To determine the final volume of the gas within the cylinder, we can use the combined gas law, which relates the pressure, volume, and temperature of a gas in two different states.
The combined gas law is stated as:
[tex]\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \][/tex]
Where:
- [tex]\( P_1 \)[/tex] is the initial pressure,
- [tex]\( V_1 \)[/tex] is the initial volume,
- [tex]\( T_1 \)[/tex] is the initial temperature,
- [tex]\( P_2 \)[/tex] is the final pressure,
- [tex]\( V_2 \)[/tex] is the final volume,
- [tex]\( T_2 \)[/tex] is the final temperature.
Given values:
- Initial volume, [tex]\( V_1 = 1.0 \)[/tex] L
- Initial pressure, [tex]\( P_1 = 950 \)[/tex] mm Hg
- Final pressure, [tex]\( P_2 = 715 \)[/tex] mm Hg
- Initial temperature, [tex]\( T_1 = 315 \)[/tex] K
- Final temperature, [tex]\( T_2 = 305 \)[/tex] K
We need to solve for the final volume [tex]\( V_2 \)[/tex]. By rearranging the combined gas law equation to solve for [tex]\( V_2 \)[/tex], we get:
[tex]\[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \][/tex]
Now, let's plug in the given values and calculate:
[tex]\[ V_2 = \frac{950 \text{ mm Hg} \times 1.0 \text{ L} \times 305 \text{ K}}{715 \text{ mm Hg} \times 315 \text{ K}} \][/tex]
Simplify the expression step-by-step:
1. Multiply the numerator:
[tex]\[ 950 \times 305 = 289750 \][/tex]
2. Multiply the denominator:
[tex]\[ 715 \times 315 = 225225 \][/tex]
3. Divide the results:
[tex]\[ V_2 = \frac{289750}{225225} \approx 1.286 \text{ L} \][/tex]
Thus, the final volume of the gas in the cylinder is approximately 1.286 liters.
