In a adiabatic process pressure is increased by `2//3%` if `C_(P)//C_(V) = 3//2`. Then the volume decreases by about

Options

A. `(4)/(9)%`
B. `(2)/(3)%`
C. `4%`
D. `(9)/(4)%`

Explanation

To solve the problem step by step, we will use the relationships defined in thermodynamics for an adiabatic process.

### Step 1: Understand the Given Information
We are given:
- The percentage increase in pressure, \( \frac{dP}{P} = \frac{2}{3} \% = \frac{2}{3} \times \frac{1}{100} = \frac{2}{300} = \frac{1}{150} \).
- The ratio of specific heats, \( \frac{C_P}{C_V} = \frac{3}{2} \).

### Step 2: Determine the Value of \( \gamma \)
The value of \( \gamma \) (gamma) is defined as:
\[ \gamma = \frac{C_P}{C_V} = \frac{3}{2} \]

### Step 3: Use the Relation for Adiabatic Processes
For an adiabatic process, we have the relation:
\[ PV^\gamma = \text{constant} \]
Differentiating both sides gives us:
\[ dP \cdot V^\gamma + P \cdot \gamma V^{\gamma - 1} dV = 0 \]
Rearranging this, we find:
\[ \frac{dP}{P} = -\gamma \frac{dV}{V} \]

### Step 4: Substitute the Known Values
Substituting the values we have:
\[ \frac{1}{150} = -\frac{3}{2} \frac{dV}{V} \]

### Step 5: Solve for \( \frac{dV}{V} \)
Rearranging the equation to solve for \( \frac{dV}{V} \):
\[ \frac{dV}{V} = -\frac{1}{150} \cdot \frac{2}{3} = -\frac{2}{450} = -\frac{1}{225} \]

### Step 6: Convert to Percentage
To find the percentage decrease in volume:
\[ \text{Percentage decrease} = \frac{dV}{V} \times 100 = -\frac{1}{225} \times 100 = -\frac{100}{225} \approx -0.444\% \]

### Step 7: Final Result
The volume decreases by approximately \( 0.444\% \).