The specific heat of the mixture of two gases at constant volume is `(13)/(6) R`. The ratio of the number of moles of the first gas to the second gas is 1:2.
The respective gases may be

Options

A. `O_(2) and N_(2)`
B. He and Ne
C. `He and N_(2)`
D. ` N_(2) and He`

Explanation

To solve the problem, we need to determine the specific heat capacities of two gases given their mole ratio and the specific heat of their mixture.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The specific heat of the mixture at constant volume is given as \( \frac{13}{6} R \).
- The ratio of the number of moles of the first gas (\( n_1 \)) to the second gas (\( n_2 \)) is \( 1:2 \).

2. **Setting Up the Equation**:
- The specific heat of a mixture at constant volume can be calculated using the formula:
\[ C_{v, \text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]
- Given the mole ratio \( n_1:n_2 = 1:2 \), we can express \( n_1 \) as \( 1 \) and \( n_2 \) as \( 2 \).

3. **Substituting Values into the Equation**:
- Substitute \( n_1 = 1 \) and \( n_2 = 2 \) into the equation:
\[ C_{v, \text{mix}} = \frac{1 \cdot C_{v1} + 2 \cdot C_{v2}}{1 + 2} \]
- This simplifies to:
\[ C_{v, \text{mix}} = \frac{C_{v1} + 2C_{v2}}{3} \]

4. **Setting Up the Equation with Given Specific Heat**:
- We know from the problem that:
\[ \frac{C_{v1} + 2C_{v2}}{3} = \frac{13}{6} R \]
- Multiplying both sides by 3 gives:
\[ C_{v1} + 2C_{v2} = \frac{13}{2} R \]

5. **Identifying the Gases**:
- Assume the first gas is monoatomic (like Helium, He) and the second gas is diatomic (like Nitrogen, N₂).
- The molar specific heat at constant volume for a monoatomic gas is:
\[ C_{v1} = \frac{3}{2} R \]
- The molar specific heat at constant volume for a diatomic gas is:
\[ C_{v2} = \frac{5}{2} R \]

6. **Substituting Specific Heats into the Equation**:
- Substitute \( C_{v1} \) and \( C_{v2} \) into the equation:
\[ \frac{3}{2} R + 2 \left(\frac{5}{2} R\right) = \frac{3}{2} R + 5R = \frac{3}{2} R + \frac{10}{2} R = \frac{13}{2} R \]
- This confirms that the assumption about the gases is correct.

7. **Conclusion**:
- The first gas is Helium (He), a monoatomic gas, and the second gas is Nitrogen (N₂), a diatomic gas.

### Final Answer:
The respective gases may be Helium (He) and Nitrogen (N₂).

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