Question

The Poisson’s ratio ($\gamma $) for ${O_2}$ is 1.4. Select incorrect option for ${O_2}$ (R = 2$cal/molK$)
(A) ${C_V} = 5{\text{ }}cal/molK$
(B) ${C_V}{\text{ or }}{{\text{S}}_V} = 0.45cal/molK$
(C) ${C_P} = \dfrac{{{R_\gamma }}}{{\gamma - 1}}$
(D) ${C_V} = \dfrac{R}{{\gamma - 1}}$

Answer 1

Hint: Poisson’s ratio is the ratio of heat capacity at constant pressure to the heat capacity at constant volume. This formula can be given as
\[\gamma = \dfrac{{{C_P}}}{{{C_V}}}\]

Complete answer:
We know that the Poisson’s ratio is the ratio of ${C_P}$ to ${C_V}$. Thus, we can write for oxygen gas that
\[\gamma = \dfrac{{{C_P}}}{{{C_V}}} = 1.4\]
We know that ${C_V}$ is the heat capacity at constant volume and ${C_P}$ is the heat capacity measured at constant pressure.
- We also know that ${C_P}$ and ${C_V}$ are related by the following equation.
\[{C_P} - {C_V} = R\]
So, we can write that ${C_P} = R + {C_V}$
Now, if we put the value of ${C_P}$ in the formula of Poisson’s ratio, we get
\[\gamma = \dfrac{{R + {C_V}}}{{{C_V}}}\]
So, $\gamma = \dfrac{R}{{{C_V}}} + \dfrac{{{C_V}}}{{{C_V}}}$
Thus, we can write that
\[\gamma = \dfrac{R}{{{C_V}}} + 1\]
So, we obtain that ${C_V} = \dfrac{R}{{\gamma - 1}}{\text{ }}.....{\text{(1)}}$
Similarly, we can write that ${C_V} = {C_P} - R$
putting this value of ${C_V}$ in Poisson’s ratio will give
\[\gamma = \dfrac{{{C_P}}}{{{C_P} - R}}\]
From this, we obtain that ${C_P} - \dfrac{{{C_p}}}{\gamma } = R$
So, we can write that ${C_P} = \dfrac{{\gamma R}}{{\gamma - 1}}$
We are given that Poisson’s ratio for oxygen (${O_2}$) is given as 1.4 and R is the ideal gas constant.
The value of R is 8.341 $J/molK$ and 1.987$cal/molK$.
- We need to give the value of ${C_V}$ in cal/molK. So, we will put the value of R in cal/molK.
Now, we can put the value of R and $\gamma $ in equation (1) to obtain
\[{C_V} = \dfrac{{1.987}}{{1.4 - 1}} = \dfrac{{1.987}}{{0.4}} = 4.9675cal/molK\]
Thus, we can take the value of ${C_V}$ as 5 cal/molK.

Thus, the option that is not correct is (B).

Note:
Note that the value of universal gas constant (R) is 8.341 J/molK. J/molK is the SI unit of R. However, if we want to express the energy in calories, then we can take the unit of R as cal/molK.