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 process of isothermal expansion followed by adiabatic compression of neon gas.

### Step 1: Understand the Isothermal Expansion
Initially, the neon gas expands isothermally to double its volume. According to Boyle's Law for isothermal processes, we have:

\[ P_1 V_1 = P_2 V_2 \]

Given that \( V_2 = 2V_1 \), we can substitute this into the equation:

\[ P_1 V_1 = P_2 (2V_1) \]

### Step 2: Simplify the Equation
Cancel \( V_1 \) from both sides:

\[ P_1 = 2P_2 \]

This indicates that the final pressure \( P_2 \) after the isothermal expansion is half of the initial pressure \( P_1 \):

\[ P_2 = \frac{P_1}{2} \]

### Step 3: Set Up for Adiabatic Compression
Next, we need to find the new pressure \( P_3 \) after a further fractional decrease in pressure, such that when the gas is adiabatically compressed, it returns to the original state (pressure \( P_1 \) and volume \( V_1 \)).

For an adiabatic process, we use the relation:

\[ P V^\gamma = \text{constant} \]

### Step 4: Apply the Adiabatic Condition
From the adiabatic process, we have:

\[ P_1 V_1^\gamma = P_3 V_3^\gamma \]

Since \( V_3 = V_2 = 2V_1 \), we can substitute this into the equation:

\[ P_1 V_1^\gamma = P_3 (2V_1)^\gamma \]

### Step 5: Rearrange the Equation
Rearranging gives:

\[ P_3 = \frac{P_1 V_1^\gamma}{(2V_1)^\gamma} = \frac{P_1}{2^\gamma} \]

### Step 6: Calculate the Fractional Decrease in Pressure
Now, we need to find the fractional decrease in pressure from \( P_2 \) to \( P_3 \):

\[ \text{Fractional Decrease} = \frac{P_2 - P_3}{P_2} \]

Substituting \( P_2 \) and \( P_3 \):

\[ \text{Fractional Decrease} = \frac{\frac{P_1}{2} - \frac{P_1}{2^\gamma}}{\frac{P_1}{2}} \]

### Step 7: Simplify the Fraction
This simplifies to:

\[ \text{Fractional Decrease} = \frac{1 - \frac{1}{2^{\gamma - 1}}}{1} = 1 - \frac{1}{2^{\gamma - 1}} \]

### Step 8: Substitute the Value of Gamma
For neon gas, which is monatomic, \( \gamma = \frac{5}{3} \):

\[ \text{Fractional Decrease} = 1 - \frac{1}{2^{\frac{5}{3} - 1}} = 1 - \frac{1}{2^{\frac{2}{3}}} \]

### Final Result
Thus, the final expression for the fractional decrease in pressure is:

\[ \text{Fractional Decrease} = 1 - 2^{-\frac{2}{3}} \]