What is the total mass of the Earth's atmosphere? (The...

What is the total mass of the Earth's atmosphere? (The radius of the Earth is $6.37 \times 10^{6} \mathrm{~m}$, and atmospheric pressure at the Earth's surface is $1.013 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2} .$ )

Key Concepts

Integration Over the Earth's Surface

Integrating the atmospheric pressure over the Earth's surface (or equivalently multiplying the uniform pressure by the total surface area) gives the total force exerted by the atmosphere. This conceptual step bridges local pressure measurements and global quantities, such as the total mass of the atmosphere, which is calculated by dividing this total force by the acceleration due to gravity.

Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted by the weight of the atmosphere above a given surface. This concept is fundamental in understanding how pressure measurements at the Earth's surface relate to the overall mass of the atmosphere, as it represents the cumulative effect of the gravitational pull on the column of air above that area.

Surface Area of a Sphere

The surface area of a sphere is calculated using the formula 4?r², where r is the radius of the sphere. In the context of Earth’s atmosphere, determining the total surface area of the Earth is essential because the atmospheric pressure measured at the surface is applied uniformly across this area, thereby allowing the estimation of the total force exerted by the atmosphere.

Relation Between Force, Mass, and Gravity

In classical mechanics, the relation between force, mass, and gravitational acceleration is given by F = mg. This relation is used to convert the total force exerted by the atmosphere (derived from atmospheric pressure times the Earth's surface area) into the mass of the atmosphere by dividing the force by the gravitational acceleration.

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