Question

${C_p}$ is always greater than ${C_v}$ for a gas. Which of the following statements provides, partly or wholly, the reason for this?
A. No work is done by a gas at a constant volume
B. When a gas absorbs heat at constant pressure, its volume must change
C. For the same change in temperature, the internal energy of a gas changes by a smaller amount at constant volume than at constant pressure
D. The internal energy of an ideal gas is function only of its temperature

Answer 1

Let us first see what are ${C_p}$ and ${C_v}$ for gases.
${C_p}$ and ${C_v}$ are both heat capacities of gases at constant pressure and constant volume.
${C_v}$ is the molar specific heat potential of a gas at constant volume is the amount of heat needed to increase the temperature of $1\,mol$ of the gas by ${1^ \circ }\,C$ at the constant volume.
${C_p}$ is the molar specific heat potential of a gas at constant pressure is the amount of heat needed to increase the temperature of $1\,mol$ of the gas by ${1^ \circ }\,C$ at the constant pressure.

Complete step by step solution:
Now we shall see why ${C_p}$ is always greater than ${C_v}$.

As the gas is heated at a constant amount, the heat is just supplied to increase the gas temperature.
The amount of the gas aside from the gas temperature increases as we heat the gas at constant pressure. It does some additional work to improve the volume as the gas expands.

So, in this case the heat is supplied to
1. Increase the gas temperature
2. Do the expansion mechanical work

As a consequence, when the pressure is kept constant, more heat is needed to raise the temperature of the gas by the same amount. Hence, ${C_p}$ is always greater than ${C_v}$.
The heat supplied at constant volume is proportional to the increase in the gas’s internal capacity. The constant pressure heat supplied is proportional to the increase in the gas’s internal energy plus the gas’s function due to increase in its length.

Hence, option B is correct.

Note: In brief we can say that ${C_p}$ is always greater than ${C_v}$ because the material expands and works as heat is applied at high pressure.