To determine by what factor the pressure must be increased to compress 8 liters of nitrogen to 2.5 liters, we start with Boyle's Law. Boyle's Law states that for a given amount of gas at a constant temperature, the pressure and volume are inversely proportional. Mathematically, this is expressed as:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
where:
- [tex]\( P_1 \)[/tex] and [tex]\( V_1 \)[/tex] are the initial pressure and volume, respectively.
- [tex]\( P_2 \)[/tex] and [tex]\( V_2 \)[/tex] are the final pressure and volume, respectively.
Given:
- Initial volume [tex]\( V_1 = 8 \, L \)[/tex]
- Final volume [tex]\( V_2 = 2.5 \, L \)[/tex]
We need to find the factor by which the pressure must be increased. This factor can be denoted as [tex]\( f \)[/tex]. Hence, we have [tex]\( P_2 = f \times P_1 \)[/tex].
Now substituting into Boyle's Law, we get:
[tex]\[ P_1 \times 8 = (f \times P_1) \times 2.5 \][/tex]
We can cancel out [tex]\( P_1 \)[/tex] from both sides since it's common:
[tex]\[ 8 = f \times 2.5 \][/tex]
Solving for [tex]\( f \)[/tex] (the factor by which the pressure needs to be increased), we divide both sides by 2.5:
[tex]\[ f = \frac{8}{2.5} \][/tex]
[tex]\[ f = 3.2 \][/tex]
Therefore, the pressure must be increased by a factor of 3.2 to compress 8 liters of nitrogen to 2.5 liters.
