The volume of air increases by `5%` in its adiabatic expansion. The precentage decrease in its pressure will be

Options

A. 0.05
B. 0.06
C. 0.07
D. 0.08

Answer

To solve the problem of finding the percentage decrease in pressure when the volume of air increases by 5% during adiabatic expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, the relationship between pressure (P), volume (V), and the adiabatic index (γ) is given by the equation:
\[ PV^\gamma = \text{constant} \]
For air, the value of γ (gamma) is approximately 1.4.

2. **Express the Change in Volume**:
Given that the volume increases by 5%, we can express this mathematically as:
\[ V' = V(1 + 0.05) = 1.05V \]
where \(V'\) is the new volume after expansion.

3. **Apply the Adiabatic Condition**:
Using the adiabatic condition, we can write:
\[ P'V'^\gamma = PV^\gamma \]
where \(P'\) is the new pressure after expansion.

4. **Substituting the New Volume**:
Substituting \(V' = 1.05V\) into the equation gives:
\[ P'(1.05V)^\gamma = PV^\gamma \]

5. **Rearranging the Equation**:
Rearranging the equation to find \(P'\):
\[ P' = P \left(\frac{V}{1.05V}\right)^\gamma = P \left(\frac{1}{1.05}\right)^\gamma \]

6. **Calculating the Change in Pressure**:
We can calculate the new pressure:
\[ P' = P \left(1.05^{-1.4}\right) \]

7. **Finding the Percentage Decrease in Pressure**:
The percentage decrease in pressure can be calculated as:
\[ \text{Percentage Decrease} = \left(\frac{P - P'}{P}\right) \times 100 \]
Substituting \(P'\):
\[ \text{Percentage Decrease} = \left(1 - 1.05^{-1.4}\right) \times 100 \]

8. **Calculating \(1.05^{-1.4}\)**:
Using a calculator:
\[ 1.05^{-1.4} \approx 0.927 \]
So,
\[ \text{Percentage Decrease} = (1 - 0.927) \times 100 \approx 7.3\% \]

### Final Answer:
The percentage decrease in pressure is approximately **7.3%**.