Energy of all molecules of a monatomic gas having a volume `V` and pressure `P` is `3//2 PV`. The total translational kinetic energy of all molecules of a diatomic gas at the same volume and pressure is

Options

A. `1//2 PV`
B. `3//2 PV`
C. `5//2 PV`
D. `3 PV`

Explanation

To find the total translational kinetic energy of all molecules of a diatomic gas at the same volume \( V \) and pressure \( P \), we will follow these steps:

### Step 1: Understand the Kinetic Energy of Gases
For gases, the total kinetic energy can be expressed in terms of the pressure \( P \) and volume \( V \). For a monatomic gas, the total kinetic energy is given by:

\[ KE_{\text{monatomic}} = \frac{3}{2} PV \]

### Step 2: Identify the Degrees of Freedom
The degrees of freedom for a gas molecule determine how many ways the molecules can store energy.

- A **monatomic gas** has 3 degrees of freedom (translational motion in x, y, and z directions).
- A **diatomic gas** has 5 degrees of freedom (3 translational + 2 rotational).

### Step 3: Calculate the Total Kinetic Energy for Diatomic Gas
The total kinetic energy for a diatomic gas can be expressed as:

\[ KE_{\text{diatomic}} = \frac{f}{2} PV \]

where \( f \) is the degrees of freedom. For a diatomic gas, \( f = 5 \):

\[ KE_{\text{diatomic}} = \frac{5}{2} PV \]

### Step 4: Determine the Translational Kinetic Energy
However, we are specifically asked for the **translational kinetic energy** of the diatomic gas. The translational kinetic energy is associated only with the translational degrees of freedom, which is 3 for diatomic gases:

\[ KE_{\text{translational}} = \frac{3}{2} PV \]

### Final Answer
Thus, the total translational kinetic energy of all molecules of a diatomic gas at the same volume \( V \) and pressure \( P \) is:

\[ \boxed{\frac{3}{2} PV} \]