Physics writing

Semester Project: Planet Earth

Making a Model for Earth’s Atmospheric Carbon Dioxide

Calculus-Based (PHYS 2425)

Background:

Scientists have been monitoring atmospheric CO2 levels for several decades. The graph below shows continuous measurements made from 1958 to 2020. The CO2 levels on the graph are in units of parts per million (1 ppm means 1 particle of a substance mixed with particles of something else to make a total of 1 million particles). Study the graph and Figure 2 and answer the questions below.

Figure 1: CO2 increase with time

Note: Figure 1 is also called the Keeling curve. It is named after the scientist Charles Keeling, who first started monitoring atmospheric CO2 at Hawaii in 1958. (https://en.wikipedia.org/wiki/Keeling_Curve ).

Figure 2: Sources and sinks of carbon.

In the year 2000, there were about 730 gigatons of carbon in the atmosphere, 2,000 gigatons in Vegetations and Soils, and 38,000 gigatons of carbon in the Oceans.

Data:

Measurement indicate that:

1. The average increase in atmospheric CO2 is 11 gigatons per year (11 x 109 tons or about 1013 kg/year). 1 gigaton = 109 tons, 1 ton = 1000 kg.

2. There are 12 gigatons of elemental carbon for every 44 gigatons of molecular CO2 (the rest is oxygen).

Note: complete the sections below, and copy-paste them into your final report.

Please show your calculations for the questions below.

Part of the evaluation will be based on following systematic, organized, problem-solving steps. This involves using the correct equation, rearranging it to find the quantity of interest, substituting the known values, and then writing the final answer with the correct units. Please use scientific notation for your final answer. For example: 1,000,000,000 tons = 1.0 x 109 tons = 1.0 gigatons = 1.0 Gt.

Questions

1. Look at Figure 2 above and list the sources that cause carbon (CO2) to increase in the atmosphere. These sources are the ones with arrows pointing up to the sky. For example, vegetation adds 119 gigatons of carbon every year. The first row has been completed as an example.

Answer: Table 1

Sources	gigatons added per year
Vegetation	119
Forming and Breaking of carbonic acid in water and respiration from aquatic plants and animals	88
Different Land Usage	1.7
Combustion of fossil fuel and industrial process	6.3
Total	215

Among all this sources, burning of fossil fuel and industrial process is the main

cause of releasing carbon element in the earth’s atmosphere.

2. Look at Figure 2 above and list the sinks for carbon (CO2), or the processes that remove carbon from the atmosphere. Processes that remove carbon have arrows pointing downwards from the sky to the Earth. For example, the oceans remove 90 gigatons of carbon from the atmosphere every year. The first row has been completed as an example.

Answer: Table 2

Sinks	gigatons removed per year
Ocean	90
Vegetation	120
Land use	1.9
Total	211.9

If we compare in the graph, vegetation play the vital role in maintaining the

carbon level in earth’s atmosphere and ocean has little contribution to the

atmosphere too.

3. Add up all the sources of carbon in Question 1 to find the total carbon input to the atmosphere in gigatons per year.

Answer: Every year 215 gigatons of carbon is being release from different

sources.

4. Add up all the sinks of carbon in Question 2 to find the total carbon removed from the atmosphere in gigatons per year.

Answer: The total carbon removed from the atmosphere every year by vegetation

and ocean is 211.9 gigatons per year.

5. Find the rate of change of carbon in gigatons per year by subtracting the total amount removed (answer to Question 4) from the amount added (answer to Question 3)

Answer: Rate of change = carbon removed – carbon added

= 215-211.9

= 3.1 gigatons of carbon every year

It means every year these sources are adding 3.1 gigatons of carbon which is

going to be accumulated one day if we don’t think of controlling.

6. What is the main cause of this increase in atmospheric carbon every year? Compare the sources and sinks of carbon to find the answer. Is this from natural, or anthropogenic (human) causes?

Answer: The main sources look like vegetation but it does remove too and

difference is lower than fossil fuel combustion and industrial process.

Regarding sources and sinks fossil fuel combustion and industrial process is the

main cause as there is no way of sinking of atmospheric carbon, it just releases.

