Four moles of hydrogen, 2 moles of helium and 1 mole of water form an ideal gas mixture. What is the molar specific heat at constant pressure of mixture ?

Options

A. `(16)/(7)R`
B. `(7 R)/(16)`
C. `R`
D. `(23)/(7)R`

Explanation

To find the molar specific heat at constant pressure (C_p) of the gas mixture, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components and Their Moles**:
- Hydrogen (H₂): 4 moles
- Helium (He): 2 moles
- Water (H₂O): 1 mole

2. **Determine the Specific Heat Capacities (C_p) of Each Gas**:
- For Hydrogen (H₂), which is diatomic:
\[ C_{p(H_2)} = \frac{7R}{2} \]
- For Helium (He), which is monoatomic:
\[ C_{p(He)} = \frac{5R}{2} \]
- For Water (H₂O), which can be treated as a polyatomic gas:
- Degrees of freedom = 3 (translation) + 3 (rotation) = 6
- Therefore,
\[ C_{v(H_2O)} = \frac{6R}{2} = 3R \]
- Hence,
\[ C_{p(H_2O)} = C_{v(H_2O)} + R = 3R + R = 4R \]

3. **Calculate the Total Contribution of Each Gas to C_p**:
- Contribution from Hydrogen:
\[ n_1 C_{p(H_2)} = 4 \times \frac{7R}{2} = 14R \]
- Contribution from Helium:
\[ n_2 C_{p(He)} = 2 \times \frac{5R}{2} = 5R \]
- Contribution from Water:
\[ n_3 C_{p(H_2O)} = 1 \times 4R = 4R \]

4. **Sum the Contributions**:
- Total contribution:
\[ \text{Total } C_p = 14R + 5R + 4R = 23R \]

5. **Calculate the Total Number of Moles**:
- Total moles:
\[ n_{total} = n_1 + n_2 + n_3 = 4 + 2 + 1 = 7 \]

6. **Calculate the Molar Specific Heat at Constant Pressure of the Mixture**:
- Molar specific heat at constant pressure (C_p) of the mixture:
\[ C_{p(mixture)} = \frac{Total \, C_p}{n_{total}} = \frac{23R}{7} \]

### Final Answer:
\[ C_{p(mixture)} = \frac{23R}{7} \]