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 regarding the fractional change in pressure during the adiabatic expansion of an ideal gas, we can follow these steps:

### Step 1: Understand the relationship for adiabatic processes
For an adiabatic process involving an ideal gas, we know that the relationship between pressure (P) and volume (V) can be expressed as:
\[ P V^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v).

### Step 2: Differentiate the equation
To find the fractional change in pressure, we need to differentiate the equation \( P V^\gamma = \text{constant} \). Taking the total differential gives:
\[ d(P V^\gamma) = 0 \]
Using the product rule, we have:
\[ V^\gamma dP + P \gamma V^{\gamma - 1} dV = 0 \]

### Step 3: Rearranging the equation
From the differentiated equation, we can rearrange it to isolate \(dP\):
\[ V^\gamma dP = -P \gamma V^{\gamma - 1} dV \]
Thus,
\[ dP = -\frac{P \gamma V^{\gamma - 1}}{V^\gamma} dV \]
This simplifies to:
\[ dP = -\frac{P \gamma}{V} dV \]

### Step 4: Finding the fractional change in pressure
Now, we can express the fractional change in pressure \( \frac{dP}{P} \):
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

### Final Result
The fractional change in pressure during the adiabatic expansion of an ideal gas is given by:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]