For the adiabatic expansion of a perfect monoatomic gas, when volume increases by 24% , what is the percentage decrease in pressure ? Given : `(25/31)^(5//3)=0.7`

Options

A. (a)`24%`
B. (b)`40%`
C. (c)`48%`
D. (d)`71%`

Answer

To solve the problem of finding the percentage decrease in pressure during the adiabatic expansion of a perfect monoatomic gas when the volume increases by 24%, we can follow these steps:

### Step 1: Understand the relationship between pressure, volume, and gamma
For an adiabatic process, the relationship between pressure (P) and volume (V) is given by the equation:
\[ P V^{\gamma} = \text{constant} \]
where \( \gamma \) (gamma) is the adiabatic index.

### Step 2: Determine the value of gamma for a monoatomic gas
For a monoatomic ideal gas, the molar heat capacities are:
- \( C_p = \frac{5}{2} R \)
- \( C_v = \frac{3}{2} R \)

Thus, we can calculate \( \gamma \):
\[ \gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2} R}{\frac{3}{2} R} = \frac{5}{3} \]

### Step 3: Relate the change in volume to the change in pressure
From the adiabatic condition, we can express the change in pressure in terms of the change in volume:
\[ P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \]
Rearranging gives:
\[ \frac{P_2}{P_1} = \left( \frac{V_1}{V_2} \right)^{\gamma} \]

### Step 4: Calculate the new volume
If the volume increases by 24%, we can express the new volume \( V_2 \) in terms of the original volume \( V_1 \):
\[ V_2 = V_1 + 0.24 V_1 = 1.24 V_1 \]

### Step 5: Substitute into the pressure equation
Now substituting \( V_2 \) into the pressure equation:
\[ \frac{P_2}{P_1} = \left( \frac{V_1}{1.24 V_1} \right)^{\frac{5}{3}} = \left( \frac{1}{1.24} \right)^{\frac{5}{3}} \]

### Step 6: Calculate \( \left( \frac{1}{1.24} \right)^{\frac{5}{3}} \)
To simplify:
\[ \frac{1}{1.24} \approx 0.8065 \]
Now raise this to the power of \( \frac{5}{3} \):
Using the given approximation \( \left( \frac{25}{31} \right)^{\frac{5}{3}} \approx 0.7 \), we can say:
\[ \frac{P_2}{P_1} \approx 0.7 \]

### Step 7: Calculate the percentage decrease in pressure
The percentage decrease in pressure can be calculated as:
\[ \text{Percentage decrease} = \left( \frac{P_1 - P_2}{P_1} \right) \times 100 \]
Substituting \( P_2 = 0.7 P_1 \):
\[ \text{Percentage decrease} = \left( \frac{P_1 - 0.7 P_1}{P_1} \right) \times 100 = (1 - 0.7) \times 100 = 0.3 \times 100 = 30\% \]

### Final Answer
The percentage decrease in pressure is **30%**.