In the Classroom
Limiting Reagent and Kinetics: Social Implications and Malthus’ Prediction L. H. Holmes, Jr. Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA 70402
The use of real-life examples to illustrate the concepts of chemistry can be of great help to students who sometimes are not used to thinking abstractly. Population growth can be used to illustrate the concepts of chemical kinetics; and if combined with the concept of limiting reagent as shown below, it can serve to illustrate how one can use concepts and mathematics to take information that is known and obtain a result that is not initially obvious but is interesting and useful—that is, an illustration of problem solving and the scientific method. In the early 1800s, Robert Malthus predicted that the human race would outstrip its food supply and would in the near future starve to death if the number of people was not controlled. Improvements in food production due to technology and science have resulted in a “stay of execution” of the result predicted by Malthus. Much has been written about this problem since that time and it is still considered by many to be of prime importance. Usually the approach is to look at such things as methods of food production, fertility rates, birth control, rate of population growth, and standard of living to indicate the extent of the problem (1). It is generally believed (with apparent good reason) that there is a limit to the number of people that the earth can sustain. That number is dependent on the average standard of living in the world. It is more efficient in terms of energy use to wear layers of clothes to keep warm than to have central heating. If we all lived a bare existence then the earth would support more people than if a much higher standard prevailed. Where the actual position of balance is, should be, or will be is to some extent a value judgment, but decisions to try to limit the population have up to now been based more on politics and religion than on scientific and technical considerations. It is possible to obtain an approximate value for the upper limit of the number of people that the earth can produce or support using the concept of limiting reagent. People, other animals, and all living things on Earth are composed of carbon compounds. At a basic chemical level, all food can be considered as a carbon source. We eat carbon (compounds) along with minerals, etc., to exist and grow. It is possible that some element other than carbon, essential to the operation of the human body, could be the limiting reagent. However, the ratio of the amount of the major elements in the human body to the amount of carbon in the human body is lower than the ratio of the amount of the same elements in the earth’s crust to the amount of carbon in the earth’s crust (2). Thus there are more than enough of these elements on the earth’s surface to combine with carbon to make human bodies and so carbon will be assumed to be the limiting reagent in this discussion. Ultimately if the number of humans continues to increase, the limiting reagent carbon will be depleted and this sets an upper limit for the number of people the earth can have. Of course, the real limit must be less than 1004
this upper limit because all the carbon on Earth cannot become incorporated into humans; some of it will have to be in the form of plants and other animals or else we will live in a world of “Soylent Green”,1 a world without the environment to support human beings. It is possible to obtain an approximate value for the upper limit of the number of people using the following data: mass of the earth = 6 × 10 24 kilograms
(1)
crust of the earth = 1% of the mass of the earth (1) (2) crust of the earth has about 0.03% carbon in it (1) (3) a human body is 18% carbon (2)
(4)
an average human will be assumed to have a mass of 75 kg (5) 0.01
kg crust kg C 24 6 × 10 kg earth 0.0003 kg earth kg crust kg C 0.18 kg person
75 kg person
=
(6)
18
1.3 × 10 people
If all the carbon on the earth were in the form of people, then there would be an upper limit of 1.3 × 1018 people on the earth. At present, the population of 5.4 × 10 9 represents 4 × 10᎑6 percent of the upper limit of 1.3 × 1018, which indicates that about the same percentage of the total amount of carbon is in the form of people. If there is trouble now getting enough carbon in the form of food, it will get much worse as the available carbon for food goes down. If the earth is assumed to be a sphere with a radius of approximately 4000 miles, then the total surface (oceans and land) area of the earth is 5.6 × 1015 ft2. It is not possible to have people on the earth so that there is one person on every square foot of surface (even if only the land surface is considered). Based on “limiting reagent” of area available, the upper limit is in the neighborhood of 1 × 1015 people and certainly a realistic limit is well below even this value. It is possible that other factors such as the ratio of the amount of forests (and other plants) to humans to maintain a habitable environment might lower the upper limit further. If one combines the above results from limiting reagent concepts with “rates of production” (i.e, kinetics), then further interesting and useful information can be gained. The increase in the number of people on the earth is usually expressed as percent increase per year. This can be expressed in the form of a differential equation as (1/N0)(dN/dt) = k
(7)
where N0 is the number of people on earth at the beginning
Journal of Chemical Education • Vol. 75 No. 8 August 1998 • JChemEd.chem.wisc.edu
