Question

The value of the gas constant $\left( R \right)$ calculated from the perfect gas equation is $8.32{\text{ Joule/gm mol K}}$ , whereas its value calculated from the knowledge of ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ of the gas is ${\text{1}}{\text{.98 cal/gm mol K}}$ . What is the value of $J$ from this data?
A. $4.16{\text{ J/cal}}$
B. $4.18{\text{ J/cal}}$
C. $4.20{\text{ J/cal}}$
D. $4.22{\text{ J/cal}}$

Answer 1

Hint:${{\text{C}}_{\text{P}}}$ is the molar heat capacity of a gas at constant pressure and ${{\text{C}}_{\text{V}}}$ is the molar heat capacity of the gas at constant volume. For an ideal gas, the relation between ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ is given by \[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R\] .



Formula used:
For an ideal gas, the relation between ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ is given by

\[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R\] .

Complete answer:
For an ideal gas, the relation between ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ is given by:
\[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R\] …(1)
Here, $R$ is the universal gas constant having a value of $8.314{\text{ J}}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}$.
However, in the given question, we are provided this value of $R$ in the unit of calories.
Hence, dividing the right-hand side of the relation in equation (1) by 1 Joule to get it in the form of calories,
\[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = \dfrac{R}{J}\]
Now, the given value of \[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}}\] is ${\text{1}}{\text{.98 cal/gm mol K}}$ .
Thus, substituting all the values, we get:
${\text{1}}{\text{.98 cal/gm mol K}} = \dfrac{{8.32{\text{ Joule/gm mol K}}}}{J}$
On simplifying further, we get:
$J = \dfrac{{8.32}}{{1.98}} = 4.20{\text{ J/cal}}$
Thus, the correct option is C.


Note: To solve the given question, just remember the relation between ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ which is given by \[{{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R\] . Note that the value of $R$ provided in the question using this formula is in units of calories while using the relation, we obtain it in units of joules. Hence, perform basic maths and convert the relation in calories to get the required answer.