The heat capacity at constant pressure is related to heat capacity at constant volume by the relation

Options

A. `C_(p)-C_(v)=R`
B. `C_(v)-R = C_(p)`
C. `C_(v)=-C_(p)-R`
D. `R-C_(p)=C_(v)`

Answer

To solve the question regarding the relationship between heat capacity at constant pressure (Cp) and heat capacity at constant volume (Cv), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Heat Capacities**:
- Heat capacity at constant pressure (Cp) is the amount of heat required to raise the temperature of a system by one degree Celsius at constant pressure.
- Heat capacity at constant volume (Cv) is the amount of heat required to raise the temperature of a system by one degree Celsius at constant volume.

2. **Using the Ideal Gas Relation**:
- For an ideal gas, there is a well-known relationship between Cp and Cv given by:
\[ Cp - Cv = R \]
where R is the universal gas constant.

3. **Analyzing the Given Options**:
- We need to evaluate the four statements provided in the question to identify which one correctly represents the relationship between Cp and Cv.

4. **Evaluating Each Statement**:
- **Statement 1**: \( Cp - Cv = R \)
- This is correct based on the ideal gas relation.
- **Statement 2**: \( Cv - R = Cp \)
- Rearranging gives \( Cv = Cp + R \), which is incorrect.
- **Statement 3**: \( Cv = -Cp - R \)
- Rearranging gives \( Cv + Cp = -R \), which is incorrect.
- **Statement 4**: \( Cv = Cp - R \)
- This is incorrect as it suggests \( Cv \) is less than \( Cp \) by \( R \) but does not follow the correct relation.

5. **Conclusion**:
- The only correct relationship is:
\[ Cp - Cv = R \]
Hence, the answer to the question is that the heat capacity at constant pressure is related to the heat capacity at constant volume by the relation \( Cp - Cv = R \).