Certainly! Let's walk through the steps needed to solve this problem:
1. Initial Condition:
- We have 4.13 moles of gas.
- The initial pressure is 2971 torr.
- The temperature and volume of the gas remain constant.
2. Change in Moles:
- Half of the moles of gas are released.
- Therefore, the number of moles remaining is:
[tex]\[
\text{Remaining moles} = \frac{4.13 \text{ moles}}{2} = 2.065 \text{ moles}
\][/tex]
3. Pressure-Volume-Temperature Relationship (Boyle's Law):
- According to Boyle's Law (for a given temperature and volume), the pressure of a gas is directly proportional to the number of moles of gas present.
- Initially we have:
[tex]\[
P_1 = 2971 \text{ torr}, \quad n_1 = 4.13 \text{ moles}
\][/tex]
- After releasing half of the gas:
[tex]\[
P_2 = \text{unknown}, \quad n_2 = 2.065 \text{ moles}
\][/tex]
- Using the direct proportion [tex]\(P_1 / n_1 = P_2 / n_2\)[/tex], we can find the new pressure:
[tex]\[
P_2 = P_1 \times \left(\frac{n_2}{n_1}\right)
\][/tex]
[tex]\[
P_2 = 2971 \text{ torr} \times \left(\frac{2.065 \text{ moles}}{4.13 \text{ moles}}\right)
\][/tex]
[tex]\[
P_2 = 2971 \times 0.5 = 1485.5 \text{ torr}
\][/tex]
4. Conversion of Pressure from Torr to kPa:
- To convert pressure from torr to kPa, we use the conversion factor [tex]\(1 \text{ torr} = 0.133322 \text{ kPa}\)[/tex]:
[tex]\[
P_2 = 1485.5 \text{ torr} \times 0.133322 \text{ kPa/torr}
\][/tex]
[tex]\[
P_2 = 198.04983099999998 \text{ kPa}
\][/tex]
5. Rounding to the Nearest kPa:
- The final pressure in kPa, rounded to the nearest whole number, is:
[tex]\[
P_2 \approx 198 \text{ kPa}
\][/tex]
Therefore, the pressure exerted by the gas inside the cylinder after releasing half of the moles, in kPa, is approximately [tex]\(198 \text{ kPa}\)[/tex].
