If one mole of a monoatomic gas `(gamma=5/3)` is mixed with one mole of a diatomic gas `(gamma=7/5)` the value of `gamma` for the mixture will be

Options

A. 1.4
B. 1.5
C. 1.53
D. 3.07

Answer

To find the value of \( \gamma \) for the mixture of one mole of a monoatomic gas and one mole of a diatomic gas, we will follow these steps:

### Step 1: Identify the specific heat capacities for each gas
For a monoatomic gas:
- \( C_V = \frac{3R}{2} \)
- \( C_P = \frac{5R}{2} \)

For a diatomic gas:
- \( C_V = \frac{5R}{2} \)
- \( C_P = \frac{7R}{2} \)

### Step 2: Calculate the mixed specific heat capacities
We will use the formulas for the mixture of gases:
- \( C_{P,\text{mix}} = \frac{N_1 C_{P1} + N_2 C_{P2}}{N_1 + N_2} \)
- \( C_{V,\text{mix}} = \frac{N_1 C_{V1} + N_2 C_{V2}}{N_1 + N_2} \)

Given that \( N_1 = N_2 = 1 \) (one mole of each gas), we can substitute the values.

### Step 3: Calculate \( C_{P,\text{mix}} \)
\[ C_{P,\text{mix}} = \frac{1 \cdot \frac{5R}{2} + 1 \cdot \frac{7R}{2}}{1 + 1} = \frac{\frac{5R}{2} + \frac{7R}{2}}{2} = \frac{\frac{12R}{2}}{2} = \frac{6R}{2} = 3R \]

### Step 4: Calculate \( C_{V,\text{mix}} \)
\[ C_{V,\text{mix}} = \frac{1 \cdot \frac{3R}{2} + 1 \cdot \frac{5R}{2}}{1 + 1} = \frac{\frac{3R}{2} + \frac{5R}{2}}{2} = \frac{\frac{8R}{2}}{2} = \frac{4R}{2} = 2R \]

### Step 5: Calculate \( \gamma_{\text{mix}} \)
Now that we have \( C_{P,\text{mix}} \) and \( C_{V,\text{mix}} \), we can calculate \( \gamma_{\text{mix}} \):
\[ \gamma_{\text{mix}} = \frac{C_{P,\text{mix}}}{C_{V,\text{mix}}} = \frac{3R}{2R} = \frac{3}{2} = 1.5 \]

### Final Answer
The value of \( \gamma \) for the mixture is \( 1.5 \).

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