One mole of ideal monoatomic gas `(gamma=5//3)` is mixed with one mole of diatomic gas `(gamma=7//5)`. What is `gamma` for the mixture? `gamma` Denotes the ratio of specific heat at constant pressure, to that at constant volume

Options

A. `3//2`
B. `23//15`
C. `35//23`
D. `4//3`

Explanation

To find the value of \( \gamma \) for the mixture of one mole of ideal monoatomic gas and one mole of diatomic gas, we can use the formula for the adiabatic exponent of a mixture of gases.

### Step-by-Step Solution:

1. **Identify the Values**:
- For the monoatomic gas:
- \( n_1 = 1 \) mole
- \( \gamma_1 = \frac{5}{3} \)
- For the diatomic gas:
- \( n_2 = 1 \) mole
- \( \gamma_2 = \frac{7}{5} \)

2. **Use the Formula for Mixture**:
The formula for the adiabatic exponent \( \gamma \) of a mixture is given by:
\[ \frac{n_1 + n_2}{\gamma_{\text{mixture}} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} \]

3. **Substitute the Values**:
Substitute \( n_1 \), \( n_2 \), \( \gamma_1 \), and \( \gamma_2 \) into the equation:
\[ \frac{1 + 1}{\gamma_{\text{mixture}} - 1} = \frac{1}{\frac{5}{3} - 1} + \frac{1}{\frac{7}{5} - 1} \]

4. **Calculate \( \gamma_1 - 1 \) and \( \gamma_2 - 1 \)**:
- For the monoatomic gas:
\[ \gamma_1 - 1 = \frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3} \]
- For the diatomic gas:
\[ \gamma_2 - 1 = \frac{7}{5} - 1 = \frac{7}{5} - \frac{5}{5} = \frac{2}{5} \]

5. **Substitute Back**:
Now substitute these values back into the equation:
\[ \frac{2}{\gamma_{\text{mixture}} - 1} = \frac{1}{\frac{2}{3}} + \frac{1}{\frac{2}{5}} \]

6. **Calculate the Right Side**:
- Calculate \( \frac{1}{\frac{2}{3}} = \frac{3}{2} \)
- Calculate \( \frac{1}{\frac{2}{5}} = \frac{5}{2} \)
- So,
\[ \frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4 \]

7. **Equate and Solve for \( \gamma_{\text{mixture}} \)**:
Now we have:
\[ \frac{2}{\gamma_{\text{mixture}} - 1} = 4 \]
Cross-multiplying gives:
\[ 2 = 4(\gamma_{\text{mixture}} - 1) \]
Expanding:
\[ 2 = 4\gamma_{\text{mixture}} - 4 \]
Rearranging gives:
\[ 4\gamma_{\text{mixture}} = 6 \quad \Rightarrow \quad \gamma_{\text{mixture}} = \frac{6}{4} = \frac{3}{2} \]

8. **Final Answer**:
Therefore, the value of \( \gamma \) for the mixture is:
\[ \gamma_{\text{mixture}} = \frac{3}{2} \]