Air is 78.1% nitrogen, 20.9% oxygen, and 0.934% argon by moles. What is the density of air at 22 °C and 760 torr? Assume ideal behavior.
0.781 x molar mass N2 = ?
0.209 x molar mass O2 = ?
0.934 x molar mass Ar = ?
(sum/22.4L) = density in g/L
To calculate the density of air at 22 °C and 760 torr, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure of the gas (in this case, 760 torr)
V = volume of the gas (we can assume 1 mole of air, so the volume will be constant)
n = number of moles of the gas
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature of the gas (in this case, 22 °C or 295 K)
We know the mole fractions of the individual gases in air, so we can calculate the number of moles for each gas based on its mole fraction and the total number of moles of air.
First, we need to calculate the total moles of air using the mole fractions:
Total moles of air = 1 mole
Moles of nitrogen = 78.1% * 1 mole = 0.781 moles
Moles of oxygen = 20.9% * 1 mole = 0.209 moles
Moles of argon = 0.934% * 1 mole = 0.00934 moles
Next, we can calculate the partial pressures of each gas using the ideal gas law:
Partial pressure of nitrogen (P(N2)) = Moles of nitrogen (n(N2)) * R * T / V
Partial pressure of oxygen (P(O2)) = Moles of oxygen (n(O2)) * R * T / V
Partial pressure of argon (P(Ar)) = Moles of argon (n(Ar)) * R * T / V
Since we assume ideal behavior, the sum of the partial pressures will be equal to the total pressure:
P = P(N2) + P(O2) + P(Ar)
Since we know the total pressure (760 torr) and the temperature (22 °C or 295 K), we can rearrange the equation to solve for the volume:
V = P / [(n(N2) * R * T) + (n(O2) * R * T) + (n(Ar) * R * T)]
Finally, we can calculate the density of air using the formula:
Density = (Total moles of air * molar mass of air) / V
The molar mass of air can be calculated by summing the product of the molar masses of each gas and their respective mole fractions:
Molar mass of air = (Molar mass of nitrogen * Mole fraction of nitrogen) + (Molar mass of oxygen * Mole fraction of oxygen) + (Molar mass of argon * Mole fraction of argon)
The molar masses of nitrogen, oxygen, and argon are approximately 28.02 g/mol, 32.00 g/mol, and 39.95 g/mol, respectively.
Using these equations, we can calculate the density of air at 22 °C and 760 torr.
To calculate the density of air at 22 °C and 760 torr, we can use the ideal gas law and the molar mass of each gas.
1. Convert the temperature from Celsius to Kelvin:
Kelvin temperature = Celsius temperature + 273.15
Kelvin temperature = 22 °C + 273.15 = 295.15 K
2. Determine the partial pressure of each gas component:
Given:
Total pressure (P) = 760 torr
Nitrogen (N2) mole fraction (Xn2) = 0.781
Oxygen (O2) mole fraction (Xo2) = 0.209
Argon (Ar) mole fraction (Xar) = 0.00934
Nitrogen partial pressure (Pn2) = Xn2 * P
Pn2 = 0.781 * 760 = 593.96 torr
Oxygen partial pressure (Po2) = Xo2 * P
Po2 = 0.209 * 760 = 158.44 torr
Argon partial pressure (Par) = Xar * P
Par = 0.00934 * 760 = 7.0924 torr
3. Calculate the number of moles of each gas:
The ideal gas law equation is PV = nRT, where:
P = pressure (in atm),
V = volume (in liters),
n = number of moles,
R = ideal gas constant (0.0821 L*atm/(mol*K)),
T = temperature (in Kelvin).
For nitrogen (N2):
nN2 = Pn2 * V / (R * T)
nN2 = (593.96 torr * V) / (0.0821 L*atm/(mol*K) * 295.15 K)
For oxygen (O2):
nO2 = Po2 * V / (R * T)
nO2 = (158.44 torr * V) / (0.0821 L*atm/(mol*K) * 295.15 K)
For argon (Ar):
nAr = Par * V / (R * T)
nAr = (7.0924 torr * V) / (0.0821 L*atm/(mol*K) * 295.15 K)
4. Calculate the total moles of gas:
nTotal = nN2 + nO2 + nAr
5. Calculate the total mass of the gas:
The molar mass of each gas can be found in the periodic table.
Nitrogen molar mass (MN2) = 28.0134 g/mol
Oxygen molar mass (MO2) = 31.9988 g/mol
Argon molar mass (MAr) = 39.948 g/mol
Total mass = (nN2 * MN2) + (nO2 * MO2) + (nAr * MAr)
Total mass = (nN2 * 28.0134 g/mol) + (nO2 * 31.9988 g/mol) + (nAr * 39.948 g/mol)
6. Calculate the density:
Density (ρ) = Total mass / Volume
Density = Total mass / (V * 0.001) [converting liters to cubic meters]
Note: The molar volume of a gas at standard temperature and pressure (STP) is 22.4 L/mol, which is equal to 0.0224 m^3/mol.
Therefore, Density = Total mass / (V * 0.001) = (nN2 * 28.0134 g/mol + nO2 * 31.9988 g/mol + nAr * 39.948 g/mol) / (V * 0.001)
Finally, substitute the values for V (volume in liters) into the equation to find the density of air.