### Number 200 of a series

Here is a nice problem, not too difficult, pertinent to a current hot topic.

Hypothetical scenario: Nothing is adding carbon dioxide to the atmosphere. Carbon dioxide has a 100-year half life in the atmosphere. We crank up a contraption that pumps 100 million tons of carbon dioxide into the atmosphere each year. How much carbon dioxide is in the atmosphere when a steady state is obtained?

Post your answer as a comment below. Extra points for describing the calculation.

A 100-year half life is a decline of 0.7% per year.

Balance is x*0.7%=100. x=14476 million tons.

A slightly more detailed derivation.

The change rate of the CO2 level obeys this equation (this form is inferred from the postulate that a half-life exists):

dx/dt = -k*x + r

where x is the CO2 level, r is the rate new CO2 is introduced (100 million tons per year), and k is a decay constant we can calculate from the half-life. Steady-state is reached when dx/dt = 0, so at that point -k*x + r = 0, which means x = r/k. The solution to the DE is an exponential e^(k*t). So by setting t equal to the half-life, we find k = 100 days / ln 2.

Thus x = 1.4427e10 tons, which is approximately Michael’s answer, the difference being due to the approximation of ln 2 = 0.7.

Edit to the above: The exponential should be e^(-k*t).

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