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 with a 5% increase in volume, we can follow these steps:

### Step 1: Understand the relationship for adiabatic processes
In an adiabatic process, the relationship between pressure (P) and volume (V) is given by the equation:
\[ PV^{\gamma} = \text{constant} \]
where \(\gamma\) (the adiabatic exponent) for air (considered as a diatomic gas) is \( \frac{7}{5} \).

### Step 2: Define the initial and final volumes
Let the initial volume be \( V \). If the volume increases by 5%, the final volume \( V' \) can be expressed as:
\[ V' = V + 0.05V = 1.05V \]

### Step 3: Apply the adiabatic condition
Using the adiabatic condition:
\[ P V^{\gamma} = P' (V')^{\gamma} \]
Substituting \( V' = 1.05V \):
\[ P V^{\gamma} = P' (1.05V)^{\gamma} \]

### Step 4: Rearranging the equation
We can rearrange the equation to find the ratio of the pressures:
\[ P' = P \left( \frac{V}{1.05V} \right)^{\gamma} \]
This simplifies to:
\[ P' = P \left( \frac{1}{1.05} \right)^{\gamma} \]

### Step 5: Substitute the value of \(\gamma\)
Substituting \(\gamma = \frac{7}{5}\):
\[ P' = P \left( \frac{1}{1.05} \right)^{\frac{7}{5}} \]

### Step 6: Calculate the ratio
Using the approximation given in the question:
\[ \left(1.05\right)^{\frac{7}{5}} \approx 1.07 \]
Thus,
\[ P' = P \left( \frac{1}{1.07} \right) \]

### Step 7: Find the percentage change in pressure
The percentage change in pressure can be calculated as:
\[ \text{Percentage change} = \frac{P' - P}{P} \times 100 \]
Substituting for \( P' \):
\[ \text{Percentage change} = \frac{\left(\frac{P}{1.07}\right) - P}{P} \times 100 \]
This simplifies to:
\[ \text{Percentage change} = \left(\frac{1 - \frac{1}{1.07}}{1}\right) \times 100 \]
Calculating the fraction:
\[ 1 - \frac{1}{1.07} = \frac{1.07 - 1}{1.07} = \frac{0.07}{1.07} \]
Thus,
\[ \text{Percentage change} \approx \frac{0.07}{1.07} \times 100 \approx 6.54\% \]

### Final Answer
The percentage change in pressure is approximately **7%** (rounded).