Question

For the specific heat of 1 mole of an ideal gas at constant pressure $\left( {{{\text{C}}_{\text{P}}}} \right)$ and at constant volume $\left( {{{\text{C}}_{\text{V}}}} \right)$ , which one of the given options is correct?
A. ${{\text{C}}_{\text{P}}}$ of hydrogen gas is $\dfrac{5}{2}R$ .
B. ${{\text{C}}_{\text{V}}}$ of hydrogen gas is $\dfrac{7}{2}R$ .
C. ${H_2}$ has very small value of ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ .
D. ${{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = 1.99{\text{ cal/mol K}}$ for ${H_2}$ .

Answer 1

Hint:To solve the given question, start checking each of the given options. Find out the value of ${{\text{C}}_{\text{P}}}$ and ${{\text{C}}_{\text{V}}}$ of an ideal diatomic gas. Also remember that ${{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R$ , where $R = 8.314{\text{ Jmo}}{{\text{l}}^{ - 1}}{{\text{K}}^{ - 1}}$ .



Complete answer:
We’ll check each option to determine the correct one.
Hydrogen gas is a diatomic gas. ${{\text{C}}_{\text{P}}}$ of an ideal diatomic gas is $\dfrac{7}{2}R$ , where $R$ is the universal gas constant.
Hence, the first option is incorrect.
${{\text{C}}_{\text{V}}}$ of an ideal diatomic gas is $\dfrac{5}{2}R$ , where $R$ is the universal gas constant.
Hence, the second option is incorrect as well.
Clearly, the third option is incorrect as well.
Now, for one mole of an ideal gas, we know that ${{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = R = 8.314{\text{ Jmo}}{{\text{l}}^{ - 1}}{{\text{K}}^{ - 1}}$ .
There are $4.186$ joules in one calorie, hence, converting $R$ in ${\text{cal mo}}{{\text{l}}^{ - 1}}{{\text{K}}^{ - 1}}$ ,
${{\text{C}}_{\text{P}}} - {{\text{C}}_{\text{V}}} = \dfrac{{8.314}}{{4.186}} = 1.99{\text{ cal mo}}{{\text{l}}^{ - 1}}{{\text{K}}^{ - 1}}$
Thus, the correct option is D.



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 options 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 per mole per kelvin to get the required answer.