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

Options

A. a. slope of adiabatic curve =`gamma` times slope of isotharmal curve
B. b.slope of isothermal curve =`gamma` times slope of adiabatic curve
C. c.slope of adiabatic curve =`gamma^(2)` times slope of isothermal curve
D. 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 can follow these steps:

### Step 1: Understand the Isothermal Process
For an isothermal process, the product of pressure (P) and volume (V) is constant. This can be expressed mathematically as:
\[ PV = \text{constant} \]

### Step 2: Differentiate the Isothermal Equation
Differentiating the equation \( PV = \text{constant} \) gives:
\[ V dP + P dV = 0 \]
Rearranging this, we find:
\[ V dP = -P dV \]
Thus, we can express the slope \( \frac{dP}{dV} \) as:
\[ \frac{dP}{dV} = -\frac{P}{V} \]
This is our **Equation 1**.

### Step 3: Understand the Adiabatic Process
For an adiabatic process, the relationship can be expressed as:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) is the heat capacity ratio (C_p/C_v).

### Step 4: Differentiate the Adiabatic Equation
Differentiating the equation \( PV^\gamma = \text{constant} \) gives:
\[ V^\gamma dP + \gamma P V^{\gamma - 1} dV = 0 \]
Rearranging this, we find:
\[ V^\gamma dP = -\gamma P V^{\gamma - 1} dV \]
Thus, we can express the slope \( \frac{dP}{dV} \) as:
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]
This is our **Equation 2**.

### Step 5: Compare the Slopes
Now, we have the slopes for both processes:
- Isothermal: \( \frac{dP}{dV} = -\frac{P}{V} \) (Equation 1)
- Adiabatic: \( \frac{dP}{dV} = -\frac{\gamma P}{V} \) (Equation 2)

From these equations, we can see that:
\[ \frac{dP}{dV} \text{ (adiabatic)} = \gamma \times \frac{dP}{dV} \text{ (isothermal)} \]

### Conclusion
Thus, the slope of the adiabatic curve is \( \gamma \) times the slope of the isothermal curve.

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