Minnesotans For Sustainability©


Sustainable:  A society that balances the environment, other life forms, and human interactions over an indefinite time period.




Forgotten Fundamentals of the Energy Crisis

Albert A. Bartlett*

Section II of II


[Note: For convenience of the reader, MFS has separated the paper into two sections.]

Section I of II

Section I Background:
    Dr. Bartlett's comments on the 20th Anniversary of "Forgotten Fundamentals of the Energy Crisis"
    Reflections on the "Fundamentals" Paper Twenty Years Later
        Table  I: Lifetimes of non-renewable resources for different rates of growth
    Horror Stories
    Part 1: Introduction
    Part 2: Background
    Part 3: The Powers of Two
        Table I. Filling the squares on the chessboard.
    Part 4:  Exponential Growth in a Finite Environment
        Table II. The last minutes in the bottle
        Table III. The effect of the discovery of three new bottles
    Part 5:  Length of Life of a Finite Resource When the Rate of Consumption Is Growing Exponentially

    Part 6: How Long will our Fossil Fuels Last?

        Table IV. United States crude oil
   Figure 1. History of U.S. crude oil production
        Table V. Exponential expiration time in years of various estimates of U.S. oil reserves
        Figure 2.  History of world crude oil production
        Table VI .World crude oil data
        Table VII. Life expectancy in years of various estimates of world oil reserves
        Table VIII. United States coal resources
   Figure. 4.  History of U.S. coal production

Section II of II

    Part 7:  What Do the Experts Say?
Table IX. Lifetime in years of United States coal
    Part 8:  A Word of Caution
    Part 9:   What do we do now?
    Part 10:  Conclusion
    Part 11:  A Postscript for Science Teachers
Figure. 6.  The delta function in the darkness
    Appendix [the math]
    About the Author


VII.  What do the Experts Say?

Now that we have seen the facts let us compare them with statements from authoritative sources.  Let us look first at a report to the Congress.

    It is clear, particularly in the case of coal, that we have ample reserves.... We have an abundance of coal in the ground.  Simply stated, the crux of the problem is how to get it out of the ground and use it in environmentally acceptable ways and on an economically competitive basis... At current levels of output and recovery these reserves can be expected to last more than 500 years.14

Here is one of the most dangerous statements in the literature.  It is dangerous because news media and the energy companies pick up the idea that "United States coal will last 500 years" while the media and the energy companies forget or ignore the important caveat with which the sentence began, "At current levels of output . . ."  The right-hand column of Table IX shows that at zero rate of growth of consumption even the low estimate of the U.S. coal resource "will last over 500 years."  However, it is absolutely clear that the government does not plan to hold coal production constant "at current levels of output."


Table IX. 

Lifetime in years of United States coal (EET).  The lifetime (EET) in years of U.S. coal reserves (both the high and low estimate of the U.S.G.S.) are shown for several rates of growth of production from the 1972 level of 0.5 (x109) metric tons per year. 

          High Estimate (yr)      Low Estimate (yr)  
Zero            2872                      680 
  1%            339                       205 
  2%            203                       134 
  3%            149                       102 
  4%            119                       83 
  5%            99                        71 
  6%            86                        62 
  7%            76                        55 
  8%            68                        50 
  9%            62                        46 
 10%            57                        42 
 11%            52                        39 
 12%            49                        37 
 13%            46                        35

    Coal reserves far exceed supplies of oil and gas, and yet coal supplies only 18 % of our total energy.  To maintain even this contribution we will need to increase coal production by 70 % by 1985, but the real goal, to increase coal's share of the energy market will require a staggering growth rate.15

While the government is telling us that we must achieve enormous increases in the rate of coal production, other governmental officials are telling us that we can increase the rate of production of coal and have the resource last for a very long time.

    The trillions of tons of coal lying under the United States will have to carry a large part of the nation's increased energy consumption, says (the) Director of the Energy Division of the Oak Ridge National Laboratories.  "He estimated America's coal reserves are so huge, they could last ‘a minimum of 300 years and probably a maximum of 1000 years."16

Compare the above statement of the life expectancy of U.S. coal reserves with the results of very simple calculations given in Table IX.

