The slope of indicator curve in adiabatic change relative to volume axis is -

Options

A. `P//V^(gamma)`
B. `(P^(gamma)//V^(gamma - 1)`
C. `(P)/(gamma(V))`
D. `- gamma ((P)/(V))`

Answer

To find the slope of the indicator curve in an adiabatic change relative to the volume axis, we will derive the expression step by step.

### Step 1: Understand the Adiabatic Process
In an adiabatic process, the relationship between pressure (P) and volume (V) can be expressed as:
\[ PV^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v).

### Step 2: Differentiate the Adiabatic Equation
To find the slope \(\frac{dP}{dV}\), we differentiate the equation \(PV^\gamma = \text{constant}\) with respect to V.

Using the product rule:
\[ \frac{d}{dV}(PV^\gamma) = P \frac{d}{dV}(V^\gamma) + V^\gamma \frac{dP}{dV} = 0 \]

### Step 3: Differentiate \(V^\gamma\)
The differentiation of \(V^\gamma\) is:
\[ \frac{d}{dV}(V^\gamma) = \gamma V^{\gamma - 1} \]

### Step 4: Substitute the Differentiation into the Equation
Substituting this back into our differentiated equation gives:
\[ P \cdot \gamma V^{\gamma - 1} + V^\gamma \frac{dP}{dV} = 0 \]

### Step 5: Rearranging the Equation
Now, rearranging the equation to isolate \(\frac{dP}{dV}\):
\[ V^\gamma \frac{dP}{dV} = -P \cdot \gamma V^{\gamma - 1} \]

### Step 6: Solve for \(\frac{dP}{dV}\)
Dividing both sides by \(V^\gamma\):
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]

### Conclusion
Thus, the slope of the indicator curve in an adiabatic change relative to the volume axis is:
\[ \frac{dP}{dV} = -\gamma \frac{P}{V} \]