Two gases X `(mol. wt. M_(X))` and Y `(mol. wt. M_(Y) , M_(Y) gt M_(X))` are at the same temperature T in two different containers. Their root mean square velocities are `C_(X) and C_(Y)` respectively. If the. average kinetic energies per molecule of two gases X and Y are `E_(X) and E_(Y)` respectively, then which of the following relation (s) is (are) true ?

Options

A. `E_(x) gt E_(y)`
B. `C_(x) gt C_(Y)`
C. `E_(x)=E_(Y)=(3)/(2)RT`
D. `E_(x)=E_(Y)=(3)/(2)k_(B)T`

Answer

To solve the problem, we need to analyze the relationship between the root mean square velocities and the average kinetic energies of the two gases X and Y.

### Step-by-Step Solution:

1. **Understanding Root Mean Square Velocity**:
The root mean square velocity (C) of a gas is given by the formula:
\[ C = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

2. **Applying the Formula for Both Gases**:
For gas X, the root mean square velocity \( C_X \) is:
\[ C_X = \sqrt{\frac{3RT}{M_X}} \]
For gas Y, the root mean square velocity \( C_Y \) is:
\[ C_Y = \sqrt{\frac{3RT}{M_Y}} \]

3. **Comparing the Velocities**:
Given that \( M_Y > M_X \), we can compare \( C_X \) and \( C_Y \):
Since \( M_Y \) is greater than \( M_X \), it follows that:
\[ C_Y < C_X \]
Therefore, the root mean square velocity of gas X is greater than that of gas Y.

4. **Average Kinetic Energy per Molecule**:
The average kinetic energy (E) per molecule of a gas is given by:
\[ E = \frac{3}{2} k_B T \]
where \( k_B \) is Boltzmann's constant.

5. **Calculating Average Kinetic Energies for Both Gases**:
For both gases X and Y, since they are at the same temperature \( T \):
\[ E_X = \frac{3}{2} k_B T \]
\[ E_Y = \frac{3}{2} k_B T \]
This means:
\[ E_X = E_Y \]

6. **Conclusion**:
From the above analysis, we can conclude:
- \( C_X > C_Y \) (the root mean square velocity of gas X is greater than that of gas Y)
- \( E_X = E_Y \) (the average kinetic energy per molecule of both gases is the same)

### Final Relations:
- \( C_X > C_Y \)
- \( E_X = E_Y \)