The work done by a thermodynamic system depends not only on the initial and final states but also on the intermediate states—that is, on the path. Like work, when heat is added to a thermodynamic system, it undergoes a change of state, and the state attained depends on the path from the initial state to the final state. Consider an ideal gas cylinder fitted with a piston. When the cylinder is heated at a constant temperature, the gas molecules absorb energy and expand slowly in a controlled isothermal manner. This pushes the piston upwards, and gas eventually attains the final volume. Here, the work is done by the gas due to heat expansion on the piston.
The gas can also attain the same final volume through a different process. Consider a cylinder surrounded by insulating walls and divided by a thin, removable partition. The lower compartment is filled with the same amount of gas at the same temperature so that the initial state is the same as mentioned above. When the partition is removed, the gas undergoes a rapid, uncontrolled expansion, with no heat passing through the insulating walls, and reaches the same final volume as in the above case. Here, no work is done by the gas during this expansion as it does not push against anything that moves. This uncontrolled expansion of a gas into the vacuum is called a free expansion.
Experimentally, there is no temperature change under a free expansion of ideal gas. This means that the final state of the gas remains the same. The intermediate states (pressures and volumes) during the transition from the initial to the final state are entirely different in the two cases. As a result, it represents two different paths connecting the same initial and final states.
