The Earth can be approximately modeled as a sphere with a radius of 3959 miles.
Assume this corresponds to "sea level" (which is changing!). Atmospheric pressure at sea
level is 14.7 psi. You may assume the atmosphere is dry and is 78.084 mole% N₂, 20.946
mole % 0₂, 0.0421 mole% CO₂, and the rest Ar. ("Dry" atmosphere, no water vapor.)
(a) (4 pts) What is the mass of the atmosphere, in kilograms, based on the data
above? What mass of CO₂, in kg, is present in it now? (Hint: every square inch of the
Earth has 14.7 lbs of atmosphere above it. Assume g is constant over height of atm.)
(b) (2 pts) If the pre-industrial CO₂ level were 0.0280 mole%, how many extra kg
of CO₂ has been added to the atmosphere since James Watt invented the steam engine and
it went into production in the beginning of 1776?
(c) (2 pts) What has been the average rate of CO₂ addition to the atmosphere, in
kg/yr since production began? (Count the year 2022 as having concluded already.)
(d) (3 pts) Suppose that all of the extra CO₂ has arisen from the burning of "100-
octane gasoline" (2,2,4-trimethylpentane). How many gallons of 100-octane gas would
have been combusted?
(e) (5 pts) In fact, the amount of CO₂ put into the atmosphere since the end of
1991 is roughly equal to all the CO₂ that had been put in from the end of 1775 until the
end of 1991. If the rate of growth can be modeled as an increasing exponential,
R(t) = Roe+at
extract the "inital rate" R, (in kg/yr) and the growth rate constant à (in yr¹'). Take the
"zero" of time to be at the end of 1775. (Hint: integral is easy.)
1991-1775
2022-1775
R(t) dt
R(t) dt
1991-1775
and the total over the whole time from "zero" until now should be that in part (b).
(f) (2 pts) According to your answers in part (e), what will be the deposition rate
in kg/yr as the year 2050 starts?
(g) (2 pts) If there are 7.87 billion people on Earth, and we continue to add CO₂
to the atmosphere just at the historical average rate of part (c), how many gallons of gas
can each of us combust over a year? N.B. Much CO₂ has gone into the ocean, so our
estimates are quite low concerning how much CO, humans have added to the biosphere.