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What volume (in mL) will a sample of [tex]$F_2$[/tex] gas occupy in a syringe at 5.5 atm, if the [tex]$F_2$[/tex] has a volume of 30.0 mL at 1.2 atm?

A. 138 mL
B. 45 mL
C. 4.8 mL
D. 1.4 mL
E. 6.5 mL

Sagot :

GinnyAnswer
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]
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