Neon gas of a given mass expands isothermally to double volume . What should be the further fractional decreases in pressure , so that the gas when adiabatically compressed from that state , reaches the original state ?

Options

A. `1 - 2^(-2//3)`
B. `1 - 3^(1//3)`
C. `2^(1//3)`
D. `3^(2//3)`

Answer

To solve the problem step by step, we will analyze the isothermal expansion and the subsequent adiabatic compression of the neon gas.

### Step 1: Understanding the Isothermal Expansion
- The neon gas expands isothermally to double its volume.
- Let the initial volume be \( V \) and the final volume after isothermal expansion be \( 2V \).
- According to the ideal gas law, for an isothermal process, the product of pressure and volume remains constant:
\[ P_1 V = P_2 (2V) \]
- From this equation, we can derive the relationship between initial pressure \( P_1 \) and final pressure \( P_2 \):
\[ P_2 = \frac{P_1}{2} \]

### Step 2: Analyzing the Adiabatic Compression
- Now, we need to find the pressure \( P_3 \) after the gas is adiabatically compressed back to its original state.
- For an adiabatic process, the relationship between pressure and volume is given by:
\[ P V^\gamma = \text{constant} \]
- For the initial state (before adiabatic compression), we have:
\[ P_2 (2V)^\gamma = P_1 V^\gamma \]
- Substituting \( P_2 = \frac{P_1}{2} \):
\[ \frac{P_1}{2} (2V)^\gamma = P_1 V^\gamma \]
- Simplifying this gives:
\[ \frac{P_1}{2} \cdot 2^\gamma V^\gamma = P_1 V^\gamma \]
- Dividing both sides by \( P_1 V^\gamma \):
\[ \frac{1}{2} \cdot 2^\gamma = 1 \]
- This leads to:
\[ P_3 = P_1 \cdot 2^{1 - \gamma} \]

### Step 3: Finding the Fractional Decrease in Pressure
- The fractional decrease in pressure from \( P_2 \) to \( P_3 \) is given by:
\[ \text{Fractional Decrease} = \frac{P_2 - P_3}{P_2} \]
- Substituting the values of \( P_2 \) and \( P_3 \):
\[ \text{Fractional Decrease} = \frac{\frac{P_1}{2} - P_1 \cdot 2^{1 - \gamma}}{\frac{P_1}{2}} \]
- Simplifying this expression:
\[ = \frac{1 - 2^{1 - \gamma}}{1} \]

### Step 4: Substitute the Value of \( \gamma \)
- For neon gas, \( \gamma = \frac{5}{3} \):
\[ \text{Fractional Decrease} = 1 - 2^{1 - \frac{5}{3}} = 1 - 2^{- \frac{2}{3}} \]

### Final Answer
- The final expression for the fractional decrease in pressure is:
\[ 1 - 2^{-\frac{2}{3}} \]