Question

The molar specific heat of hydrogen at constant volume is $5cal\,mol\,{}^{o}C$. Heat required to raise temperature of $1gm\,{{H}_{2}}$ gas by $10\,{}^{o}C$ at constant volume is-
(A). $20\,cal$
(B).$22\,cal$
(C).$25\,cal$
(D). $40\,cal$

Answer 1

Hint: Heat is required to raise the temperature of gas. The heat required or change in heat depends on the number of moles and temperature. The specific heat at constant volume gives us the heat required to raise temperature by one unit. Number of moles is calculated by dividing given mass by molecular mass.

Formulas used:
$\Delta Q=n{{C}_{V}}\Delta T$

Complete step-by-step solution:
In order to raise the temperature without any change in volume the gas must absorb some heat. The heat absorbed or change in heat of the gas is given by-
$\Delta Q=n{{C}_{V}}\Delta T$ - (1)
Here,
$\Delta Q$ is change in heat
$n$ is the number of moles of the gas
${{C}_{V}}$ is molar specific heat at constant volume
$\Delta T$ is change in temperature
The number of moles is given by-
$n=\dfrac{m}{M}$ - (2)
$m$ is the given mass
$M$ is the molecular mass of ${{H}_{2}}$ which is $2gm\,mo{{l}^{-1}}$
Given mass of ${{H}_{2}}$ is $1gm$
Substituting values in eq (2), we get,
$n=\dfrac{1}{2}=0.5\,mol$ - (3)
The change in temperature $\Delta T$ is $10\,{}^{o}C=10K$
Substituting values in eq (1), we get,
$\begin{align}
  & \Delta Q=0.5\times 5\times 10 \\
 & \Rightarrow \Delta Q=25\,cal \\
\end{align}$
Therefore, in order to raise the temperature of the gas by $10\,{}^{o}C$, heat absorbed by the gas is $25\,cal$.

Hence, the correct option is (C).

Note:
At constant volume, temperature changes when heat is absorbed or given out. Whereas, when temperature is constant, volume expands or contracts when heat changes take place. The heat capacity of a substance is given by its specific heat. The specific heat of a substance is the amount of heat required to raise its temperature by one unit. While latent heat is the heat required to change the state of a substance at constant temperature.