    In the three-hour CBS television special on energy (August 31, 1977) a reporter stressed the great efforts that are being made to increase the rate of production of U.S. coal, and he summarized the situation in these words, "By the lowest estimate, we have enough (coal) for 200 years.  By the highest, enough for more than a thousand years."

Again, compare the above statement with the results of simple calculations shown in Table IX.

While we read these news stories we are bombarded by advertisements by the energy companies which say that coal will last a long time at present rates of consumption and which say at the same time that we must dramatically increase our rate of production of coal.

    At the rate the United States uses coal today, these reserves could help keep us in energy for the next two hundred years . . . Most coal used in America today is burned by electric power plants –(which)– consumed about 400 million tons of coal last year.  By 1985 this figure could jump to nearly 700 million tons.17

Other advertisements stress just the 500 years (no caveat): "We are sitting on half the world's known supply of coal –enough for over 500 years."18  Some ads stress the idea of self-sufficiency without stating for how long a period we might be self-sufficient.  "Coal, the only fuel in which America is totally self-sufficient."19  Other ads suggest a deep lack of understanding of the fundamentals of the exponential function.

    Yet today there are still those who shrill (sic) for less energy and no growth... Now America is obligated to generate more energy - not less - merely to provide for its increasing population... With oil and gas in short supply, where will that energy come from?  Predominately from coal.  The U.S. Department of the Interior estimates America has 23 %  more coal than we dreamed of,  4,000,000,000,000 (trillion!) tons of it.  Enough for over 500 years.  (The non-sentences are in the original.)20

A simple calculation of the EET based on a current production rate of  0.6 x 109 tons / yr shows that the growth in the rate of production of coal can't exceed  0.8 % / yr if the ad's 4 x 1012 tons of coal is to last for the ad's 500 yr.  However, it should be noted that the 4 x 1012 tons cited in the ad is  2.8  times the size of the large estimate of U.S. coal reserves and is 12 times the size of the small estimate of U.S. coal reserves as cited by Hubbert.

When we view the range of creative information that is offered to the public we cannot wonder that people are confused.  We may wish that we could have rapid growth of the rate of consumption and have the reserves of U.S. coal last for a large number of years, but very simple calculations are all that is needed to prove that these two goals are incompatible.  At this critical time in our nation's history we need to shift our faith to calculations (arithmetic) based on factual data and give up our belief in Walt Disney's First Law: "Wishing will make it so."21

On the broad aspects of the energy problem we note that the top executive of one of our great corporations is probably one of the world's authorities on the exponential growth of investments and compound interest.  However, he observes that "the energy crisis was made in Washington."  He ridicules "the modern-day occult prediction" of "computer print-outs" and warns against extrapolating past trends to estimate what may happen in the future.  He then points out how American free-enterprise solved the great "Whale Oil Crisis" of the 1850s.  With this single example as his data base he boldly extrapolates into the future to assure us that American ingenuity will solve the current energy crisis if the bureaucrats in Washington will only quit interfering.22   It is encouraging to note that the person who made these statements in 1974, suggesting that the energy crisis was contrived rather than real, has now signed his name on an advertisement in Newsweek Magazine (Sept. 12, 1977) saying that, "Energy is not a political issue.  It's an issue of survival. Time is running out."  However, the same issue of Newsweek Magazine carried two advertisements for coal which said: "We've limited our use of coal while a supply that will last for centuries sits under our noses... Coal –can provide our energy needs for centuries to come."

Carefully read this ad by the Edison Electric Institute for the Electric Companies telling us that: "There is an increasing scarcity of certain fuels.  But there is no scarcity of energy.  There never has been.  There never will be.  There never could be. Energy is inexhaustible."  (Emphasis is in the original.)23  We can read that a professor in a school of mining technology offers "proof" of the proposition: "Mankind has the right to use the world's resources as it wishes, to the limits of its abilities . . ."24

We have the opening sentence of a major scientific study of the energy problem: "The United States has an abundance of energy resources; fossil fuels (mostly coal and oil shale) adequate for centuries, fissionable nuclear fuels adequate for millennia and solar energy that will last indefinitely."25  We can read the words of an educated authority who asserts that there is no problem of shortages of resources: "It is not true that we are running out of resources that can be easily and cheaply exploited without regard for future operations."  His next sentence denies that growth is a serious component of the energy problem, "It is not true that we must turn our back on economic growth" (emphasis is in the original).   Three sentences later he says that there may be a problem: "We must face the fact that the well of nonrenewable natural resources is not bottomless."26  He does suggest that lack of "leadership" is part of the problem.

