Molar heat capacity

Definition:
Molar heat capacity is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin).

State function:
A state function is a property whose value depends only on the current state or condition of a system and not on the path taken to reach that state. In other words, the value of a state function is independent of the process or the path by which the system reaches that state.

Molar heat capacity as a state function:
Molar heat capacity is a state function because it only depends on the initial and final states of the system, regardless of the path taken between them. The heat energy absorbed or released by a substance when its temperature changes depends only on the initial and final temperatures, not on how the temperature change was achieved.

Example:
Consider a sample of a gas at a certain temperature and pressure. If the gas is heated, its temperature will increase and it will absorb a certain amount of heat energy. The amount of heat energy absorbed will be the same whether the gas was heated directly or if it was cooled and then heated again.

Path function:
A path function is a property or quantity that depends on the path taken to reach a particular state. The value of a path function is not only determined by the initial and final states but also by the process or path followed to reach those states.

Molar heat capacity as a path function:
Molar heat capacity is not a path function because it does not depend on the specific process or path taken to reach a particular temperature change. It only depends on the initial and final states of the system, making it a state function.

Conclusion:
Molar heat capacity is a state function because it only depends on the initial and final states of the system. It is independent of the path taken to reach those states. This distinction is important in thermodynamics as it helps in simplifying calculations and understanding the behavior of substances during temperature changes.