According to the ideal gas law, a 1.005 mol sample of oxygen gas in a 1.609 L container at 269.2 K should exert a pressure of 13.80 atm. What is the percent difference between the pressure calculated using the van der Waals' equation and the ideal pressure?
To calculate the percent difference between the pressure calculated using the van der Waals' equation and the ideal pressure, we need to compute the pressure using both equations and then find the difference.
The ideal gas law is given by:
PV = nRT
Where:
P = Pressure (in atm)
V = Volume (in L)
n = Number of moles
R = Ideal gas constant (0.0821 L*atm/(mol*K))
T = Temperature (in K)
Rearranging the equation, we can solve for P:
P = (nRT) / V
Substituting the given values into the formula, we get:
P_ideal = (1.005 mol * 0.0821 L*atm/(mol*K) * 269.2 K) / 1.609 L
P_ideal = 14.0283 atm
The van der Waals' equation is given by:
[P + a(n/V)^2] [V - nb] = nRT
Where:
P = Pressure (in atm)
V = Volume (in L)
n = Number of moles
R = Ideal gas constant (0.0821 L*atm/(mol*K))
T = Temperature (in K)
a = van der Waals' constant for the gas (0.084 L^2*atm/(mol^2) for oxygen)
b = van der Waals' constant for the gas (0.0318 L/mol for oxygen)
Rearranging the equation, we can solve for P:
P = [nRT/V - a(n/V)^2] / (V - nb)
Substituting the given values into the formula, we get:
P_vanderwaals = [1.005 mol * 0.0821 L*atm/(mol*K) * 269.2 K / 1.609 L - 0.084 L^2*atm/(mol^2) (1.005 mol / 1.609 L)^2] / (1.609 L - 0.0318 L/mol * 1.005 mol)
P_vanderwaals = 13.034 atm
To find the percent difference, we can use the formula:
Percent difference = |(P_ideal - P_vanderwaals) / P_ideal| * 100
Substituting the values, we get:
Percent difference = |(14.0283 atm - 13.034 atm) / 14.0283 atm| * 100
Percent difference = |0.9943 atm / 14.0283 atm| * 100
Percent difference = 0.0709 * 100
Percent difference = 7.09%
Therefore, the percent difference between the pressure calculated using the van der Waals' equation and the ideal pressure is 7.09%.
To calculate the percent difference between the pressure calculated using the van der Waals' equation and the ideal pressure, we need to follow these steps:
Step 1: Calculate the ideal pressure using the Ideal Gas Law equation.
According to the Ideal Gas Law equation:
PV = nRT
where P = pressure, V = volume, n = number of moles, R = gas constant, and T = temperature.
Given:
n = 1.005 mol
V = 1.609 L
T = 269.2 K
The gas constant R is 0.08206 L*atm/(mol*K).
Substituting the values into the equation:
P_ideal = (n * R * T) / V
P_ideal = (1.005 mol * 0.08206 L*atm/(mol*K) * 269.2 K) / 1.609 L
P_ideal = 13.755 atm (rounded to three decimal places)
The ideal pressure is 13.755 atm.
Step 2: Calculate the pressure using the van der Waals' equation.
The van der Waals' equation for real gases is:
(P + an^2/V^2)(V - nb) = nRT
where "a" and "b" are van der Waals' constants.
For oxygen, the van der Waals' constants are approximately:
a = 1.364 L^2 * atm/mol^2
b = 0.0318 L/mol
Substituting the values into the equation:
(P + (1.364 L^2 * atm/mol^2) * (1.005 mol)^2 / (1.609 L)^2)(1.609 L - 0.0318 L * 1.005 mol) = 1.005 mol * 0.08206 L*atm/(mol*K) * 269.2 K
Simplifying the equation:
(P + 0.005540 atm(L/mol)^2)(1.57405 L) = 21.805 atm*L
P + 0.008726 atm = 13.850 atm
P = 13.841 atm (rounded to three decimal places)
The pressure using the van der Waals' equation is 13.841 atm.
Step 3: Calculate the percent difference between the two pressures.
Percent difference = (|P_vdw - P_ideal| / P_ideal) * 100
Percent difference = (|13.841 atm - 13.755 atm| / 13.755 atm) * 100
Percent difference = (0.086 atm / 13.755 atm) * 100
Percent difference = 0.625% (rounded to three decimal places)
The percent difference between the pressure calculated using the van der Waals' equation and the ideal pressure is approximately 0.625%.