The volume of air increases by 10% in the adiabatic expansion. The approximate percentage decrease in its pressure will be: (Assume `gamma = 1.4` )

Options

A. `5%`
B. `14%`
C. `8%`
D. `10%`

Explanation

To solve the problem of finding the approximate percentage decrease in pressure when the volume of air increases by 10% during adiabatic expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure (P), volume (V), and the adiabatic constant (γ) is given by:
\[ PV^\gamma = \text{constant} \]

2. **Differentiate the Equation**:
We can differentiate the equation \(PV^\gamma = \text{constant}\) with respect to pressure and volume. Using the product rule, we have:
\[ d(PV^\gamma) = P d(V^\gamma) + V^\gamma dP = 0 \]
This simplifies to:
\[ P \gamma V^{\gamma - 1} dV + V^\gamma dP = 0 \]

3. **Rearranging the Equation**:
Rearranging gives us:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

4. **Substituting Values**:
We know from the problem that the volume increases by 10%. This means:
\[ \frac{dV}{V} = \frac{10}{100} = 0.1 \]
Given that \(\gamma = 1.4\), we can substitute these values into the equation:
\[ \frac{dP}{P} = -1.4 \times 0.1 = -0.14 \]

5. **Calculating the Percentage Decrease in Pressure**:
To find the percentage decrease in pressure, we multiply by 100:
\[ \text{Percentage decrease in pressure} = -0.14 \times 100 = -14\% \]
The negative sign indicates a decrease in pressure.

### Final Answer:
The approximate percentage decrease in pressure is **14%**.

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