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 use the relationship between pressure and volume in an adiabatic process.

### 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) and volume (V) can be described by the equation:
\[ P V^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio, given as 1.4 in this problem.

2. **Initial and Final Volumes**: Let the initial volume be \(V\). If the volume increases by 10%, the final volume \(V_f\) can be expressed as:
\[ V_f = V + 0.1V = 1.1V \]

3. **Using the Adiabatic Condition**: According to the adiabatic condition:
\[ P_i V^\gamma = P_f (1.1V)^\gamma \]
where \(P_i\) is the initial pressure and \(P_f\) is the final pressure.

4. **Rearranging the Equation**: We can rearrange this equation to find the relationship between the initial and final pressures:
\[ P_f = P_i \left(\frac{V}{1.1V}\right)^\gamma = P_i \left(\frac{1}{1.1}\right)^\gamma \]

5. **Calculating the Pressure Ratio**: Now we can calculate the ratio:
\[ P_f = P_i \left(\frac{1}{1.1}\right)^{1.4} \]

6. **Calculating \(\left(\frac{1}{1.1}\right)^{1.4}\)**: We can compute this value:
\[ \left(\frac{1}{1.1}\right)^{1.4} \approx 0.8686 \]

7. **Finding the Percentage Decrease in Pressure**: The percentage decrease in pressure can be calculated as:
\[ \text{Percentage Decrease} = \left(1 - \frac{P_f}{P_i}\right) \times 100\% \]
Substituting the value we found:
\[ \text{Percentage Decrease} = \left(1 - 0.8686\right) \times 100\% \approx 13.14\% \]

8. **Final Result**: Therefore, the approximate percentage decrease in pressure is about 13.14%.

### Summary:
The approximate percentage decrease in pressure when the volume of air increases by 10% in an adiabatic expansion, assuming \(\gamma = 1.4\), is approximately **13.14%**.