Question

The heat capacity of a certain amount of a particular gas at constant pressure is greater than that at constant volume by $29.1\text{J/K}$. Match the items given in Column I with items given in Column II.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i. If the gas is monoatomic, heat capacity at constant volume}& \text{a.}131\text{J/K}\\ \text{ii. If the gas is monoatomic, heat capacity at constant pressure}& \text{b.}43.7\text{J/K}\\ \text{iii. If the gas is rigid diatomic, heat capacity at constant pressure}& \text{c.}72.7\text{J/K}\\ \text{iv. If the gas is vibrating diatomic, heat capacity at constant pressure}& \text{d.}102\text{J/K}\end{array}$

• A. i - b ii -a iii- b iv-c
• B. i - b ii- c iii- d iv- a
• C. i- a ii-d iii- b iv- a
• D. i- d ii-a iii- d iv- c

Answer 1

The correct option is B i - b ii- c iii- d iv- a

$i\to b;ii\to c;iii\to d;iv\to a$$C_p-C_v=29.1\text{J/K}=nR$
For molecular heat capacities
$C_p-C_v=R$
Number of moles in the gas,
$n=\frac{29.1}{8.31}=3.5\text{mol}$
For monoatomic gas,
$\frac{C_p}{C_v}=\frac{5}{3}$
$\frac{5}{3}{C}_v-C_v=29.1\text{J/K}$
$C_v=(\frac{3}{2}\times 29.1)\text{J/K}=43.7\text{J/K}$
$C_p=72.7\text{J/K}$
For rigid diatomic gas the number of degree of freedom is $5$.
$\frac{C_p}{C_v}=\frac{7}{5}$
$\frac{7}{5}{C}_v-C_v=29.1$
Solving this we get $C_v$ and then further we get
$C_p=102\text{J/K}$
For vibrating diatomic ideal gas, the total number of degrees of freedom is $7$.

Internal energy,
$U=\frac{7}{2}nRT$
$C_v=\frac{7}{2}nR$ and $C_p=\frac{9}{2}nR$
$C_p-C_v=nR\Rightarrow C_p=\frac{9}{2}\times 29.1=131\text{J/K}$