Question 4.2: The average radius of the earth is R=6371.22 km. Let us mode......
The average radius of the earth is R=6371.22 km. Let us model the troposphere as an 8-km-thick homogeneous layer at standard conditions of pressure and temperature ( http://access.inrp.fr ). The average concentrations of {\mathrm{CO}}_{2} in 1991 and 2001 were 355.67 and 370.9 ppm, respectively. Between the period 1991 and 2001, it is estimated that 64 Gt of carbon was emitted in human activities .With this simplified model calculate the amount of carbon contained in the atmosphere in 1991 and 2001. Calculate the increase in {\mathrm{CO}}_{2} concentration during this period using the estimations of the quantity of {\mathrm{CO}}_{2} emitted into the atmosphere and discuss the results.
The volume of the troposphere is given by (\textstyle{\frac{4}{3}})\pi\big[(R+8)^{3}\!-\!R^{3}\big]\,. The volume of {\mathrm{CO}}_{2} in the troposphere is just that number multiplied by the concentration (number of ppm/ 10^{6} ). The number of moles is this volume (in liters) divided by 22.4 liters, the volume of 1 mol under standard conditions. Finally the mass of carbon is obtained by multiplying by the mass of 1 mol of carbon (12 g). This gives the following figures:
In 1991: Volume of the troposphere: 4\times10^{9} \mathrm{km}^{3}=4\times10^{21} liters. Volume of {\mathrm{CO}}_{2} : 1.42\times10^{18} liters. Number of moles: 6.35\times10^{16}. Mass of carbon: 7.6\times10^{17} g=766\,\mathrm{Gt}\mathrm{~of~CO}_{2}.
In 2001: Volume of {\mathrm{CO}}_{2} : 1.48\times10^{18} liters. Number of moles: 6.62\times10^{16}. Mass of carbon: 7.95\times10^{17}\displaystyle\mathrm{g}=795\,\mathrm{Gt~of~CO_{2}}.
The difference (795 − 766 = 29 Gt) is smaller than the actual value (64 Gt) by a little more than a factor of 2, showing the crudeness of the model, although it provides some understanding of the underlying physics.