We have a statement by Ralph Nader, "The supply of oil, gas, and coal in this country is enormous and enough for hundreds of years.  It is not a question of supply but a question of price and profits, of monopolies and undue political influence."27

Expert analysis of the problem can yield unusual recommendations.  We have the opening paper in an energy conference in which a speaker from a major energy company makes no mention of the contribution of growth to the energy crisis when he asserts that: "The core of the energy problem both U.S. and worldwide [is] our excessive dependence on our two scarcest energy resources - oil and natural gas."  For him continued growth is not part of the problem, it is part of the solution! More energy must be made available at a higher rate of growth than normal – in the neighborhood of 6 percent per year compared to a recent historical growth rate of 4 percent per year.28

The patient is suffering from cancer, and after a careful study, the doctor prescribes the remedy; give the patient more cancer.  Here is a second case where cancer is prescribed as the cure for cancer.  The National Petroleum Council in its report to the energy industry on the energy crisis: observed that "Restrictions on energy demand growth could prove (to be) expensive and undesirable. . . The Council ‘flatly rejected' any conservation-type measures proposing instead the production of more energy sources domestically and the easing of environmental controls."29

Study this statement carefully: "Energy industries agree that to achieve some form of energy self-sufficiency the U.S. must mine all the coal that it can."30  The plausibility of this statement disappears and its real meaning becomes apparent when we paraphrase it: "The more rapidly we consume our resources, the more self-sufficient we will be."  David Brower has referred to this as the policy of "Strength through Exhaustion." 31 This policy has many powerful adherents.  For example, on the three-hour CBS television special on energy (Aug. 31, 1977) William Simon, energy adviser to President Ford said: "We should be "trying to get as many holes drilled as possible to get the proven (oil) reserve . . ."

Is it in the national interest to get and use these reserves as rapidly as possible?  We certainly get no sense of urgency from the remarks of the Board Chairman of a major multinational energy corporation who concludes the discussion "Let's Talk Frankly About Energy" with his mild assessment of what we must do.  "Getting on top of the energy problem won't be easy.  It will be an expensive and time-consuming task.  It will require courage, creativeness and discipline . . ."32

If one searches beyond the work of Hubbert for an indication of others who understand the fundamental arithmetic of the problem one finds occasional encouraging evidence.33  However, when one compares the results of the simple exponential calculations with news stories, with statements from public officials, and with assertions in advertisements of the energy companies it is hard to imagine that this arithmetic is widely understood.

The arithmetic of growth is the forgotten fundamental of the energy crisis.

VIII.  A Word of Caution

We must note that these calculations of the EET of fossil fuels are not predictions of the future.  They simply give us first-order estimates of the life expectancies of known quantities of several fuels under the conditions of steady growth which our society and our government hold sacred.  These estimates are emphasized as aids to understanding the consequences of any particular growth scenario that the reader may want to consider or to evaluate.

The rate of production of our mineral resources will not rise exponentially until the EET is reached and then plunge abruptly to zero, as modeled in these calculations and as shown in curve A of Fig. 5 even though our national goals are predicated on uninterrupted growth.  The rate of production of our nonrenewable mineral resources will not follow the classical S-shaped transition from an early period of exponential growth to a horizontal curve representing a constant rate of production, curve B.  Such a curve can be achieved in the production of renewable resources such as food, forest products, or the production of solar energy, provided the rate of production of the renewable resource is not dependent on fossil fuels.