In the Classroom Table 1. World Population, 1650 –1991 Year
World Population/ Value of k/year ᎑1 Doubling Time/ 109 to fit eq 8 year
1650
0.55
–
1750
0.725
0.00276
– 251
1850
1.175
0.00483
143
1900
1.6
0.00617
112
1950
2.564
0.00943
73.5
1980
4.478
0.0186
37
1991
5.423
0.0174
40
Table 2. Time Required for 1991 Population to Reach Nt Assuming Different Values of k No. of Years To Reach Nt if k is No. of People Nt 0.0001/yr 0.0085/yr 0.017/yr 0.034/yr 1.3 × 1018
193,000
2270
1135
567
1 × 1015
121,000
1516
713
357
1 × 1014
98,000
1156
578
289
1 × 1012
52,000
614
307
153
1 × 1011
29,000
343
172
86
of the year and k is the fractional increase in population per year. If k is a constant, then eq 7 indicates that the fractional increase in population per year is constant; that is, it is a firstorder process. Then k is the equivalent of the specific rate constant in first-order chemical (and nuclear decay) kinetics and has units of reciprocal time, usually (years)᎑1 in the case of population; 100k is the called the percent increase in the population per year. If eq 7 is integrated assuming k is a constant, then one gets an equation very similar to the one obtained in firstorder chemical kinetics (except that the sign of the exponent of e is positive here): Nt = N0e kt
(8)
where Nt is population at a time t if the population is N0 at time t = 0. Equation 8 can be used to calculate the time it takes for the population to reach any numerical value Nt starting with N0 at time 0. One time that is used often is the time required for the population to double (the doubling time, td), as shown in eq 9. The doubling time here is equivalent to the half-life in first-order chemical reactions and nuclear transformation processes. td = (ln 2)/k
(9)
Equation 8 can also be solved for k to give k = (1/t )ln(Nt /N0)
(10)
Table 1 gives the population of the world as a function of time from 1650 to 1991 (3). Using the data in columns 1 and 2, values for k for each interval of time (assuming eq 8 is satisfied in the interval) can be calculated using eq10. These values are shown in Table 1 as well as the doubling times (calculated using eq 9) for each value of k.
A graph of population versus time appears to be approximately exponential. However, the values of k in Table 1 indicate that the rate of growth has been increasing with time and so the graph is not exactly exponential. The decrease in k from 1980 to 1991, while it is hoped to be real, might only be an apparent decrease due to the short interval and uncertainties in the data. The values here are for the world population growth; some countries have rates of population growth higher and some lower than that of the world as a whole. Taking the current world population to be 5.4 × 109 (to be N0 in eq 8), then eq 8 allows the calculation of the time required for the population to reach a given number if a value of k is assumed. The results for several numbers of people and four k values are given in Table 2. A combination of kinetic and limiting reagent considerations together with several assumptions produced the numbers in Table 2, which provides perspective on the problem of population growth in a somewhat quantitative fashion. The time frame in which the problem must be “solved” will not be larger than indicated by Table 2 for a given rate of growth. If the growth continues at the present rate then we have considerably less than a thousand years to gain control of the situation. Slowing the rate will increase the time needed and this can be made rather long as indicated by the data for k = 0.0001 yr ᎑1 in Table 2. However, the times for k = 0.0001 are not long compared to the time the earth will continue to exist much as it does today (several billion years). The data indicate that a desired value for k is zero (a condition called zero population growth, ZPG). There are efforts to reduce population growth rates, such as China’s restricting couples to having one child. In many developed countries the rate of population increase is now decreasing. An optimist would believe that this will become more prevalent and the problem can solve itself by natural processes. Realistically, however, it is likely that some effort will have to be expended. It might be argued that there is a big universe out there with possibly or probably millions or billions of planets that have carbon and the other elements needed as well as space. Calculations similar to those above show that if k = 0.017 yr ᎑1 (the current value), the time required for the mass of all human beings that would exist to become equal to the mass of the known universe (at the present time) is less than 5500 years. So expansion is not the answer. In the appendices of ref 1 are given names and addresses of federal and international organizations from which information can be obtained. Also in the appendices of ref 1 are references for further reading on this topic and related ones. Note 1. Soylent Green is a movie set in the future. It is revealed as the story progresses that the major protein source (Soylent Green) that people eat is processed from human flesh.
Literature Cited 1. Miller, G. T., Jr. Living in the Environment, 7th ed.; Wadsworth: Belmont, CA, 1992; pp 195–221. 2. Lange’s Handbook of Chemistry, rev. 10th ed.; Lange, N. A., Ed.; McGraw-Hill: New York, 1967; pp 163, 816. 3. The World Almanac and Book of Facts; Funk and Wagnalls, Mahwah, NJ, 1994; p 828.
JChemEd.chem.wisc.edu • Vol. 75 No. 8 August 1998 • Journal of Chemical Education
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