Equal masses of methane and ethane have their total translational kinetic energy in the ratio `3:1`, then their temperature are in the ratio is

Options

A. `5:8`
B. `45:8`
C. `15:8`
D. `8:5`

Answer

To solve the problem, we need to find the ratio of temperatures of methane and ethane given that their translational kinetic energy is in the ratio of 3:1.

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy Formula**:
The translational kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

2. **Setting Up the Equations**:
Let:
- \( KE_M \) = translational kinetic energy of methane
- \( KE_E \) = translational kinetic energy of ethane

According to the problem, we have:
\[ \frac{KE_M}{KE_E} = \frac{3}{1} \]

Using the kinetic energy formula for both gases, we can write:
\[ \frac{KE_M}{KE_E} = \frac{\frac{3}{2} n_M R T_M}{\frac{3}{2} n_E R T_E} \]

This simplifies to:
\[ \frac{n_M T_M}{n_E T_E} = 3 \]

3. **Finding the Number of Moles**:
The number of moles \( n \) can be calculated using the formula:
\[ n = \frac{W}{M} \]
where \( W \) is the weight and \( M \) is the molar mass.

For methane (CH₄):
- Molar mass \( M_M = 16 \, \text{g/mol} \)
- Number of moles of methane:
\[ n_M = \frac{W_M}{16} \]

For ethane (C₂H₆):
- Molar mass \( M_E = 30 \, \text{g/mol} \)
- Number of moles of ethane:
\[ n_E = \frac{W_E}{30} \]

4. **Substituting into the Ratio**:
Since the problem states that equal masses of methane and ethane are used, we have \( W_M = W_E = W \). Thus:
\[ n_M = \frac{W}{16} \quad \text{and} \quad n_E = \frac{W}{30} \]

Now substituting these values into the kinetic energy ratio:
\[ \frac{\frac{W}{16} T_M}{\frac{W}{30} T_E} = 3 \]

The \( W \) cancels out:
\[ \frac{30 T_M}{16 T_E} = 3 \]

5. **Solving for the Temperature Ratio**:
Rearranging gives:
\[ 30 T_M = 48 T_E \]
Thus:
\[ \frac{T_M}{T_E} = \frac{48}{30} = \frac{8}{5} \]

### Final Answer:
The ratio of temperatures \( T_M : T_E \) is:
\[ \boxed{8 : 5} \]