The relation between the slope of isothemal curve and slope of adiabatic curve

Options

A. slope of adiabatic curve =`gamma` times slope of isotharmal curve
B. slope of isothermal curve =`gamma` times slope of adiabatic curve
C. slope of adiabatic curve =`gamma^(2)` times slope of isothermal curve
D. slope of isothermal curve =`gamma^(2)` times slope of adiabatic curve

Explanation

To find the relation between the slope of the isothermal curve and the slope of the adiabatic curve, we will analyze both processes step by step.

### Step 1: Understand the Isothermal Process
The isothermal process is defined by the equation:
\[ PV = nRT \]
where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature (constant during the process).

Since \( T \) is constant, the product \( PV \) remains constant.

### Step 2: Differentiate the Isothermal Equation
To find the slope of the isothermal curve, we differentiate the equation:
\[ P = \frac{nRT}{V} \]
Taking the derivative with respect to \( V \):
\[ dP = -\frac{nRT}{V^2} dV \]
Thus, the slope \( \frac{dP}{dV} \) for the isothermal process is:
\[ \frac{dP}{dV} = -\frac{nRT}{V^2} \]

### Step 3: Understand the Adiabatic Process
The adiabatic process is described by the equation:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( \frac{C_p}{C_v} \).

### Step 4: Differentiate the Adiabatic Equation
To find the slope of the adiabatic curve, we differentiate the equation:
\[ P = \frac{C}{V^\gamma} \]
where \( C \) is a constant. Taking the derivative with respect to \( V \):
\[ dP = -\gamma \frac{C}{V^{\gamma + 1}} dV \]
Thus, the slope \( \frac{dP}{dV} \) for the adiabatic process is:
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]

### Step 5: Relate the Slopes
Now we have the slopes for both processes:
- Isothermal slope: \( \frac{dP}{dV} = -\frac{nRT}{V^2} \)
- Adiabatic slope: \( \frac{dP}{dV} = -\frac{\gamma P}{V} \)

To find the relationship between the slopes, we can express the slope of the adiabatic curve in terms of the slope of the isothermal curve:
\[ \text{Slope of adiabatic} = \gamma \times \text{Slope of isothermal} \]

### Conclusion
The relation between the slope of the isothermal curve and the slope of the adiabatic curve is:
\[ \text{Slope of adiabatic curve} = \gamma \times \text{Slope of isothermal curve} \]