A perfect gas is compressed adiabatically. In that state the value of `Delta P//P` will be :

Options

A. `(1)/(gamma). (Delta V)/(V)`
B. `(Delta V)/(V)`
C. `- gamma (Delta V)/(V)`
D. `+ gamma (Delta V)/(V)`

Answer

To solve the problem of finding the value of \(\Delta P / P\) for a perfect gas that is compressed adiabatically, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process for a perfect gas, the relationship between pressure (P) and volume (V) is given by:
\[ P V^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio \(C_p/C_v\).

2. **Differentiate the Adiabatic Condition**:
To find the relationship between changes in pressure and volume, we differentiate the equation \(P V^\gamma = K\) (where K is a constant):
\[ d(P V^\gamma) = 0 \]
Using the product rule, we have:
\[ P \cdot d(V^\gamma) + V^\gamma \cdot dP = 0 \]

3. **Differentiate \(V^\gamma\)**:
We can express \(d(V^\gamma)\) as:
\[ d(V^\gamma) = \gamma V^{\gamma - 1} dV \]
Substituting this into the differentiated equation gives:
\[ P \cdot (\gamma V^{\gamma - 1} dV) + V^\gamma \cdot dP = 0 \]

4. **Rearranging the Equation**:
Rearranging the equation leads to:
\[ P \gamma V^{\gamma - 1} dV = -V^\gamma dP \]
Dividing both sides by \(PV^\gamma\) gives:
\[ \frac{\gamma dV}{V} = -\frac{dP}{P} \]

5. **Expressing \(\Delta P / P\)**:
This can be rewritten as:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]
Thus, integrating gives:
\[ \Delta P = -\gamma \frac{P \Delta V}{V} \]

6. **Final Expression for \(\Delta P / P\)**:
Therefore, we can express \(\Delta P / P\) as:
\[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]

### Final Result:
The value of \(\Delta P / P\) for a perfect gas compressed adiabatically is:
\[ \frac{\Delta P}{P} = -\gamma \frac{\Delta V}{V} \]