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 misture? `gamma` Denotes the ratio of specific heat at constant pressure, to that at constant volume

Options

A. `35//23`
B. `23//15`
C. `3//2`0
D. `4//3`

Explanation

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

\[ \frac{n_1 + n_2}{\gamma_{\text{mixture}} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} \]

Where:
- \( n_1 \) and \( n_2 \) are the number of moles of the monoatomic and diatomic gases, respectively.
- \( \gamma_1 \) is the heat capacity ratio for the monoatomic gas, which is \( \frac{5}{3} \).
- \( \gamma_2 \) is the heat capacity ratio for the diatomic gas, which is \( \frac{7}{5} \).

### Step 1: Identify the values
Given:
- \( n_1 = 1 \) mole (monoatomic gas)
- \( n_2 = 1 \) mole (diatomic gas)
- \( \gamma_1 = \frac{5}{3} \)
- \( \gamma_2 = \frac{7}{5} \)

### Step 2: Substitute the values into the equation
Substituting the values into the equation:

\[ \frac{1 + 1}{\gamma_{\text{mixture}} - 1} = \frac{1}{\frac{5}{3} - 1} + \frac{1}{\frac{7}{5} - 1} \]

### Step 3: Simplify the left side
The left side simplifies to:

\[ \frac{2}{\gamma_{\text{mixture}} - 1} \]

### Step 4: Simplify the right side
Now, simplify the right side:

1. For the monoatomic gas:
\[ \frac{1}{\frac{5}{3} - 1} = \frac{1}{\frac{5}{3} - \frac{3}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

2. For the diatomic gas:
\[ \frac{1}{\frac{7}{5} - 1} = \frac{1}{\frac{7}{5} - \frac{5}{5}} = \frac{1}{\frac{2}{5}} = \frac{5}{2} \]

Adding these two results together:

\[ \frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4 \]

### Step 5: Set the equation
Now we have:

\[ \frac{2}{\gamma_{\text{mixture}} - 1} = 4 \]

### Step 6: Solve for \( \gamma_{\text{mixture}} \)
Cross-multiplying gives:

\[ 2 = 4(\gamma_{\text{mixture}} - 1) \]

Expanding this:

\[ 2 = 4\gamma_{\text{mixture}} - 4 \]

Adding 4 to both sides:

\[ 6 = 4\gamma_{\text{mixture}} \]

Dividing by 4:

\[ \gamma_{\text{mixture}} = \frac{6}{4} = \frac{3}{2} \]

### Conclusion
Thus, the value of \( \gamma \) for the mixture is:

\[ \gamma_{\text{mixture}} = \frac{3}{2} \]