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Notes by Category University Engineering

Mechanics & Stress Analysis*
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The Second Law of Thermodynamics

In this notes sheet...

  1. Reversible & Irreversible Processes

  2. The Second Law

  3. Heat Engines & Efficiency

  4. Clausius' Statement

  5. Actual vs Ideal Efficiencies

  6. Improving Efficiency

  7. The Carnot Cycle


The second law is used for a number of important calculations: finding temperature differences, reversibility, and efficiency of reversible vs irreversible heat engines.

Reversible & Irreversible Processes

Let us take the classic example of a quasi-equilibrium expansion:

If the mass on the piston is made up of a vast number of minuscule weights, like sand, removing a single grain pushes the piston up ever so slightly. The change is so small, that the return to equilibrium is almost instant.

For each grain of sand removed, we have an additional point on the P-V diagram – as pressure decreases, volume increases.

This process is reversible

It is reversible, because if we return the grains of sand one at a time, the piston will move down by the same amount. We neglect friction or leakage.

If, however, the weight consist of one large mass, and this is removed, the process is not in quasi-equilibrium.

The process is irreversible

In this example, the expansion itself is quasi-equilibrium, as we have returned to the grains of sand. However, when the piston hits the top or bottom stoppers, it makes a sound, releasing energy. This energy cannot be gotten back, so

The process is irreversible

What makes a process irreversible?

  • Friction

  • Unresisted expansion

  • Heat transfer to surroundings

  • Combustion

  • Mixing of different fluids

Clearly, then, the majority of real-life processes are irreversible.

The Second Law

Take the example of a spindle rotating in a container:

Spindle rotates

  • Gas in container heats up

  • Heat from gas is conducted through container and heats up the surroundings

If we reverse this:

Surroundings are heated

  • Gas inside container heats up

  • Spindle does not start rotating

We know from experience that this is the case, but why?

This is where the second law comes in:

Heat cannot be transferred from a cold body to a hotter body without a work input.

This is the all-important idea of direction: some things are allowed backwards as well as forwards, but many things are only allowed forwards.

Heat Engines & Efficiency

A heat engine is a device that connects a hot and cold reservoir with a work input or output.

From the first law:

The second law is often also used to describe efficiency, η:

Using the equation from the first law above, the efficiency can also be written as: