For an adiabatic expansion of a perfect gas, the value of `(DeltaP)/(P)` is equal to

Options

A. `-gamma^((1)/(2))(DeltaV)/(V)`
B. `-(DeltaV)/(V)`
C. `-gamma(DeltaV)/(V)`
D. `gamma^2(DeltaV)/(V)`

Explanation

To find the value of \(\frac{\Delta P}{P}\) for an adiabatic expansion of a perfect gas, we can follow these steps:

### Step 1: Understand Adiabatic Process
An adiabatic process is one in which no heat is exchanged with the surroundings. For a perfect gas undergoing an adiabatic process, we can use the relation:
\[ P V^\gamma = \text{constant} \]
where \(P\) is pressure, \(V\) is volume, and \(\gamma\) (gamma) is the heat capacity ratio (\(\frac{C_p}{C_v}\)).

### Step 2: Differentiate the Adiabatic Condition
Starting from the equation \(P V^\gamma = \text{constant}\), we can differentiate both sides with respect to volume \(V\):
\[ \frac{d(P V^\gamma)}{dV} = 0 \]
Using the product rule, we have:
\[ V^\gamma \frac{dP}{dV} + P \gamma V^{\gamma - 1} \frac{dV}{dV} = 0 \]
This simplifies to:
\[ V^\gamma \frac{dP}{dV} + \gamma P V^{\gamma - 1} = 0 \]

### Step 3: Rearranging the Equation
Rearranging the above equation gives:
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]

### Step 4: Express \(\Delta P\) in Terms of \(\Delta V\)
Now, we can express the change in pressure \(\Delta P\) in terms of the change in volume \(\Delta V\):
\[ \Delta P = \frac{dP}{dV} \Delta V = -\frac{\gamma P}{V} \Delta V \]

### Step 5: Find \(\frac{\Delta P}{P}\)
Now, we can find \(\frac{\Delta P}{P}\):
\[ \frac{\Delta P}{P} = -\frac{\gamma P \Delta V}{PV} = -\gamma \frac{\Delta V}{V} \]
Thus, we have:
\[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]

### Conclusion
The value of \(\frac{\Delta P}{P}\) for an adiabatic expansion of a perfect gas is:
\[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]