Consider a mixture of 3 moles of helium gas and 2 moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its `C_P//C_V` value will be:

Options

A. 67/45
B. 19/13
C. 29/19
D. 40/27

Explanation

To find the value of \( \frac{C_P}{C_V} \) for a mixture of 3 moles of helium gas and 2 moles of oxygen gas, we can follow these steps:

### Step 1: Identify the specific heat capacities
- For helium (He), which is a monatomic gas, the specific heat at constant volume \( C_V \) is given by:
\[ C_V(\text{He}) = \frac{3R}{2} \]
The specific heat at constant pressure \( C_P \) is:
\[ C_P(\text{He}) = C_V + R = \frac{3R}{2} + R = \frac{5R}{2} \]

- For oxygen (O₂), which is a diatomic gas, the specific heat at constant volume \( C_V \) is:
\[ C_V(\text{O}_2) = \frac{5R}{2} \]
The specific heat at constant pressure \( C_P \) is:
\[ C_P(\text{O}_2) = C_V + R = \frac{5R}{2} + R = \frac{7R}{2} \]

### Step 2: Calculate the total \( C_V \) and \( C_P \) for the mixture
- The total number of moles in the mixture is:
\[ N = 3 + 2 = 5 \text{ moles} \]

- The total \( C_V \) for the mixture can be calculated as:
\[ C_V(\text{mixture}) = \frac{N_{\text{He}} C_V(\text{He}) + N_{\text{O}_2} C_V(\text{O}_2)}{N} \]
Substituting the values:
\[ C_V(\text{mixture}) = \frac{3 \cdot \frac{3R}{2} + 2 \cdot \frac{5R}{2}}{5} = \frac{\frac{9R}{2} + \frac{10R}{2}}{5} = \frac{\frac{19R}{2}}{5} = \frac{19R}{10} \]

- The total \( C_P \) for the mixture can be calculated similarly:
\[ C_P(\text{mixture}) = \frac{N_{\text{He}} C_P(\text{He}) + N_{\text{O}_2} C_P(\text{O}_2)}{N} \]
Substituting the values:
\[ C_P(\text{mixture}) = \frac{3 \cdot \frac{5R}{2} + 2 \cdot \frac{7R}{2}}{5} = \frac{\frac{15R}{2} + \frac{14R}{2}}{5} = \frac{\frac{29R}{2}}{5} = \frac{29R}{10} \]

### Step 3: Calculate \( \frac{C_P}{C_V} \)
Now we can find the ratio:
\[ \frac{C_P}{C_V} = \frac{C_P(\text{mixture})}{C_V(\text{mixture})} = \frac{\frac{29R}{10}}{\frac{19R}{10}} = \frac{29}{19} \]

Thus, the value of \( \frac{C_P}{C_V} \) for the mixture is:
\[ \frac{C_P}{C_V} = \frac{29}{19} \]

### Final Answer:
\[ \frac{C_P}{C_V} = \frac{29}{19} \]