Fig. 5.  Three patterns of growth.  Curve A represents steady exponential growth in the rate of production of a non-renewable resource until the resource is exhausted at Te, the exponential expiration time (EET).  The area under the curve from the present (t  =  0)  to  t  =  Te  is equal to the known size of the resource.  Curve C represents Hubbert's model of the way in which the rate of production of a nonrenewable resource rises and falls.  This model is based on studies of the rate of use of resources which have been nearly completely consumed.  The area under the curve from the present to  t =  infinity is equal to the size of the resource.. Curve B represents the rate of production of a renewable resource such as agricultural or forest products, where a constant steady-state production can be maintained for long periods of time provided this production is not dependent on the use of a nonrenewable resource (such as petroleum) whose production is following a curve such as C.

Reference has already been made to the dependence of modern agriculture on petroleum, and as long as this dependence continues, the curve of agricultural production would be expected to follow curve C, (the curve for nonrenewable petroleum) rather than curve B.  Although the rate of production of mineral resources has been growing exponentially, one knows that at some time in the future the resource will be exhausted and the rate of production will return to zero.

The past history, this one future datum and a careful study of the rate versus time of production of resources that have expired has led Dr. M. King Hubbert to the conclusion that the rate of production of a nonrenewable resource will rise and fall in the symmetrical manner of a Gaussian error curve as shown in curve C of Fig. 5.  When he fits the data for U.S. oil production in the lower 48 states to a curve such as C, Hubbert finds that we are now just to the right of the peak.  We have used one-half of the recoverable petroleum that was ever in the ground in the U.S. and in the future the rate of production can only go downhill.  However, our national demand for petroleum has continued to grow exponentially and the difference between our demands and our production has been made up by imports.  Bold initiatives by the Congress could temporarily reverse the trend and could put a small bump on the downhill side of the curve.  Alaskan oil can put a little bump on the downhill side of the curve.  The downhill trend on the right side of the curve was noted clearly by Deputy Energy Secretary John O'Leary under the headline, "U.S. Energy ‘Disaster' Inevitable by 1985,"34

    Although U.S. oil and gas production hit their peak several years ago and are declining by about 8 percent per year, O'Leary said, the nation has avoided serious problems by using more foreign oil...We are walking into a disaster in the next three or four years with our eyes wide open.

The most dramatic conclusion that Hubbert draws from his curve for the complete cycle of U.S. oil production is that the consumption of the central  80 % of the resource will take place in only 67 yr!

It is very sobering to face the downhill side of the curve and to note that in the past the rise in our annual per capita consumption of energy has gone hand-in-hand with the increase of our standard of living.  It is more sobering to note the close coupling between our production of food and our use of petroleum.  It is even more sobering to note that on March 7, 1956 (over 22 years ago) Dr. Hubbert, addressing the conference in San Antonio, Texas, of a large group of petroleum engineers and geologists said:

    According to the best currently available information, the production of petroleum and natural gas on a world scale will probably pass its climax within the order of half a century, while for both the United States and for Texas, the peaks of production can be expected to occur within the next 10 or 15 years. (i.e., between 1966 and 1971)

Pazik tells33 of the shock this statement and the related analysis caused in oil industry circles and he tells about the efforts that were made by the "experts" to ignore this and the other results of the analysis made by Hubbert.

IX.  What do we do Now?

The problems are such that we have rather few options.  All of the following points are vital:

(i)  We must educate all of our people to an understanding of the arithmetic and consequences of growth, especially in terms of the earth's finite resources.  David Brower has observed that, "The promotion of growth is simply a sophisticated way to steal from our children."

(ii)  We must educate people to the critical urgency of abandoning our religious belief in the disastrous dogma that "growth is good," that "bigger is better," that "we must grow or we will stagnate," etc., etc.  We must realize that growth is but an adolescent phase of life which stops when physical maturity is reached.  If growth continues in the period of maturity it is called obesity or cancer.  Prescribing growth as the cure for the energy crisis28, 29 has all the logic of prescribing increasing quantities of food as a remedy for obesity.  The recent occasion of our nation's 200th anniversary would be an appropriate time to make the transition from national adolescence to national maturity.

