Question

Given,\[{{\text{C}}_{p}}\text{ and }{{\text{C}}_{v}}\] are specific heat at constant pressure and constant volume respectively. It is observed that –
\[{{\text{C}}_{\text{p}}}\text{ - }{{\text{C}}_{\text{v}}}=a\] for hydrogen gas
\[{{\text{C}}_{p}}\text{ - }{{\text{C}}_{v}}=b\] for nitrogen gas.
The correct relation between a and b is –
\[\begin{align}
  & \text{A) }a=28b \\
 & \text{B) }a=\dfrac{1}{14}b \\
 & \text{C)}\text{ }a=b \\
 & \text{D) }a=14b \\
\end{align}\]

Answer 1

Hint: We are provided with the specific heats of hydrogen and nitrogen gas at constant pressures and constant volumes. The value for the difference between these quantities for each gas is also given. We can find the relation between them.

Complete answer:
The specific heat capacity at constant pressure (\[{{C}_{p}}\]) and the specific heat capacity at constant volume (\[{{C}_{v}}\]) of the hydrogen and nitrogen gas is given. We know that the difference between the specific heat capacity at constant pressure and the constant volume of 1 mole of the ideal gas gives the gas constant, R.
i.e.,
\[{{C}_{p}}-{{C}_{v}}=R\]
Now, we are given that for two different gases the difference is ‘a’ and ‘b’. This means that we have to use the molar mass of the gas to get the required relation between the constants ‘a’ and ‘b’. We can relate them as –
For Hydrogen gas,
\[{{m}_{H}}({{C}_{p}}-{{C}_{v}})=R\]
For Nitrogen gas,
\[{{m}_{N}}({{C}_{p}}-{{C}_{v}})=R\]
We know the molar masses of hydrogen gas and nitrogen gas to be 2 and 28 respectively.
i.e.,
\[\begin{align}
  & {{m}_{H}}=2g \\
 & {{m}_{N}}=28g \\
\end{align}\]
We can relate the constants ‘a’ and ‘b’ as –
\[\begin{align}
  & ({{C}_{p}}-{{C}_{v}})=a=\dfrac{R}{{{m}_{H}}} \\
 & \text{and,} \\
 & ({{C}_{p}}-{{C}_{v}})=b=\dfrac{R}{{{m}_{N}}} \\
\end{align}\]
The relation between ‘a’ and ‘b’ is given by equating the difference in the specific heat capacities –
i.e.,
\[\begin{align}
  & {{C}_{p}}-{{C}_{v}}=a=\dfrac{R}{2} \\
 & \text{and,} \\
 & {{C}_{p}}-{{C}_{v}}=b=\dfrac{R}{28} \\
 & \Rightarrow 2a=R \\
 & \Rightarrow 28b=R \\
 & \Rightarrow \dfrac{a}{b}=14 \\
 & \therefore a=14b \\
\end{align}\]
Therefore, the difference between the specific heat capacities at constant pressure and constant volume for the hydrogen gas and nitrogen gas is fourteen times for nitrogen as that of hydrogen.

The correct answer is option D.

Note:
The difference in the specific heat capacities for any ideal gas of a unit molar mass is a constant, i.e., the gas constant, R. Here, the heat capacities for the gas are not in unit molar masses, which needed to be simplified using the relation between the constants.