If 2 moles of diatomic gas and 1 mole of monatomic gas are mixed, then the ratio of specific heats for the mixture is

Options

A. `(7)/(3)`
B. `(5)/(4)`
C. `(19)/(13)`
D. `(15)/(19)`

Answer

To find the ratio of specific heats (γ) for a mixture of gases, we will use the formula:

\[ \gamma_{\text{mix}} = \frac{N_1 C_{P1} + N_2 C_{P2}}{N_1 C_{V1} + N_2 C_{V2}} \]

Where:
- \(N_1\) and \(N_2\) are the number of moles of the diatomic and monatomic gases, respectively.
- \(C_{P1}\) and \(C_{P2}\) are the molar specific heats at constant pressure for the diatomic and monatomic gases, respectively.
- \(C_{V1}\) and \(C_{V2}\) are the molar specific heats at constant volume for the diatomic and monatomic gases, respectively.

### Step 1: Identify the specific heats for each gas type
For a diatomic gas:
- \(C_{P1} = \frac{7}{2} R\)
- \(C_{V1} = \frac{5}{2} R\)

For a monatomic gas:
- \(C_{P2} = \frac{5}{2} R\)
- \(C_{V2} = \frac{3}{2} R\)

### Step 2: Substitute the values into the formula
Given:
- \(N_1 = 2\) (for diatomic gas)
- \(N_2 = 1\) (for monatomic gas)

Substituting the values into the equation:

\[ \gamma_{\text{mix}} = \frac{2 \left(\frac{7}{2} R\right) + 1 \left(\frac{5}{2} R\right)}{2 \left(\frac{5}{2} R\right) + 1 \left(\frac{3}{2} R\right)} \]

### Step 3: Simplify the numerator and denominator
Numerator:
\[ = 2 \cdot \frac{7}{2} R + 1 \cdot \frac{5}{2} R = 7R + \frac{5}{2} R = \frac{14R + 5R}{2} = \frac{19R}{2} \]

Denominator:
\[ = 2 \cdot \frac{5}{2} R + 1 \cdot \frac{3}{2} R = 5R + \frac{3}{2} R = \frac{10R + 3R}{2} = \frac{13R}{2} \]

### Step 4: Calculate the ratio
Now substituting back into the formula:

\[ \gamma_{\text{mix}} = \frac{\frac{19R}{2}}{\frac{13R}{2}} = \frac{19}{13} \]

### Final Answer
The ratio of specific heats for the mixture is:

\[ \gamma_{\text{mix}} = \frac{19}{13} \]