(iii)  We must conserve in the use and consumption of everything.  We must outlaw planned obsolescence.  We must recognize that, as important as it is to conserve, the arithmetic shows clearly that large savings from conservation will be wiped out in short times by even modest rates of growth.  For example, in one or two dozen years a massive federal program might result in one-half of the heat for the buildings where we live and work being supplied by solar energy instead of by fossil fuels.  This would save 10 %  of our national use of fossil fuels, but this enormous saving could be completely wiped out by two years of  5 %  growth.  Conservation alone cannot do the job!  The most effective way to conserve is to stop the growth in consumption.

As we consider the absolute urgency of conservation we must recognize that some powerful people are hostile to the concept of conservation.  One of our great multinational oil companies has advertised that conservation is: "Good for you –but not if there's too much."  And in the same ad they noted that: "Conservation does no harm."35

In his message to the American people President Carter proposed a tax on large "gas guzzling" cars.  General Motors Chairman Thomas Murphy had the following reaction to this proposal to conserve energy: Murphy calls the excise tax on big cars, coupled with rebates on small cars "one of the most simplistic irresponsible and short-sighted ideas ever conceived by the hip-shooting marketeers of the Potomac."36

Big labor is hostile to this same conservation measure.  Leonard Woodcock, President of the United Auto Workers said of the tax: "I respectfully suggest that the proposal is wrong. It is not properly thought through and should be withdrawn."37

Congress is not enthusiastic about conservation: "Look for Senate leaders on both sides of the aisle –including Chairman Russell Long of the Finance Committee and Minority Leader Howard Baker– to gang up on Carter's energy package.  The two influential lawmakers want more stress on the production of oil, not so much on conservation."38

Closer to home we can note that our governors don't show much enthusiasm for conservation: "The nation's governors told President Carter that the federal government is placing too much emphasis on conservation and not enough on developing new resources."39

With all this influential opposition one can see how difficult it will be to launch major national programs of energy conservation.

(iv) We must recycle almost everything.  Except for the continuous input of sunlight the human race must finish the trip with the supplies that were aboard when the "spaceship earth" was launched.

(v) We must invest great sums in research (a) to develop the use of solar, geothermal, wind, tidal, biomass, and alternative energy sources; (b) to reduce the problems of nuclear fission power plants; (c) to explore the possibility that we may be able to harness nuclear fusion.  These investments must not be made with the idea that if these research programs are successful the new energy sources could sustain growth for a few more doubling times.  The investments must be made with the goal that the new energy sources could take over the energy load in a mature and stable society in which fossil fuels are used on a declining exponential curve as chemical raw materials and are not used as fuel for combustion.  One great area of responsibility of our community of scientists and engineers is vigorous pursuit of research and development in all these areas.  These areas offer great opportunity to creative young people.

Perhaps the most critical things that we must do is to decentralize, and consequently humanize, the scale and scope of our national industrial and utility enterprises.40

(iv) We must recognize that it is exceedingly unscientific to promote ever-increasing rates of consumption of our fuel resources based on complete confidence that science, technology, and the economics of the marketplace will combine to produce vast new energy resources as they are needed.  Note the certainty that characterizes this confidence.

Coal could help fight a rear-guard action to provide time for scientific breakthroughs which will move the world from the fossil fuel era of wood, gas, oil, and coal to the perpetual energy era of infinitely renewable energy resources.41  The supply (of coal) is adequate to carry the U.S. well past the transition from the end of the oil and gas era to new, possibly not discovered sources of energy in the 2000s.42

There seems to be an almost complete absence of the caution that would counsel us to stop the growth of our national energy appetite until these "unlimited energy resources" are proven to be capable of carrying the national energy load.  We must recognize that it is not acceptable to base our national future on the motto "When in doubt, gamble."

