Answer
The total mass of the atmosphere is $5.29\times 10^{18}~kg$
Work Step by Step
The atmospheric pressure comes from the weight of the atmosphere pushing down on the earth's surface area. Let $M$ be the mass of the atmosphere.
$\frac{F}{A} = P$
$Mg = P~A$
$M = \frac{P~A}{g}$
$M = \frac{P~(4\pi~R^2)}{g}$
$M = \frac{(1.013\times 10^5~N/m^2)~(4\pi)(6.38\times 10^6~m)^2}{9.80~m/s^2}$
$M = 5.29\times 10^{18}~kg$
The total mass of the atmosphere is $5.29\times 10^{18}~kg$.
