For an ideal gas `C_(p)` and `C_(v)` are related as

Options

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

Answer

To find the relationship between the heat capacities \( C_p \) (heat capacity at constant pressure) and \( C_v \) (heat capacity at constant volume) for an ideal gas, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Definitions**:
- \( C_p \) is the heat capacity at constant pressure, defined as:
\[ Q = nC_p \Delta T \]
- \( C_v \) is the heat capacity at constant volume, defined as:
\[ Q = nC_v \Delta T \]

2. **Relating Heat Capacities to Internal Energy and Enthalpy**:
- At constant volume, the heat added is equal to the change in internal energy (\( \Delta U \)):
\[ Q_v = nC_v \Delta T = \Delta U \]
- At constant pressure, the heat added is equal to the change in enthalpy (\( \Delta H \)):
\[ Q_p = nC_p \Delta T = \Delta H \]

3. **Using the First Law of Thermodynamics**:
- The relationship between enthalpy and internal energy can be expressed as:
\[ \Delta H = \Delta U + \Delta (PV) \]
- For an ideal gas, using the ideal gas equation \( PV = nRT \):
\[ \Delta (PV) = R \Delta T \]

4. **Substituting into the Enthalpy Equation**:
- Substitute \( \Delta (PV) \) into the enthalpy equation:
\[ \Delta H = \Delta U + R \Delta T \]
- Replacing \( \Delta U \) with \( nC_v \Delta T \) gives:
\[ nC_p \Delta T = nC_v \Delta T + R \Delta T \]

5. **Simplifying the Equation**:
- Dividing through by \( n \Delta T \) (assuming \( \Delta T \neq 0 \)):
\[ C_p = C_v + R \]

6. **Final Relation**:
- Rearranging gives the final relationship:
\[ C_p - C_v = R \]

### Conclusion
The relationship between the heat capacities for an ideal gas is:
\[ C_p - C_v = R \]