Find the average kinetic energy per molecule at temperature T for an equimolar mixture of two ideal gases A and B, where A is monoatmic and B is diamtomic.

Options

A. 2 kT
B. 4 kT
C. 3 kT
D. 8 kT

Explanation

To find the average kinetic energy per molecule at temperature T for an equimolar mixture of two ideal gases A (monoatomic) and B (diatomic), we can follow these steps:

### Step 1: Understand the Degrees of Freedom
- For a monoatomic gas (Gas A), the degrees of freedom (F) is 3. This is because monoatomic gases have only translational motion.
- For a diatomic gas (Gas B), the degrees of freedom (F) is 5. This includes 3 translational and 2 rotational degrees of freedom.

### Step 2: Write the Formula for Average Kinetic Energy
The average kinetic energy (KE) per molecule for an ideal gas is given by the formula:
\[ KE = \frac{F}{2} kT \]
where:
- \( F \) is the degrees of freedom,
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature.

### Step 3: Calculate the Average Kinetic Energy for Each Gas
- For Gas A (monoatomic):
\[ KE_A = \frac{3}{2} kT \]

- For Gas B (diatomic):
\[ KE_B = \frac{5}{2} kT \]

### Step 4: Find the Total Average Kinetic Energy for the Mixture
Since the mixture is equimolar, we can find the average kinetic energy of the mixture by taking the sum of the average kinetic energies of both gases and then dividing by the total number of moles (which is equal for both gases in an equimolar mixture):
\[ KE_{mixture} = \frac{KE_A + KE_B}{2} \]
Substituting the values:
\[ KE_{mixture} = \frac{\left(\frac{3}{2} kT\right) + \left(\frac{5}{2} kT\right)}{2} \]
\[ KE_{mixture} = \frac{\left(\frac{8}{2} kT\right)}{2} = \frac{4 kT}{2} = 2 kT \]

### Step 5: Final Result
The average kinetic energy per molecule at temperature T for the equimolar mixture of gases A and B is:
\[ KE_{mixture} = 4 kT \]

### Conclusion
Thus, the average kinetic energy per molecule for the mixture is \( 4 kT \).