In an adiabatic expansion of air , the volume increases by 5% What is the the percentage change in pressure ? `[(1.05)^(7/5) =1.07]`

Options

A. `7%`
B. `5%`
C. `4%`
D. `3%`

Answer

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

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, the relationship between pressure (P) and volume (V) is given by:
\[ P V^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio. For air, \(\gamma = \frac{C_p}{C_v} = 1.4\).

2. **Express the Change in Volume**:
Given that the volume increases by 5%, we can express this mathematically as:
\[ \frac{dV}{V} = 0.05 \]

3. **Differentiate the Adiabatic Relation**:
Taking the logarithm of the adiabatic relation and differentiating gives:
\[ \frac{dP}{P} + \gamma \frac{dV}{V} = 0 \]
Rearranging this, we find:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

4. **Substituting the Values**:
Now, substituting \(\gamma = 1.4\) and \(\frac{dV}{V} = 0.05\):
\[ \frac{dP}{P} = -1.4 \times 0.05 \]

5. **Calculate the Change in Pressure**:
\[ \frac{dP}{P} = -0.07 \]
This indicates that the pressure decreases by 7%.

6. **Convert to Percentage**:
To express this as a percentage change in pressure:
\[ \text{Percentage change in pressure} = -0.07 \times 100 = -7\% \]

### Final Answer:
The percentage change in pressure is **-7%**, indicating a decrease in pressure.