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 total translational kinetic energy is in the ratio of 3:1.

### Step-by-step Solution:

1. **Understanding the Kinetic Energy Formula**:
The translational kinetic energy (TKE) of a gas can be expressed as:
\[ \text{TKE} = \frac{3}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Setting Up the Ratio**:
Given that the total translational kinetic energy of methane (TKE₁) to ethane (TKE₂) is in the ratio of 3:1, we can write:
\[ \frac{\text{TKE}_1}{\text{TKE}_2} = \frac{3}{1} \]
Substituting the formula for TKE, we have:
\[ \frac{\frac{3}{2} n_1 R T_1}{\frac{3}{2} n_2 R T_2} = \frac{3}{1} \]

3. **Simplifying the Equation**:
The constants \(\frac{3}{2}\) and \(R\) cancel out, leading to:
\[ \frac{n_1 T_1}{n_2 T_2} = \frac{3}{1} \]

4. **Considering Equal Masses**:
Since we have equal masses of methane and ethane, we can denote the mass of each gas as \(w\). The number of moles \(n\) can be expressed as:
\[ n = \frac{w}{M} \]
where \(M\) is the molar mass of the gas. Thus, we have:
\[ n_1 = \frac{w}{M_1} \quad \text{and} \quad n_2 = \frac{w}{M_2} \]
Therefore, substituting \(n_1\) and \(n_2\) into the equation gives:
\[ \frac{\frac{w}{M_1} T_1}{\frac{w}{M_2} T_2} = \frac{3}{1} \]
The mass \(w\) cancels out:
\[ \frac{M_2 T_1}{M_1 T_2} = 3 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{T_1}{T_2} = \frac{3 M_1}{M_2} \]

6. **Substituting Molar Masses**:
The molar mass of methane (CH₄) is \(M_1 = 16 \, \text{g/mol}\) and the molar mass of ethane (C₂H₆) is \(M_2 = 30 \, \text{g/mol}\). Substituting these values:
\[ \frac{T_1}{T_2} = \frac{3 \times 16}{30} = \frac{48}{30} = \frac{8}{5} \]

7. **Final Result**:
Thus, the ratio of temperatures \(T_1 : T_2\) is:
\[ T_1 : T_2 = 8 : 5 \]