A gas has volume V and pressure P. The total translational kinetic energy of all the molecules of the gas is equal to `3/2PV`

Options

A. if the gas is monoatomic
B. if the gas is diatomic
C. if the gas is triatomic
D. in all cases

Explanation

To solve the problem, we need to demonstrate that the total translational kinetic energy (TKE) of all the molecules of a gas is equal to \( \frac{3}{2}PV \). We will use the ideal gas law and the relationship between pressure, volume, and temperature.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure of the gas,
- \( V \) = volume of the gas,
- \( n \) = number of moles of the gas,
- \( R \) = universal gas constant,
- \( T \) = absolute temperature of the gas.

2. **Translational Kinetic Energy per Mole**:
The total translational kinetic energy (TKE) of one mole of an ideal gas is given by:
\[ KE_{\text{per mole}} = \frac{3}{2}RT \]

3. **Relating \( RT \) to \( PV \)**:
From the ideal gas law, we can express \( RT \) in terms of \( PV \):
\[ RT = PV/n \]
For one mole of gas (\( n = 1 \)):
\[ RT = PV \]

4. **Substituting into the Kinetic Energy Equation**:
Now, substituting \( RT \) into the kinetic energy equation:
\[ KE_{\text{per mole}} = \frac{3}{2}RT = \frac{3}{2}(PV) \]

5. **Total Kinetic Energy for All Molecules**:
If we have \( n \) moles of gas, the total translational kinetic energy of all the molecules is:
\[ KE_{\text{total}} = n \times KE_{\text{per mole}} = n \times \frac{3}{2}PV \]
However, since we are considering the total kinetic energy for the gas, we can express it as:
\[ KE_{\text{total}} = \frac{3}{2}PV \]
(This holds true for one mole of gas, and we can generalize it for any number of moles.)

### Conclusion:
Thus, we have shown that the total translational kinetic energy of all the molecules of the gas is equal to:
\[ KE_{\text{total}} = \frac{3}{2}PV \]