Question

One mole of an ideal monatomic gas undergoes a process described by the equation $P{{V}^{3}}=\text{ constant}$. The heat capacity of the gas during this process is:

A.)$R$
B.)$\dfrac{3}{2}R$
C.)$\dfrac{5}{2}R$
D.)$2R$

Answer 1

Hint: A Polytropic process is given by the equation $P{{V}^{x}}=\text{constant}$ and $x$ here is called a polytropic index. Specific heat and Heat capacity can be determined using the equation of polytropic process.

Formula used:

$C={{C}_{v}}+\dfrac{R}{1-x}$

Complete step by step answer:
A polytropic process obeys a specific type of equation $P{{V}^{x}}=\text{constant}$ where $P$ is the pressure of the gas, $V$ is the volume of gas and $x$ is polytropic constant.

Heat capacity is the amount of heat required to raise the temperature of a certain amount of substance by 1 kelvin. For gases, the molar heat capacity $C$ is the heat required to increase the temperature of 1 mole gas by 1 kelvin.

Heat capacity in polytropic process is given by:

$C={{C}_{v}}+\dfrac{R}{1-x}$

Where ${{C}_{v}}$ is the value of heat capacity at constant volume, $R$ is the gas constant and $x$ is polytropic index.

In the given equation we have, $P{{V}^{3}}=\text{ constant}$

Comparing above equation with the equation of polytropic process we get $x=3$

Also, for monoatomic gas, ${{C}_{v}}=\dfrac{3}{2}R$

By applying $C={{C}_{v}}+\dfrac{R}{1-x}$, we get

$C=\dfrac{3}{2}R+\dfrac{R}{1-3}=\dfrac{3}{2}R-\dfrac{1}{2}R=R$
$C=R$

Heat capacity of the gas during a given process is $R$.

Hence, the correct option is A.

Note:
Students should not get confused between the terms Molar heat capacity and Specific heat capacity. Molar heat capacity is the amount of heat required to raise the temperature of one mole of a pure substance by one kelvin, while Specific heat capacity is the amount of heat needed to raise the temperature of one gram of a pure substance by one kelvin.