Question

The internal energy $\left(U\right)$, pressure $\left(P\right)$ and volume $\left(V\right)$ of an ideal gas are related as $U=3PV+4$. The gas is:

• A. Polyatomic only
• B. Monoatomic only
• C. Either monoatomic or diatomic
• D. Diatomic only

Answer 1

The correct option is A

Polyatomic only

Step 1: Given that:

The given ideal gas equation $U=3PV+4$

Where internal energy$\left(U\right)$, pressure$\left(P\right)$, and volume$\left(V\right)$.

Step 2: Formula used:

For ideal gas-

$PV=nRT$……….(1)

Where $n$ is the number of moles, $R$ is the universal gas constant,

The degree of freedom will be-

$$\begin{gathered} f=\frac{2\times U}{nRT} \\ f=\frac{2U}{PV}..............\left(2\right) \end{gathered}$$ From equation (1)

Step 3: Calculation of degree of freedom:

Substituting the known values in equation (2)-

$$\begin{gathered} f=\frac{2\times \left(3PV+4\right)}{PV} \\ f=6+\frac{8}{PV} \end{gathered}$$

Since the degree of freedom is more than $6$ therefore the gas is polyatomic.

A polyatomic gas has $3$ translational or $3$ rotational degree of freedom other than this it has a certain number of vibrational modes. Thus its degree of freedom will be more than $6$.

Hence, option A is the correct answer.