Question

Consider a gas of triatomic molecules. The molecules are assumed to be triangular and made of massless rigid rods whose vertices are occupied by atoms. The internal energy of a mole of the gas at temperature $T$ is:

• A. $\frac{3}{2}RT$
• B. $3RT$
• C. $\frac{5}{2}RT$
• D. $\frac{9}{2}RT$

Answer 1

The correct option is B

$3RT$

Step 1. Given Data,

The temperature of a mole of gas is $T$,

Let $U$ be the internal energy of a mole of gas.

Step 2. Calculate the Internal Energy $U$ of a mole of gas,

We know that Internal Energy,

$U=\left(\frac{f}{2}\right)nRT$ (where, $f$ is Degree of Freedom, $n$ is the mole of gas, $R$ is the universal gas constant, $T$ is temperature)

Since the gas is of triatomic molecules then the Degree of Freedom $f$,

$f=3N-K$ (where, $N$ is number of particles in system, $K$ is independent relation among the particle.)

Since, the gas is of triatomic molecules hence, $N=3$

$$\begin{gathered} \Rightarrow f=3N-K \\ \Rightarrow f=\left(3\times 3\right)-3 \\ \Rightarrow f=6 \end{gathered}$$

We have to calculate the Internal Energy $U$ of a mole of gas then, $n=1$because the amount of gas is one mole.

Then, Internal Energy,

$$\begin{gathered} U=\left(\frac{f}{2}\right)nRT \\ \Rightarrow U=\left(\frac{6}{2}\right)\times 1\times RT \\ \Rightarrow U=3RT \end{gathered}$$

Hence, option B is correct.