Answer
$5.29\times10^{18}kg$.
Work Step by Step
Assume that the atmospheric pressure is due to the total weight of the atmosphere, distributed over the surface area of the Earth.
$$P=\frac{F}{A}=\frac{m_{atm}g}{4\pi r_{Earth}^2}$$
$$m=\frac {4\pi r_{Earth}^2 P}{g}$$
$$=\frac {4\pi (6.38\times 10^6\;m)^2 (1.013\times10^5\;N/m^2)}{9.80\;m/s^2}$$
$$=5.29\times10^{18}kg$$
