For adiabatic process of an ideal gas the value of `(dP)/P` is equals to

Options

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

Explanation

To find the value of \(\frac{dP}{P}\) for an adiabatic process of an ideal gas, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, the relationship between pressure \(P\), volume \(V\), and the adiabatic index \(\gamma\) (which is the ratio of specific heats \(C_p/C_v\)) is given by:
\[ PV^\gamma = k \]
where \(k\) is a constant.

### Step 2: Differentiate the Adiabatic Condition
We differentiate the equation \(PV^\gamma = k\) with respect to \(V\):
\[ \frac{d}{dV}(PV^\gamma) = 0 \]
Using the product rule, we have:
\[ dP \cdot V^\gamma + P \cdot \gamma V^{\gamma-1} \cdot dV = 0 \]

### Step 3: Rearranging the Equation
From the differentiated equation, we can isolate \(dP\):
\[ dP \cdot V^\gamma = -P \cdot \gamma V^{\gamma-1} \cdot dV \]
Now, we can solve for \(\frac{dP}{P}\):
\[ \frac{dP}{P} = -\gamma \frac{V^{\gamma-1}}{V^\gamma} dV \]

### Step 4: Simplifying the Expression
This simplifies to:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

### Final Result
Thus, the value of \(\frac{dP}{P}\) for an adiabatic process of an ideal gas is:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]