Analyzing the Process Relationship

The given process for an ideal gas is defined by the relationship $P = kT$, where $P$ is pressure, $T$ is temperature, and $k$ is a constant.

We use the ideal gas law: $PV = nRT$ where $V$ is volume, $n$ is the number of moles, and $R$ is the universal gas constant.

Substitute the given process condition ($P = kT$) into the ideal gas law:

Dividing both sides by $kT$ (assuming $T \neq 0$ and $k \neq 0$), we get:

Since $n$, $R$, and $k$ are constants, this equation shows that the volume $V$ remains constant throughout the process.

Determining Molar Heat Capacity

A process where volume is constant is known as an isochoric process.

The molar heat capacity ($C$) for any process is defined as:

where $dQ$ is the heat added to the gas.

According to the first law of thermodynamics:

where $dU$ is the change in internal energy and $dW$ is the work done by the gas.

For an ideal gas, the change in internal energy depends only on temperature:

where $C_V$ is the molar heat capacity at constant volume.

The work done by the gas is given by:

Since the volume is constant in this process ($V = \text{constant}$), the change in volume $dV = 0$. Therefore, the work done is:

Substituting $dU$ and $dW$ into the first law:

Now, substitute this expression for $dQ$ into the definition of molar heat capacity $C$:

Simplifying the expression gives:

Thus, for an ideal gas undergoing the process $P = kT$, the molar heat capacity $C$ is equal to the molar heat capacity at constant volume ($C_V$).