Entropy Balance in a Closed System

In a closed system, the entropy balance can be given as:

$$\Delta S_{\text{system}}=S_{\text{in}}-S_{\text{out}}+S_{\text{gen}}$$

where:

• $\Delta S_{\text{system}}$ is the change in total entropy of the system
• $S_{in}-S_{\text{out}}$ is the total entropy entering - total entropy leaving
• $S_{\text{gen}}$ is the total entropy generated

The above can also be written in differential form:

$$\frac{dS}{dt}={\dot{S}}_{\text{in}}-{\dot{S}}_{\text{out}}+{\dot{S}}_{\text{gen}}$$

We can also write:

$$\Delta S_{\text{sys}}=\sum \frac{Q_k}{T_k}+S_{\text{gen}}$$

where $\sum Q_k/T_k$ is the sum of all differential amounts of heat transfer divided by temperature $T_k$ at location $k$ on the boundary. We can interpret this equation as:

• $\Delta S_{\text{sys}}$ is the change in number of microstates
• $\Sigma Q_k/T_k$ is the change in number of microstates due to heat addition
• $\Delta S_{\text{gen}}$ is the change in number of microstates due to irreversibilities

Recall that we always have:

$$S_{\text{gen}}\ge 0$$

Some more things to note:

• Adding/subtracting work $W$ does not change the number of microstates (does not change entropy).
• Adding heat $Q$ results in a large increase in number of microstates if the current temperature is low, or a small increase in number of microstates if the current temperature is already high.

Example