Cyclic Process

The Meaning of a Cyclic Process

A thermodynamic process is called cyclic when the system, after undergoing a series of changes, returns exactly to its initial state. This means every state variable — pressure, volume, temperature, internal energy — comes back to its original value.

Because the system returns to its starting point, the net change in any state function over one complete cycle is zero. The most important consequence is for internal energy $U$, a state function:

This single fact drives all the work and heat relations for a cyclic process.

Work Done in a Cycle

On a $P$–$V$ diagram, a cyclic process is represented by a closed curve. The system may expand (doing work on the surroundings) and then contract (work being done on the system). The net work done by the system in one complete cycle equals the area enclosed by the closed curve on the $P$–$V$ diagram.

If the cycle is traversed clockwise, the net work done by the system is positive. If traversed anticlockwise, the net work is negative (work is done on the system).

The First Law for a Cyclic Process

Apply the first law of thermodynamics to one complete cycle:

Since $\Delta U = 0$, we get:

Here $Q$ is the net heat supplied to the system over the cycle, and $W$ is the net work done by the system over the cycle.

This is a powerful result: in a cyclic process, the net heat absorbed by the system is entirely converted into net work output. No net change in internal energy occurs.

Sign Convention in a Cycle

Heat may be added to the system during some parts of the cycle and rejected during others. Similarly, work may be done by the system during expansion and on the system during compression. The equation $Q = W$ refers to the algebraic sum of all heat transfers and all work transfers over the complete cycle.

Let $Q_1$ be the heat absorbed, $Q_2$ the heat rejected (so $Q_2$ is negative). Then:

And the net work done by the system is:

Since $Q_2$ is negative, $W$ is less than $Q_1$. This is the basis of heat engine efficiency.

Efficiency of a Cyclic Process (Heat Engine)

For a cyclic process that operates as a heat engine:

• Heat $Q_1$ is absorbed from a hot reservoir.
• Heat $Q_2$ is rejected to a cold reservoir ($Q_2$ is negative in the sign convention where $Q$ is heat added to the system).
• Net work done by the system: $W = Q_1 - |Q_2|$.

The thermal efficiency $\eta$ is defined as:

Substituting $W = Q_1 - |Q_2|$:

Efficiency is always less than 1 because some heat must always be rejected (a consequence of the second law of thermodynamics).

Example: A Simple Cyclic Process

Consider a gas taken through the following cycle:

1. Isothermal expansion at temperature $T_1$ from volume $V_1$ to $V_2$, absorbing heat $Q_1$.
2. Adiabatic expansion from $V_2$ to $V_3$, temperature falling to $T_2$.
3. Isothermal compression at $T_2$ from $V_3$ to $V_4$, rejecting heat $Q_2$.
4. Adiabatic compression from $V_4$ back to $V_1$, temperature rising to $T_1$.

This is the Carnot cycle. For an ideal gas, one can show:

Hence the efficiency becomes:

$\eta = 1 - \frac{T_2}{T_1}$ …