# Entropy & The Second Law

In this notes sheet...

As seen in the notes sheet on the second law, we can compare a heat engine’s actual efficiency with that of its reversible counterpart. If we want to compare these in greater detail, we need to use **entropy**.

## The Clausius Inequality

In the notes sheet on the second law, we looked at heat engine connected to two thermal reservoirs, modelling two heat transfers only:

In reality, however, this is not the case. Instead, each of these heat transfers occurs in infinitely many infinitesimally small steps, dQ:

We can integrate to sum the total positive and negative heat transfers in one complete cycle:

However, this on its own is not particularly helpful. Instead, we want to be able to find the heat transfer at a specific temperature. Therefore, the Clausius inequality looks at Q/T instead:

### Reversible Cycles

The Carnot cycle is the ideal reversible cycle:

Applying the Clausius inequality:

Since the process is reversible, the two ratios must equal each other. **Therefore, for a reversible process:**

### Irreversible Cycles

We know that the efficiency of the reversible engine is greater than that of the irreversible one, so the heat output of the irreversible engine is greater than that of the reversible:

This means that

But since they are both taking heat from the same hot reservoir:

Therefore, plugging these into the equation derived above:

**We find that for an irreversible cycle:**