Assertion. For`CO_(2)` , the ratio of the molar heat capacities `( C_(p) //C_(v))` is 1.40.
Reason . For `CO_(2) , C_(v) = (5)/(2) R` and `C_(p) = ( 7)/( 2) R`

Options

A. If both A and R are true,andR is the true explanation of A.
B. If both A and R are true,but R is not the true explanation of A.
C. If A is true, but R is false
D. If both A and R are false.

Answer

To solve the problem, we need to analyze the assertion and the reason provided regarding the molar heat capacities of carbon dioxide (CO₂).

### Step-by-Step Solution:

1. **Understanding the Heat Capacities**:
- The heat capacities at constant pressure (Cₚ) and constant volume (Cᵥ) are related by the equation:
\[ Cₚ - Cᵥ = R \]
- Here, R is the universal gas constant.

2. **Degrees of Freedom for CO₂**:
- CO₂ is a linear molecule, which means it has:
- Translational degrees of freedom (Fₜ): 3 (movement in x, y, z directions)
- Rotational degrees of freedom (Fᵣ): 2 (rotation around two axes perpendicular to the molecular axis)
- Vibrational degrees of freedom (Fᵥ): 4 (considering the two bonds and the linear structure)
- The total degrees of freedom (F) can be calculated as:
\[ F = Fₜ + Fᵣ + Fᵥ = 3 + 2 + 4 = 9 \]

3. **Calculating Cᵥ**:
- The molar heat capacity at constant volume (Cᵥ) can be calculated using the formula:
\[ Cᵥ = \frac{F}{2} R \]
- Substituting the total degrees of freedom:
\[ Cᵥ = \frac{9}{2} R = \frac{9R}{2} \]

4. **Calculating Cₚ**:
- Using the relationship between Cₚ and Cᵥ:
\[ Cₚ = Cᵥ + R = \frac{9R}{2} + R = \frac{9R}{2} + \frac{2R}{2} = \frac{11R}{2} \]

5. **Calculating the Ratio (Cₚ/Cᵥ)**:
- Now, we can calculate the ratio of Cₚ to Cᵥ:
\[ \frac{Cₚ}{Cᵥ} = \frac{\frac{11R}{2}}{\frac{9R}{2}} = \frac{11}{9} \approx 1.22 \]

6. **Evaluating the Assertion and Reason**:
- The assertion states that the ratio \( \frac{Cₚ}{Cᵥ} \) for CO₂ is 1.40, which we calculated to be approximately 1.22. Thus, the assertion is **false**.
- The reason states that \( Cᵥ = \frac{5}{2} R \) and \( Cₚ = \frac{7}{2} R \), which is also incorrect based on our calculations. Therefore, the reason is **false** as well.

### Conclusion:
Both the assertion and the reason are false.