Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from `57.4kPa` and `1.0` to a final volume of `2.0L` . Take `gamma = 1.4`

Options

A. `1 kPa`
B. `10 kPa`
C. `20 kPa`
D. `22kPa`

Explanation

To solve the problem of calculating the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, the relationship between pressure and volume is given by the equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( P_1 \) and \( V_1 \) are the initial pressure and volume, \( P_2 \) and \( V_2 \) are the final pressure and volume, and \( \gamma \) (gamma) is the heat capacity ratio.

### Step 2: Identify Given Values
From the problem statement, we have:
- Initial Pressure, \( P_1 = 57.4 \, \text{kPa} \)
- Initial Volume, \( V_1 = 1.0 \, \text{L} \)
- Final Volume, \( V_2 = 2.0 \, \text{L} \)
- \( \gamma = 1.4 \)

### Step 3: Substitute Values into the Equation
Using the adiabatic equation, we can substitute the known values:
\[ 57.4 \, \text{kPa} \times (1.0 \, \text{L})^{1.4} = P_2 \times (2.0 \, \text{L})^{1.4} \]

### Step 4: Rearrange to Solve for \( P_2 \)
Rearranging the equation to solve for \( P_2 \):
\[ P_2 = \frac{57.4 \, \text{kPa} \times (1.0 \, \text{L})^{1.4}}{(2.0 \, \text{L})^{1.4}} \]

### Step 5: Calculate the Final Pressure
Now, we can calculate \( P_2 \):
1. Calculate \( (2.0)^{1.4} \):
\[ (2.0)^{1.4} \approx 2.639 \]
2. Substitute this value back into the equation:
\[ P_2 = \frac{57.4 \, \text{kPa}}{2.639} \]
3. Calculate \( P_2 \):
\[ P_2 \approx 21.7 \, \text{kPa} \]

### Step 6: Round the Final Answer
Rounding to two significant figures, we get:
\[ P_2 \approx 22.0 \, \text{kPa} \]

### Final Answer
The final pressure of the carbon dioxide after the adiabatic expansion is approximately **22.0 kPa**.

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