Question

For real gases ${C_P} = {C_V}$ at
(A) Critical temperature
(B )Triple point
(C) All temperature
(D) Absolute zero

Answer 1

To solve this we must know that ${C_P}$ and ${C_V}$ are the heat capacities at constant pressure and constant volume respectively. The total amount of energy in the form of heat that is required to increase the temperature of one mole of a substance by one unit is known as heat capacity.

Complete step by step solution:

We know that the total amount of energy in the form of heat that is required to increase the temperature of one mole of a substance by one unit is known as heat capacity.

The heat capacity depends on the nature, size and the composition of the substance in the system.
The equation for heat capacity is as follows:

$q = nC\Delta T$

Where $q$ is the heat supplied,

          $n$ is the number of moles,  

         $C$ is the molar heat capacity,  

        $\Delta T$ is the change in temperature.  

We know that ${C_P}$ is the heat capacity at constant pressure. ${C_P}$ is the amount of heat absorbed or released by unit mass of a substance with change in temperature at constant pressure. The change in temperature causes a change in the enthalpy of the system.

The equation for ${C_P}$ is as follows:

${C_P} = {\left( {\dfrac{{dH}}{{dT}}} \right)_P}$

Where, $dH$ is the change in enthalpy.

We know that ${C_V}$ is the heat capacity at constant volume. ${C_V}$ is the amount of heat absorbed or released by unit mass of a substance at constant volume.

The equation for ${C_V}$ is as follows:

${C_V} = {\left( {\dfrac{{dU}}{{dT}}} \right)_V}$

Where, $dU$ is the change in internal energy.

The value of ${C_P}$ is always greater than ${C_V}$.

But at absolute zero temperature ${C_P} = {C_V}$.

Thus, for real gases ${C_P} = {C_V}$ at absolute temperature.

Thus, the correct option is (D) absolute temperature.

Note: The value of ${C_P}$ is always greater than ${C_V}$. This is because $W = P\Delta V$ and at constant volume $\Delta V = 0$ and thus, work done is also equal to zero.Ratio of both the entities is known as adiabatic index.both are the heat capacity at constant pressure and heat capacity at constant volume.