To solve the problem of finding the final pressure of a gas when it expands, we can use Boyle's Law, which states that the pressure of a gas times its volume is a constant value as long as the temperature remains constant. Mathematically, Boyle's Law is expressed as:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
Where:
- [tex]\( P_1 \)[/tex] is the initial pressure
- [tex]\( V_1 \)[/tex] is the initial volume
- [tex]\( P_2 \)[/tex] is the final pressure
- [tex]\( V_2 \)[/tex] is the final volume
Given the values:
- [tex]\( V_1 = 1.17 \)[/tex] liters
- [tex]\( P_1 = 264.5 \)[/tex] kPa
- [tex]\( V_2 = 11.25 \)[/tex] liters
We need to find [tex]\( P_2 \)[/tex].
Step-by-step solution:
1. Start with Boyle's Law formula:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
2. Substitute the known values into the equation:
[tex]\[ 264.5 \text{ kPa} \times 1.17 \text{ L} = P_2 \times 11.25 \text{ L} \][/tex]
3. Solve for [tex]\( P_2 \)[/tex] by isolating it on one side of the equation:
[tex]\[ P_2 = \frac{264.5 \text{ kPa} \times 1.17 \text{ L}}{11.25 \text{ L}} \][/tex]
4. Perform the multiplication of the numerator first:
[tex]\[ 264.5 \times 1.17 = 309.465 \][/tex]
5. Now divide by the final volume to get [tex]\( P_2 \)[/tex]:
[tex]\[ P_2 = \frac{309.465}{11.25} \][/tex]
6. Complete the division:
[tex]\[ P_2 = 27.5 \text{ kPa} \][/tex]
Therefore, the pressure in the container after the gas has expanded to 11.25 liters, keeping the temperature constant, is 27.5 kPa.
