The slope of isothermal and adiabatic curves are related as

Options

A. isothermal curve slope = adiabatic curve slope
B. isothermal curve slope = `gamma xx `adiabatic slope
C. adiabatic curve slope = `gamma xx ` isothermal slope
D. adiabatic curve slope = (1/2) `xx` isothermal curve slope

Answer

To find the relationship between the slopes of isothermal and adiabatic curves, we will analyze the equations governing these processes for an ideal gas.

### Step-by-Step Solution:

1. **Understand Isothermal Process**:
- For an isothermal process (constant temperature), the ideal gas law can be expressed as:
\[ PV = nRT \quad \text{(where T is constant)} \]
- Rearranging gives:
\[ PV = \text{constant} \]
- The slope of the isothermal curve can be derived by differentiating this equation.

2. **Differentiate Isothermal Equation**:
- From \(PV = \text{constant}\), we can differentiate:
\[ P \, dV + V \, dP = 0 \]
- Rearranging gives:
\[ dP = -\frac{P}{V} \, dV \]
- Thus, the slope \( \frac{dP}{dV} \) for the isothermal process is:
\[ \frac{dP}{dV} = -\frac{P}{V} \quad \text{(Equation 1)} \]

3. **Understand Adiabatic Process**:
- For an adiabatic process, the relationship is given by:
\[ PV^\gamma = \text{constant} \quad \text{(where } \gamma = \frac{C_p}{C_v} \text{)} \]
- Differentiating this gives:
\[ P \, dV + V^\gamma \, dP = 0 \]
- Rearranging gives:
\[ dP = -\frac{P}{V} \gamma \, dV \]
- Thus, the slope \( \frac{dP}{dV} \) for the adiabatic process is:
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \quad \text{(Equation 2)} \]

4. **Relate the Slopes**:
- From Equations 1 and 2, we can see that:
\[ \frac{dP}{dV} \text{ (adiabatic)} = \gamma \times \frac{dP}{dV} \text{ (isothermal)} \]
- This shows that the slope of the adiabatic curve is \(\gamma\) times the slope of the isothermal curve.

5. **Conclusion**:
- Therefore, we conclude that:
\[ \text{Slope of adiabatic curve} = \gamma \times \text{Slope of isothermal curve} \]

### Final Answer:
The slope of the adiabatic curve is related to the slope of the isothermal curve by the equation:
\[ \text{Slope}_{\text{adiabatic}} = \gamma \times \text{Slope}_{\text{isothermal}} \]