22 g of carbon dioxide at `27^(@)C` is mixed in a closed container with 16 g of oxygen at `37^(@)C`. If both gases are considered as ideal gases, then the temperature of the mixture is

Options

A. `24.2^(@)C`
B. `28.5^(@)C`
C. `32^(@)C`
D. `33.5^(@)C`

Answer

To solve the problem of finding the final temperature of a mixture of carbon dioxide and oxygen, we can use the principle of conservation of energy, which states that the heat lost by the hotter gas will be equal to the heat gained by the cooler gas.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Mass of CO₂ (m₁) = 22 g
- Initial temperature of CO₂ (T₁) = 27°C
- Mass of O₂ (m₂) = 16 g
- Initial temperature of O₂ (T₂) = 37°C

2. **Calculate Molar Masses:**
- Molar mass of CO₂ = 44 g/mol
- Molar mass of O₂ = 32 g/mol

3. **Calculate Number of Moles:**
- Number of moles of CO₂ (n₁) = m₁ / Molar mass of CO₂ = 22 g / 44 g/mol = 0.5 mol
- Number of moles of O₂ (n₂) = m₂ / Molar mass of O₂ = 16 g / 32 g/mol = 0.5 mol

4. **Determine Specific Heat Capacities:**
- Specific heat capacity at constant volume for CO₂ (Cᵥ₁) = 3R (where R is the universal gas constant)
- Specific heat capacity at constant volume for O₂ (Cᵥ₂) = (5/2)R

5. **Set Up the Energy Balance Equation:**
- Heat gained by CO₂ = Heat lost by O₂
- \( n₁ \cdot Cᵥ₁ \cdot (T_f - T₁) = n₂ \cdot Cᵥ₂ \cdot (T₂ - T_f) \)

6. **Substitute the Values:**
- \( 0.5 \cdot (3R) \cdot (T_f - 27) = 0.5 \cdot \left(\frac{5}{2}R\right) \cdot (37 - T_f) \)

7. **Cancel Common Terms:**
- Cancel \( 0.5 \) and \( R \) from both sides:
- \( 3(T_f - 27) = \frac{5}{2}(37 - T_f) \)

8. **Expand and Rearrange the Equation:**
- \( 3T_f - 81 = \frac{5}{2} \cdot 37 - \frac{5}{2}T_f \)
- \( 3T_f + \frac{5}{2}T_f = \frac{5}{2} \cdot 37 + 81 \)

9. **Combine Like Terms:**
- Convert \( 3T_f \) to a fraction: \( \frac{6}{2}T_f + \frac{5}{2}T_f = \frac{11}{2}T_f \)
- \( \frac{11}{2}T_f = \frac{185}{2} \)
- \( T_f = \frac{185}{11} \approx 16.82 \)

10. **Final Calculation:**
- \( T_f \approx 31.5°C \)

11. **Conclusion:**
- The final temperature of the mixture is approximately 32°C.