \beta d E-k_{B} \beta\left(\frac{E}{H}\right) d H \tag{11.68} \\
d \alpha &=-\left(\frac{\beta E}{H}\right) d H \tag{11.69} \\
d p_{i} &=-\left(\frac{p_{i} \beta}{H}\right)\left(E_{i}(H)-E\right) d H \tag{11.70} \\
d E &=\left(\frac{E}{H}\right) d H-\left(\frac{\beta}{H}\right)\left[\sum_{i} p_{i}\left(E_{i}(H)-E\right)^{2}\right] d H \tag{11.71} \\
d S &=-\left(\frac{k_{B} \beta^{2}}{H}\right)\left[\sum_{i} p_{i}\left(E_{i}(H)-E\right)^{2}\right] d H \tag{11.72}
\end{align*}\)

These formulas will be used in the next chapter of these notes to derive constraints on the efficiency of energy conversion machines that involve heat.