Question

Define molar specific heat capacity at constant volume and at constant pressure. Derive the relationship between these two

Answer 1

Heat Capacity

It is the amount of heat energy required to raise the temperature of a substance through 1 K or 1°C. It is the amount of heat energy required per unit temperature. It is denoted by the letter $C.$

By definition,

$C=\dfrac {\Delta q}{\Delta T}$

If $C$ is calculated for a very small rise in temperature,

$C=\dfrac {dq}{dT}$

Molar Heat Capacity at Constant Volume

It is the amount of heat energy required to raise the temperature of one mole of a substance through 1 K or 1°C at constant volume. It is denoted by $C_V.$

By definition,

$C_V=\dfrac {dq_V}{dT}$

But, $q_V=U,$ internal energy.

Then,

$C_V=\dfrac {dU}{dT}$

Molar Heat Capacity at Constant Pressure

It is the amount of heat energy required to raise the temperature of one mole of a substance through 1 K or 1°C at constant pressure. It is denoted by $C_P.$

By definition,

$C_P=\dfrac {dq_P}{dT}$

But, $q_P=H,$ enthalpy.

Then,

$C_P=\dfrac {dH}{dT}$

Relationship between Molar Heat Capacities at Constant Volume and Pressure - In the Case of One Mole of an Ideal Gas

We have the equation for enthalpy.

$H=U+PV$

In the case of one mole of an ideal gas, we have,

$PV=RT$

Then,

$H=U+RT$

Differentiating both sides of the equation with temperature,

$\dfrac {dH}{dT}=\dfrac {dU}{dT}+\dfrac {d}{dT}(RT)\\\\\\C_P=C_V+R\cdot\dfrac {dT}{dT}\\\\\\C_P=C_V+R\\\\\\\boxed{\boxed{C_P-C_V=R}}$

Thus, the difference between the molar heat capacities at constant volume and pressure always equals R, the universal gas constant.

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