Question

What will be the ratio of adiabatic to isothermal elasticity of a diatomic gas?
$\begin{align}
  & A.1.67 \\
 & B.1.4 \\
 & C.1.33 \\
 & D.1.27 \\
\end{align}$

Answer 1

Hint: The elasticity can be found by taking the ratio of the specific heat capacity at constant pressure to the specific heat capacity at constant volume. The specific heat at constant pressure is the sum of the universal gas constant and the specific heat capacity at constant volume. This information will help you in solving this question.

Complete answer:
The ratio of adiabatic to isothermal elasticity of a diatomic gas can be found using the equation which can be written as,
$\gamma =\dfrac{{{C}_{P}}}{{{C}_{V}}}$
Where ${{C}_{P}}$ be the specific heat at constant pressure and ${{C}_{V}}$ be the specific heat capacity at constant volume.
As we all know that the specific heat at constant pressure is the sum of the universal gas constant and the specific heat capacity at constant volume. That is,
${{C}_{P}}={{C}_{V}}+R$
Substituting this in the above equation will give,
\[\gamma =\dfrac{{{C}_{V}}+R}{{{C}_{V}}}\]
The specific heat at constant volume for a diatomic gas can be found by the equation,
\[{{C}_{V}}=\dfrac{5}{2}R\]
Substituting this in the equation obtained above will give,
\[\begin{align}
  & \gamma =\dfrac{\dfrac{5}{2}R+R}{\dfrac{5}{2}R}=\dfrac{\dfrac{7}{2}}{\dfrac{5}{2}}=\dfrac{7}{5} \\
 & \Rightarrow \gamma =1.4 \\
\end{align}\]

So, the correct answer is “Option B”.

Note:
Adiabatic elasticity is defined as the gas which has been compressed in such a way that heat has been neither allowed to enter the system nor did it allow it to leave the system. Truly speaking the elasticity which is in correspondence to the adiabatic condition is known as the adiabatic elastic. If the gas is being compressed such that the temperature is kept at constant which is basically the isothermal conditions then the corresponding volume elasticity is referred to as the isothermal elasticity.