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

The correct option is 3PV/2.

Step-by-step Explanation:

1. Translational Kinetic Energy per Molecule:
According to the kinetic theory of gases, the average translational kinetic energy of a single gas molecule depends solely on the absolute temperature of the gas and is given by the formula:
$E_{\mathrm{molecule}}=\frac{3}{2}kT$
where $k$ is the Boltzmann constant and $T$ is the absolute temperature.

2. Total Translational Kinetic Energy of the Gas:
For a gas containing $N$ molecules, the total translational kinetic energy ($E_t$) is the product of the number of molecules and the average translational kinetic energy per molecule:
$E_t=N\times \frac{3}{2}kT$
Using the relation $Nk=nR$ (where $n$ is the number of moles and $R$ is the universal gas constant), this expression simplifies to:
$E_t=\frac{3}{2}nRT$
From the ideal gas law, we know that:
$PV=nRT$
Substituting $PV$ into the energy equation, we obtain:
$E_t=\frac{3}{2}PV$

3. Independence of Atomicity:
The expression for the total translational kinetic energy, $E_t=\frac{3}{2}PV$, depends only on the pressure ($P$) and volume ($V$) of the gas. It is completely independent of the atomicity of the gas (whether it is monoatomic, diatomic, or polyatomic).
While a diatomic gas has additional degrees of freedom (rotational and vibrational) that contribute to its total internal energy, its translational kinetic energy remains exactly the same as that of a monoatomic gas under identical pressure and volume conditions.

Therefore, the total translational kinetic energy of all molecules of the diatomic gas at the same volume $V$ and pressure $P$ is:
$E_t=\frac{3}{2}PV$