To solve for the volume of [tex]\( F_2 \)[/tex] gas under different pressure conditions, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional when temperature and the amount of gas are held constant. This relationship can be mathematically 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 data:
- The initial pressure ([tex]\( P_1 \)[/tex]) is [tex]\( 1.2 \)[/tex] atm
- The initial volume ([tex]\( V_1 \)[/tex]) is [tex]\( 30.0 \)[/tex] mL
- The final pressure ([tex]\( P_2 \)[/tex]) is [tex]\( 5.5 \)[/tex] atm
We need to find the final volume ([tex]\( V_2 \)[/tex]) at the final pressure of [tex]\( 5.5 \)[/tex] atm. Rearranging the equation to solve for [tex]\( V_2 \)[/tex]:
[tex]\[ V_2 = \frac{P_1 \times V_1}{P_2} \][/tex]
Substitute the given values into the equation:
[tex]\[ V_2 = \frac{1.2 \text{ atm} \times 30.0 \text{ mL}}{5.5 \text{ atm}} \][/tex]
Now solve for [tex]\( V_2 \)[/tex]:
[tex]\[ V_2 = \frac{36.0 \text{ atm} \cdot \text{mL}}{5.5 \text{ atm}} \][/tex]
[tex]\[ V_2 = 6.545454545454546 \text{ mL} \][/tex]
The result, rounded to one decimal place, is approximately:
[tex]\[ V_2 \approx 6.5 \text{ mL} \][/tex]
Therefore, the volume the [tex]\( F_2 \)[/tex] gas will occupy in the syringe at 5.5 atm is:
[tex]\[ \boxed{6.5 \text{ mL}} \][/tex]
