For a p-V plot, the slope of an adiabatic curve = x × slope of isothermal at the same point. Here x is :

Options

A. `R/(lamda - 1)`
B. `1/lamda`
C. `lamda`
D. `1/(lamda-1)`

Explanation

To find the value of \( x \) in the relationship between the slopes of the adiabatic and isothermal curves on a p-V plot, we can follow these steps:

### Step 1: Understand the Isothermal Process
In an isothermal process, the temperature remains constant. The equation governing this process is given by:
\[ PV = \text{constant} \]
Differentiating this equation with respect to volume \( V \) gives us:
\[ d(PV) = 0 \implies P \, dV + V \, dP = 0 \]
Rearranging this, we find:
\[ \frac{dP}{dV} = -\frac{P}{V} \]
This expression represents the slope of the isothermal curve.

### Step 2: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. The relationship for an adiabatic process is given by:
\[ PV^\lambda = \text{constant} \]
where \( \lambda \) is the adiabatic constant. Differentiating this equation gives:
\[ d(PV^\lambda) = 0 \implies V^\lambda \, dP + P \lambda V^{\lambda - 1} \, dV = 0 \]
Rearranging this leads to:
\[ \frac{dP}{dV} = -\frac{\lambda P}{V} \]
This expression represents the slope of the adiabatic curve.

### Step 3: Relate the Slopes of the Two Processes
From the expressions derived, we have:
- Slope of isothermal process: \( \frac{dP}{dV} = -\frac{P}{V} \)
- Slope of adiabatic process: \( \frac{dP}{dV} = -\frac{\lambda P}{V} \)

### Step 4: Set Up the Relationship
We can set the slopes of the two processes equal to each other in the form:
\[ \text{slope of adiabatic} = x \times \text{slope of isothermal} \]
Substituting the expressions we derived:
\[ -\frac{\lambda P}{V} = x \left(-\frac{P}{V}\right) \]

### Step 5: Solve for \( x \)
Cancelling the common terms \( -\frac{P}{V} \) from both sides (assuming \( P \) and \( V \) are not zero):
\[ \lambda = x \]
Thus, we find that:
\[ x = \lambda \]

### Final Answer
The value of \( x \) is equal to \( \lambda \), the adiabatic constant.

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