Question

We give$\mathbf{70}\mathit{J}$as heat to a diatomic gas, which then expands at constant pressure. The gas molecules rotate but do not oscillate. By how much does the internal energy of the gas increase?

Short answer

.

Step-by-step solution

Given data

Heat energy given to a diatomic gas is$\text{Q}=70\text{J}$

Understanding the concept

The expression for the amount of energy transferred to the gas is given by,

$Q=n{C}_V\Delta T$

Here Q is the amount of energy transferred to gas,n is the number of moles, $C_v$is the specific heat capacity at constant volume and $\Delta T$is the temperature difference.

Calculate by how much the internal energy of the gas increases

$\Delta {\text{E}}_{int}={\text{nC}}_V\Delta \text{T}$

But, for diatomic gas$,{\text{C}}_V=\frac{5}{2}\text{R}$

Hence,

$\Delta E_{int}=\frac{5}{2}nR\Delta T$ …… (i)

Also,$Q=n{C}_P\Delta T$

But, for diatomic gas$C_P=\frac{7}{2}R$

Hence,

$\begin{array}{c}Q=\frac{7}{2}nR\Delta T\\ nR\Delta T=\frac{2}{7}Q\end{array}$

Substitute$\frac{2}{7}Q$for $nR\Delta T$into the equation (i)

$\Delta {\text{E}}_{int}=\frac{5}{2}\left(\frac{2}{7}\text{Q}\right)$

Substitute$70J$ for Q into the above equation,

$\Rightarrow \Delta {\text{E}}_{int}=50\text{\hspace{0.17em}J}$

Therefore, the increase in the internal energy is$50\text{J}.$