Question

A certain mass of a gas undergoes a process given by $dU = \dfrac{{dW}}{2}$. If the molar heat capacity of the gas for this process is $\dfrac{{15}}{2}R$ , then gas is:
(A) monoatomic
(B) polyatomic
(C) diatomic
(D) data insufficient

Answer 1

In this question, we are given a relation between internal energy and work done for a process and molar heat capacity of a gas for this process. Now, for determining whether this gas is monatomic, diatomic or polyatomic, we need to find the specific heat capacity of a gas using the given data by using the given information.

Formulas used:
${C_m} = \dfrac{1}{n}\dfrac{{dQ}}{{dT}}$, where ${C_m}$is molar heat capacity of a gas, $n$ is number of moles of a gas, $dQ$ is heat transfer and $dT$ is change in temperature.
$dU = n{C_v}dT$, where, $dU$ is change in internal energy, $n$ is number of moles of a gas, ${C_v}$ is specific heat capacity of a gas and $dT$is change in temperature.
The first law of thermodynamics: $dQ = dU + dW$, where, $dQ$ is heat transfer, $dU$ is change in internal energy and $dW$is work done during the process.

Complete step by step answer: The given process is
$dU = \dfrac{{dW}}{2}$
$\Rightarrow dW = 2dU$
It is also given that molar heat capacity of the gas for this process is
${C_m} = \dfrac{1}{n}\dfrac{{dQ}}{{dT}} = \dfrac{{15}}{2}R$
As per the first law of thermodynamics
$dQ = dU + dW$
But, $dW = 2dU$ for the process $3R$.
$dQ = dU + 2dU \\\ \Rightarrow dQ = 3dU \\\$
We know that $dU = n{C_v}dT$
$dQ = 3n{C_v}dT \\\ \Rightarrow \dfrac{1}{n}\dfrac{{dQ}}{{dT}} = 3{C_v} \\\$
It is given that molar heat capacity of the gas for this process
${C_m} = \dfrac{1}{n}\dfrac{{dQ}}{{dT}} = \dfrac{{15}}{2}R$
$\Rightarrow \dfrac{{15}}{2}R = 3{C_v} \\\ \Rightarrow {C_v} = \dfrac{5}{2}R \\\$
This is the specific heat capacity of diatomic gas.

Thus, in the gas given in the question is diatomic gas.Hence, option C is the right choice.

Note: In the given case, we got the value ${C_v} = \dfrac{5}{2}R$ which is the heat capacity of diatomic gas.If the gas is monatomic, its specific heat capacity is
${C_v} = \dfrac{3}{2}R$ and If the gas is monatomic, its specific heat capacity is
${C_v} = \dfrac{6}{2}R$ which is $3R$.