Equal volumes of two monoatomic gases, `A,B` at the same temperature and pressure are mixed.The ratio of specific heats`(C_(p)//C_(v))`of the mixture will be

Options

A. `0.83`
B. `1.50`
C. `3.3`
D. `1.67`

Answer

To find the ratio of specific heats \( \frac{C_p}{C_v} \) for the mixture of two monoatomic gases \( A \) and \( B \), we can follow these steps:

### Step 1: Understand the specific heats of monoatomic gases
For a monoatomic ideal gas:
- The specific heat at constant volume \( C_v \) is given by:
\[ C_v = \frac{3}{2} R \]
- The specific heat at constant pressure \( C_p \) is given by:
\[ C_p = C_v + R = \frac{3}{2} R + R = \frac{5}{2} R \]

### Step 2: Calculate the specific heats for the mixture
Since both gases \( A \) and \( B \) are monoatomic and are mixed in equal volumes:
- The specific heat at constant volume \( C_v \) for the mixture can be calculated as:
\[ C_v (\text{mixture}) = \frac{C_{vA} + C_{vB}}{2} = \frac{\frac{3}{2} R + \frac{3}{2} R}{2} = \frac{3}{2} R \]
- The specific heat at constant pressure \( C_p \) for the mixture can be calculated as:
\[ C_p (\text{mixture}) = \frac{C_{pA} + C_{pB}}{2} = \frac{\frac{5}{2} R + \frac{5}{2} R}{2} = \frac{5}{2} R \]

### Step 3: Calculate the ratio \( \frac{C_p}{C_v} \)
Now, we can find the ratio of specific heats for the mixture:
\[ \frac{C_p}{C_v} = \frac{\frac{5}{2} R}{\frac{3}{2} R} = \frac{5}{3} \]

### Step 4: Conclusion
Thus, the ratio of specific heats \( \frac{C_p}{C_v} \) for the mixture of the two monoatomic gases \( A \) and \( B \) is:
\[ \frac{C_p}{C_v} = \frac{5}{3} \approx 1.67 \]