To determine the resulting pressure of a gas sample when it is expanded, we can use Boyle's Law, which states that the product of the initial pressure and volume of a gas is equal to the product of the final pressure and volume, provided the temperature and the amount of gas remain constant. Mathematically, this is expressed as:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
Here:
- [tex]\( P_1 \)[/tex] is the initial pressure of the gas.
- [tex]\( V_1 \)[/tex] is the initial volume of the gas.
- [tex]\( P_2 \)[/tex] is the final pressure of the gas that we need to find.
- [tex]\( V_2 \)[/tex] is the final volume of the gas.
Given:
- The initial volume [tex]\( V_1 \)[/tex] is 19.7 L.
- The initial pressure [tex]\( P_1 \)[/tex] is 3.72 atm.
- The final volume [tex]\( V_2 \)[/tex] is 32.3 L.
Our goal is to find the final pressure [tex]\( P_2 \)[/tex]. Using the formula:
[tex]\[ P_2 = \frac{P_1 \times V_1}{V_2} \][/tex]
Substitute the given values into the equation:
[tex]\[ P_2 = \frac{3.72 \ \text{atm} \times 19.7 \ \text{L}}{32.3 \ \text{L}} \][/tex]
Perform the multiplication in the numerator:
[tex]\[ P_2 = \frac{73.284}{32.3} \][/tex]
Next, perform the division:
[tex]\[ P_2 \approx 2.27 \ \text{atm} \][/tex]
Therefore, the final pressure [tex]\( P_2 \)[/tex] of the gas sample when its volume is expanded to 32.3 L is approximately 2.27 atm.
So the correct answer is [tex]\( 2.27 \ \text{atm} \)[/tex].
