Certainly! To solve this problem, we can use Boyle's Law, which states that the pressure of a gas times its volume is constant when the temperature remains the same. Mathematically, Boyle's Law is written as:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
Where:
- [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 after compression.
- [tex]\( V_2 \)[/tex] is the final volume of the gas after compression.
We are given the following information:
- Initial pressure, [tex]\( P_1 \)[/tex], is [tex]\( 22.4 \)[/tex] psi.
- The final volume, [tex]\( V_2 \)[/tex], is one-third of the original volume, [tex]\( V_1 \)[/tex].
Since [tex]\( V_2 = \frac{V_1}{3} \)[/tex]:
Substitute [tex]\( V_2 \)[/tex] in Boyle's Law:
[tex]\[ P_1 \times V_1 = P_2 \times \frac{V_1}{3} \][/tex]
We can simplify this by canceling out [tex]\( V_1 \)[/tex] from both sides:
[tex]\[ P_1 = \frac{P_2}{3} \][/tex]
To isolate [tex]\( P_2 \)[/tex] (the final pressure), multiply both sides of the equation by 3:
[tex]\[ P_2 = 3 \times P_1 \][/tex]
Now, substitute [tex]\( P_1 \)[/tex] with the given initial pressure:
[tex]\[ P_2 = 3 \times 22.4 \text{ psi} \][/tex]
Perform the multiplication:
[tex]\[ P_2 = 67.2 \text{ psi} \][/tex]
Thus, the pressure of the compressed gas is [tex]\( 67.2 \)[/tex] psi.
