For an adiabatic expansion of an ideal gas the fractional change in its pressure is equal to

Options

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

Answer

To solve the problem of finding the fractional change in pressure during an adiabatic expansion of an ideal gas, we can follow these steps:

### Step 1: Understand the Adiabatic Process
For an adiabatic process involving an ideal gas, the relationship between pressure (P) and volume (V) is given by the equation:
\[ PV^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v).

### Step 2: Differentiate the Adiabatic Condition
We need to differentiate the equation \(PV^\gamma = \text{constant}\) with respect to volume (V). Using the product rule for differentiation, we get:
\[ \frac{d}{dV}(PV^\gamma) = 0 \]
This gives us:
\[ V^\gamma \frac{dP}{dV} + P \cdot \gamma V^{\gamma - 1} \frac{dV}{dV} = 0 \]
Simplifying this, we have:
\[ V^\gamma \frac{dP}{dV} + \gamma P V^{\gamma - 1} = 0 \]

### Step 3: Solve for dP
Rearranging the above equation to isolate \(\frac{dP}{dV}\):
\[ V^\gamma \frac{dP}{dV} = -\gamma P V^{\gamma - 1} \]
Dividing both sides by \(V^\gamma\):
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]

### Step 4: Find the Fractional Change in Pressure
The fractional change in pressure is defined as:
\[ \frac{dP}{P} \]
Substituting the expression we found for \(\frac{dP}{dV}\):
\[ \frac{dP}{P} = -\frac{\gamma dV}{V} \]

### Step 5: Conclusion
Thus, the fractional change in pressure during an adiabatic expansion of an ideal gas is given by:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

### Final Answer
The correct answer is:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

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