Question

Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom. The gas is maintained at a temperature of$T$. The total internal energy, $U$of a mole of this gas, and the value of $\gamma =\left(\frac{C_p}{C_v}\right)$ being given, respectively by:

• A. $U=\left(\frac{5}{2}\right)RT,\gamma =\frac{7}{5}$
• B. $U=5RT,\gamma =\frac{6}{5}$
• C. $U=5RT,\gamma =\frac{7}{5}$
• D. $U=\left(\frac{5}{2}\right)RT,\gamma =\frac{6}{5}$

Answer 1

The correct option is A

$U=\left(\frac{5}{2}\right)RT,\gamma =\frac{7}{5}$

Step 1. Given data:

1. The temperature of gas=$T$.
2. The total internal energy of a mole of this gas= $U$
3. $\gamma =\left(\frac{C_p}{C_v}\right)$

Step 2. Calculating the total internal energy$U$:

As we know, the molecules of an ideal gas are having three translational degrees of freedom and two rotational degrees of freedom.

So the total degrees of freedom,

$\begin{array}{rcl}f& =& 3+2\\ & =& 5\end{array}$

now,

$$\begin{gathered} dU=C_{v}dT \\ \Rightarrow U=C_{v}T \\ \Rightarrow U=\left(\frac{f}{2}\right)RT \end{gathered}$$(where $dU=$Change in internal energy,$C_v=$ The specific heat capacity at constant volume,$C_v=\left(\frac{f}{2}\right)R$)

Then,

$\Rightarrow U=\left(\frac{5}{2}\right)RT$($R=$Universal gas constant)

Step 2. Calculating the ratio of two specific heat capacities:

we know that,

$\begin{array}{rcl}{C}_p-C_v& =& R\\ & \Rightarrow & \frac{C_p}{C_v}-1=\frac{R}{C_v}\\ & \Rightarrow & \gamma =\frac{R}{C_v}+1\\ & =& \frac{2}{f}+1\end{array}$

Then, by putting the values we will get,

$\begin{array}{rcl}& \Rightarrow & \gamma =1+\frac{2}{5}\\ & =& \frac{7}{5}\end{array}$

Thus,$U=\left(\frac{5}{2}\right)RT,\gamma =\frac{7}{5}$.

Hence, option A is the correct answer.