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I have been given the task of calulating the mass of the atmosphere given the following (not quite correct) information. It is intended to be a thought experiment for an undergraduate course in atmospheric science. I recognize that I have chosen a difficult way of doing this but I believe it should still work, and I am confused as to where I may be going wrong in my reasoning.

Radius of Earth: 6000 km.

Density of Atmosphere at sea level: $3times10^{19}$ $frac{molecules}{m^3}$.

Now, taking the composition of the atmosphere to be approximately 78.1% N2 (28.014 g/mol), 20.9% 02 (31.998 g/mol), and 0.9% Ar (39.948 g/mol), we get an average molecular weight of 28.926 g/mol.

Converting the density (Avogadro's and average molecular weight from above), we get a density of 0.00144 $frac{g}{m^3}$.

Now I am assuming that the density of the atmosphere decreases exponentially proportionate to height. That is, $d(h)=e^{lambda h}$ and calculated lambda using my density and the radius of the earth, $d(6000000)=0.00144=e^{6000000lambda}$. Solving for the proportionality constant we get $lambda = -1.09times10^{-6}$. Note, prior to writing this question I had been using my original given density to calculate this proportionality constant and that was not valid. However, this is also wrong.

Taking the 'top' of the atmosphere to be 500 km above the Earth, and plugging this into a triple integral we get:

$int_0^{2pi} int_0^pi int_{6000000}^{6500000} r^2e^{(-1.09times10^{-6}r)}sinphi, dr, dphi, dtheta$

I have been using Wolfram Alpha to compute, but have not been able to finagle an answer that is even close to correct.

Is there anything glaringly wrong with what I have done thus far? I am deeply suspicious of my treatment of the density function, and also of my units. Intuitively, this feels like a valid (albeit complicated) approach to the problem. I am not very interested in a simpler approach, but rather want to know where I have gone wrong. My calculus is also quite rusty.

This is my first StackExchange post, so thank you beforehand for any help.

Two issues that I see:

1. You calculated $lambda$ incorrectly


2. You can simplify the integral

First, the Earth is not made up of gas. The exponential drop in density applies to distances above the Earth. If the density of the atmosphere is $0.00144 frac{g}{m^3}$ at sea level, this implies:

$$Ce^{lambda 0} = 0.00144$$

not

$$Ce^{lambda 6,000,000} = 0.0014$$

Unfortunately, this means that you need one more data point (at some distance above sea level) to calculate $lambda$ (since below sea level we have water/rock, not gas).

Second, there is a symmetry about $r$ if we treat the Earth as a sphere. You can take advantage of this to integrate the mass of the atmosphere using "shells" of radius $6000000 + h$ ($h$ is height above sea level). This reduces a triple integral to a single integral:

$$M_{atm} = int_{6,000,000}^{6,500,000} Ce^{lambda h}4pi (6000000+h)^2dh$$

This is a realtively easy one for WolframAlpha...just need more data to get $lambda$.