Since air is a gas, its weight will depend upon the volume and pressure in which it is contained.
The simplest approach would be to use the ideal gas law, which relates pressure P and volume V to temperature T and the number of moles of gas n, with a constant of proportionality called R:
P V = n R T
We need some figures for pressure and volume. If we're interested in the density of air at the surface, we can use an arbitrary volume (say 1 cubic meter) and set the pressure equal to atmospheric pressure, which is about 100 kilopascals.
You're given the temperature, but you didn't mention it; so I'm going to solve it for 27 C (80 F) and you can do the same for whatever you actually have.
Remember that T in this equation must be given in kelvin, so 27 C is 300 K. The constant R is 8.3 J/K/mol.
P V = n R T
(10^5 Pa) (1 m) = n (8.3 J/K/mol) (300 K)
This will tell us how many moles we've got in each cubic meter:
n = (10^5)(1)/(8.3)/(300) = 4.00 mol
Then to get mass, we need the molar mass.
Air is a mixture of gases, but the really important ones are 78% nitrogen and 21% oxygen and 1% argon. A weighted average of these molar masses will give us the effective molar mass of air.
The molar mass of nitrogen is 14 g/mol, so N2 (nitrogen gas) is 28 g/mol.
The molar mass of oxygen is 16 g/mol, so O2 (oxygen gas) is 32 g/mol.
The molar mass of argon is 40 g/mol, and argon is a monatomic gas.
Weighted average of these is:
(0.78)(28) + (0.21)(32) + (0.01)(40)
21.84 + 6.72 + 0.40 = 28.96 g/mol
Multiply this by the 4.00 mol of gas we have:
(28.96 g/mol)(4.00 mol/m^3) = 116 g/m^3 = 0.116 kg/m^3
That at least is what we get for a temperature of 27 C; if we use a different temperature, the density we get will be different, but the process of calculation will be the same.
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