Answer:
To solve this problem, we can use the equation for the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
First, we need to find the total number of moles of gas in the bottle. We know that the bottle contains 0.270 mol of N2, and we can find the number of moles of CO2 by using the partial pressure and the total pressure:
P(CO2) / P(total) = n(CO2) / n(total)
0.250 atm / 4.90 atm = n(CO2) / n(total)
n(CO2) = 0.0128 mol
The total number of moles is then:
n(total) = n(O2) + n(N2) + n(CO2) = n(N2) + n(CO2) + n(O2) = 0.270 mol + 0.0128 mol + n(O2)
n(total) = 0.2828 mol + n(O2)
Next, we can use the ideal gas law to find the number of moles of O2:
PV = nRT
n(O2) = PV / RT
We can rearrange this equation to solve for P(O2):
P(O2) = n(O2)RT / V
We know that R = 0.08206 L·atm/mol·K and T = 273 K. We also know that P(total) = 4.90 atm, and we can find the volume of the bottle by using the given information that it contains 3.00 L:
n(O2) = P(O2) V / RT
0.2828 mol + n(O2) = P(O2) (3.00 L) / (0.08206 L·atm/mol·K) (273 K)
0.2828 mol + n(O2) = 35.75 P(O2)
We can use the information that the partial pressure of CO2 is 0.250 atm to find the partial pressure of N2:
P(N2) = P(total) - P(O2) - P(CO2) = 4.90 atm - P(O2) - 0.250 atm
P(N2) = 4.65 atm - P(O2)
Finally, we can use the fact that the total pressure is the sum of the partial pressures to find P(O2):
P(total) = P(O2) + P(N2) + P(CO2)
4.90 atm = P(O2) + (4.65 atm - P(O2)) + 0.250 atm
4.90 atm = 4.90 atm
Solving for P(O2), we get:
P(O2) = 0.990 atm
Therefore, the partial pressure of O2 is 0.990 atm.