7. Write the rate of change of atmospheric carbon (answer to Question 5), as a differential equation of the form dC/dt = some number. Where “some number” is the amount by which the amount of atmospheric carbon changes in the units of gigatons per year.

Answer: dC/dt= 3.1 gigatons

8. Integrate the differential equation in Question 7 above. The integral will be of the form:

Where “some number” is the same number from Question 7.

After integrating the above expression, you will get an equation relating the amount of carbon on the left-hand side to how much it changes with time on the right-hand side. Show your steps of integration, and the resulting equation below.

Answer: ∫dC = ∫(3.1) dt

=3.1 t

∫dC = 3.1 t +A

9. Next, we will find the constant of integration (A). We will assume that the total atmospheric carbon is 730 gigatons in the year 2000 (the value listed next to “Atmosphere” in Figure 2 above. Substitute this value for the left-hand side in the equation you got in Question 8 and use time (t) = 2000. This will allow you to find the constant of integration (A). Show your calculations and answer below.

Answer: ∫dC = 3.1 * t +A

730= 3.1 * 2000 +A

A = - 5470

10. Now that we have found the constant of integration (A), substitute this back into the equation you got in Question 8, to find the equation relating the total atmospheric carbon to time in years.

Answer: Required equation is

∫dC = 3.1*t -5470

11. We will now convert our equation for carbon to an equation for CO2. We know that 44 gigatons of CO2 contain 12 gigatons of elemental carbon. Therefore, there is 44/12 or 3.7 times as much CO2 compared to carbon in the atmosphere.

Multiply all the quantities on the right-hand side of the equation you got in Question 10 by 3.7 to get an equation relating the amount of atmospheric CO2 as a function of time.

Answer: required equation relating CO2 is

∫dC = 11.47 * t – 20239

12. We now have a model that tells us how much CO2 gas we will have in the atmosphere if conditions do not change (this is the equation you got as the answer to Question 11). Use this model to predict the amount of atmospheric CO2 in the years 2020 and 2030 (substitute t = 2020, or t = 2030 in your equation and solve).

Answer: In year 2020, amount of atmospheric CO2 was 2,930.4 gigatons.

In the year 2030, amount of atmospheric CO2 will be 3,045.1 gigatons.

13. From Figure 1 above, we see that the concentration of atmospheric CO2 in the year 2020 is 410 ppm (parts per million). Use your model to predict the CO2 concentration in the year 2030 if conditions do not change.

Note: Figure 1 is also called the Keeling curve. It is named after the scientist Charles Keeling, who first started monitoring atmospheric CO2 at Hawaii in 1958. (https://en.wikipedia.org/wiki/Keeling_Curve ).

Hint: use proportions:

Substitute the known values (from answers to Question 12) and find the ppm of CO2 in 2030. Show your calculations below.

Answer: 3045.1/2930.4 =ppmof CO2 in 2030/410

ppm of CO2 in 2030 will be 426,04 ppm

14. Find the slope of the curve in Figure 1. Use the following values:

CO2 concentration in 1960 = 315 ppm

CO2 concentration in 2020 = 410 ppm

The slope (Rise/Run) will give you the amount in ppm by which the atmospheric CO2 increases per year. Show your calculations for the slope below.

Answer: Slope = 410-315/2020-1960

= 1.6

15. Given the atmospheric CO2 concentration of 410 ppm in the year 2020, use your value of the slope from Question 14, to estimate the CO2 concentration in ppm in the year 2030. Does this value obtained from the “Keeling Curve” (Figure 1), agree with the value predicted by your model (derived from Figure 2)?

Answer: Estimation using slope from 2020 to 2030 is given by

=ppm in 2020+ (2030-2020) * slope

= 410+ 16

= 426 ppm

After you finish, please copy-paste all the questions and your answers to the “Quantitative Section” of the “Project_Report_Outline” document, save the Outline report. You will then need to submit this completed “Outline” report on eCampus to the semester project link. Thank you.

Sources:

1. National Oceanic and Atmospheric Administration (n.d.). Trends in Atmospheric Carbon Dioxide. Retrieved from https://www.esrl.noaa.gov/gmd/ccgg/trends/

2. Odenwald, S. (n.d.). Space Math @ NASA. Retrieved from https://spacemath.gsfc.nasa.gov/SpaceMath.html

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