Answer :
To determine the gas pressure during the compression, we need to use the work-energy principle for a gas under constant pressure. The work (W) done on the gas by the piston is given by the formula:
[tex]\[ W = P \Delta V \][/tex]
where:
- [tex]\( W \)[/tex] is the work done,
- [tex]\( P \)[/tex] is the constant pressure,
- [tex]\( \Delta V \)[/tex] is the change in volume.
First, let's convert the initial volume ([tex]\( V_{initial} \)[/tex]) and the final volume ([tex]\( V_{final} \)[/tex]) from liters to cubic meters:
- 1 liter = [tex]\( 10^{-3} \)[/tex] cubic meters
- [tex]\( V_{initial} = 125 \)[/tex] liters = [tex]\( 125 \times 10^{-3} \)[/tex] cubic meters = [tex]\( 0.125 \)[/tex] cubic meters
- [tex]\( V_{final} = 90 \)[/tex] liters = [tex]\( 90 \times 10^{-3} \)[/tex] cubic meters = [tex]\( 0.09 \)[/tex] cubic meters
Next, calculate the change in volume ([tex]\( \Delta V \)[/tex]):
[tex]\[ \Delta V = V_{final} - V_{initial} \][/tex]
[tex]\[ \Delta V = 0.09 \, \text{m}^3 - 0.125 \, \text{m}^3 \][/tex]
[tex]\[ \Delta V = -0.035 \, \text{m}^3 \][/tex]
Since the volume is being compressed, [tex]\( \Delta V \)[/tex] is negative. For the purposes of calculating pressure, we consider the magnitude of [tex]\( \Delta V \)[/tex]:
[tex]\[ |\Delta V| = 0.035 \, \text{m}^3 \][/tex]
Given that the work done on the gas ([tex]\( W \)[/tex]) is [tex]\( 10^4 \)[/tex] joules, we can solve for the pressure [tex]\( P \)[/tex] using the formula [tex]\( W = P |\Delta V| \)[/tex]:
[tex]\[ P = \frac{W}{|\Delta V|} \][/tex]
[tex]\[ P = \frac{10^4 \, \text{J}}{0.035 \, \text{m}^3} \][/tex]
[tex]\[ P = \frac{10^4}{0.035} \][/tex]
[tex]\[ P \approx 2.857 \times 10^5 \, \text{Pa} \][/tex]
Since the pressure value we calculated is approximately [tex]\( 2.857 \times 10^5 \)[/tex] pascals, the closest answer choice is:
B. [tex]\( 3 \times 10^5 \)[/tex] pascals
Thus, the correct answer is [tex]\( \text{B. } 3 \times 10^5 \text{ pascals} \)[/tex].
[tex]\[ W = P \Delta V \][/tex]
where:
- [tex]\( W \)[/tex] is the work done,
- [tex]\( P \)[/tex] is the constant pressure,
- [tex]\( \Delta V \)[/tex] is the change in volume.
First, let's convert the initial volume ([tex]\( V_{initial} \)[/tex]) and the final volume ([tex]\( V_{final} \)[/tex]) from liters to cubic meters:
- 1 liter = [tex]\( 10^{-3} \)[/tex] cubic meters
- [tex]\( V_{initial} = 125 \)[/tex] liters = [tex]\( 125 \times 10^{-3} \)[/tex] cubic meters = [tex]\( 0.125 \)[/tex] cubic meters
- [tex]\( V_{final} = 90 \)[/tex] liters = [tex]\( 90 \times 10^{-3} \)[/tex] cubic meters = [tex]\( 0.09 \)[/tex] cubic meters
Next, calculate the change in volume ([tex]\( \Delta V \)[/tex]):
[tex]\[ \Delta V = V_{final} - V_{initial} \][/tex]
[tex]\[ \Delta V = 0.09 \, \text{m}^3 - 0.125 \, \text{m}^3 \][/tex]
[tex]\[ \Delta V = -0.035 \, \text{m}^3 \][/tex]
Since the volume is being compressed, [tex]\( \Delta V \)[/tex] is negative. For the purposes of calculating pressure, we consider the magnitude of [tex]\( \Delta V \)[/tex]:
[tex]\[ |\Delta V| = 0.035 \, \text{m}^3 \][/tex]
Given that the work done on the gas ([tex]\( W \)[/tex]) is [tex]\( 10^4 \)[/tex] joules, we can solve for the pressure [tex]\( P \)[/tex] using the formula [tex]\( W = P |\Delta V| \)[/tex]:
[tex]\[ P = \frac{W}{|\Delta V|} \][/tex]
[tex]\[ P = \frac{10^4 \, \text{J}}{0.035 \, \text{m}^3} \][/tex]
[tex]\[ P = \frac{10^4}{0.035} \][/tex]
[tex]\[ P \approx 2.857 \times 10^5 \, \text{Pa} \][/tex]
Since the pressure value we calculated is approximately [tex]\( 2.857 \times 10^5 \)[/tex] pascals, the closest answer choice is:
B. [tex]\( 3 \times 10^5 \)[/tex] pascals
Thus, the correct answer is [tex]\( \text{B. } 3 \times 10^5 \text{ pascals} \)[/tex].
