Question

The average degree of freedom per molecule for a gas is 6. The gas performs $25\,{\text{J}}$ of work when it expands at constant pressure. The heat absorbed by the gas is
A.$75\,{\text{J}}$
B.$100\,{\text{J}}$
C.$150\,{\text{J}}$
D.$125\,{\text{J}}$

Answer 1

Use the formulae for specific heat of the gas at constant pressure and volume. Also use the formula for heat absorbed by the gas in terms of the gas constant and change in temperature of the gas. Substitute the formula for work done by the gas in this formula to determine the heat absorbed by the gas.

Formulae used:
The specific heat ${C_V}$ at constant volume is given by
${C_V} = \dfrac{{fR}}{2}$ …… (1)
Here, $f$ is the degrees of freedom for the gas and $R$ is the gas constant.
The specific heat ${C_P}$ at constant pressure is given by
${C_P} = {C_V} + R$ …… (2)
Here, ${C_V}$ is the specific heat at constant volume and $R$ is the gas constant.
work done $W$ by the gas is given by
$W = nR\Delta T$ …… (3)
Here, $n$ is the number of moles of the gas, $R$ is the gas constant and $\Delta T$ is the change in temperature of the gas.
The heat $Q$ absorbed by the gas is given by
$Q = n{C_P}\Delta T$ …… (4)
Here, $n$ is the number of moles of gas, ${C_P}$ is the specific heat at constant pressure and $\Delta T$ is the change in temperature of gas.

Complete step by step answer:
We have given that the average degrees of freedom per molecule for a gas is 6 and the work done by the gas when it expands is $25\,{\text{J}}$.
$f = 6$
$\Rightarrow W = 25\,{\text{J}}$

Let us first determine the specific heat of the gas at constant volume using equation (1).
Substitute $6$ for $f$ in equation (1).
${C_V} = \dfrac{{6R}}{2}$
$\Rightarrow {C_V} = 3R$
Hence, the specific heat of the gas at constant volume is $3R$.

Now let us determine the specific heat of the gas at constant pressure using equation (2).
Substitute $3R$ for ${C_V}$ in equation (2).
${C_P} = 3R + R$
$\Rightarrow {C_P} = 4R$
Hence, the specific heat of the gas at constant pressure is $4R$.

Now we can determine the heat absorbed by the gas during its expansion.
Substitute $4R$ for ${C_P}$ in equation (4).
$Q = n4R\Delta T$
$\Rightarrow Q = 4nR\Delta T$
Substitute $W$ for $nR\Delta T$ in the above equation.
$\Rightarrow Q = 4W$
Substitute $25\,{\text{J}}$ for $W$ in the above equation.
$\Rightarrow Q = 4\left( {25\,{\text{J}}} \right)$
$\therefore Q = 100\,{\text{J}}$
Therefore, the heat absorbed by the gas is $100\,{\text{J}}$.

Hence, the correct option is B.

Note: One can also solve the same question by another way. One can use the formula for adiabatic index in terms of degrees of freedom of the gas and then use the formula for work heat absorbed by the gas in terms of the work done by the gas and the adiabatic index of the gas.