Fusion is most commonly mentioned as being an unlimited energy source.  The optimism that leads some people to believe that fusion power will be ready whenever it is needed should be balanced against this opening statement in a report on fusion from MIT.  "Designing a fusion reactor in 1977 is a little like planning to reach heaven: theories abound on how to do it, and many people are trying, but no one alive has ever succeeded."43

If the generation of electric power from fusion was achieved today, we could ask how long would it then be before fusion could play a significant role in our national energy picture.  The time-constant for the replacement of one major energy source by another can be estimated from the fact that the first nuclear fission reactor was operated in December 1942.  Even though the recent growth of nuclear energy in the U.S. has been spectacular, it was not until around 1972 that annual energy consumption equaled our annual energy consumption from firewood!  By 1973 nuclear energy had climbed to the point where it supplied  1.3 %  of our U.S. total annual energy consumption and  4.6 %  of our electrical power.44 Thus in 31 years nuclear energy has grown to provide only a small fraction of our energy needs.  Had there been no growth of our national electrical needs since 1942, today's nuclear plants would be supplying  41 %  of our national electrical power.

(vii) We can no longer sit back and deplore the lack of "leadership" and the lack of response of our political system.  In the immortal words of Pogo "We have met the enemy, and they's us."  We are the leaders, we are vital parts of the political system and we have an enormous responsibility.

The arithmetic makes clear what will happen if we hope that we can continue to increase our rate of consumption of fossil fuels.  Some experts suggest that the system will take care of itself and that growth will stop naturally, even though they know that cancer, if left to run its natural course, always stops when the host is consumed.  My seven suggestions are offered in the spirit of preventive medicine.

X. Conclusion

The preceding calculations are offered as guideposts which must be understood by those who would deal constructively with the energy crisis.  The role and limitations of science in analyzing and in solving our problems was beautifully expressed by Gustav Lebon (1841-1931).

    "Science has promised us truth; an understanding of such relationships as our minds can grasp.  It has never promised us either peace or happiness."

Perhaps the most succinct conclusion that is indicated by the analysis above is taken from the immortal words of Pogo, "The future ain't what it used to be!"  The American system of free enterprise has flourished for 200 years with spectacular achievements.  Until recently it flourished in a world whose energy resources were essentially infinite.  Whenever one fossil fuel came into short supply, another could always be found to take its place.  We are now close enough that we can see the end of the world's total supply of fossil fuels.  The challenge that we must meet is set forth clearly in the question, "Can free enterprise survive in a finite world?"  President Carter observed (April 18, 1977) that: "If we fail to act soon we will face an economic, social, and political crisis that will threaten our free institutions." (See Fig. 6)

XI.  A Postscript for Science Teachers

For decades physics teachers throughout the world have discussed the  RC  circuit and the decay of radioactive atoms and have thus introduced the simple differential equation that gives rise to exponential decay of the charge on the capacitor or of the number of remaining radioactive nuclei.  These provide a wonderful opportunity for us to digress and to point out that exponential arithmetic has great value outside of these two special examples in physics and to show our students that exponential arithmetic is probably the most important mathematics they will ever see.  It is especially important for students to see how the change in the sign of the exponent can make an enormous difference in the behavior of the function.  But we will need to do more.  We must integrate the study of energy and of the exponential arithmetic into our courses as has been done, for example, in one new text.44   In addition, we have an even larger task.  As science teachers we have the great responsibility of participating constructively in the debates on growth and energy.  We must be prepared to recognize opinions such as the following, which was expressed in a letter to me that was written by an ardent advocate of "controlled growth" in our local community: "I take no exception to your arguments regarding exponential growth. I don't think the exponential argument is valid on the local level."

Figure. 6.  The delta function in the darkness.  Redrawn from Hubbert's Fig. 69, Ref. 7.  The epoch of the world's use of its fossil fuels is shown on a time scale of human history from 5000 yr ago to 5000 yr in the future.  The vertical axis is the rate of consumption of fossil fuels measured in units of  1014  kW h / yr.  The vertical scale is a linear scale.

We must bring to these debates the realism of arithmetic and the new concept of precision in the use of language.  We must convey to our students the urgency of analyzing all that they read for realism and precision.  We must convey to our students the importance of making this analysis even though they are reading the works of an eminent national figure who is writing in one of the world's most widely circulated magazines.  (The emphasis in the following quotations is in the original.

    The simple truth is that America has an abundance of energy resources... An estimated 920 trillion cubic feet of natural gas still lies beneath the United States.  Even at present consumption rates, this should last at least 45 years... About 160 billion barrels of oil still lie below native ground or offshore.  That's enough to last us into the next century at present rates of consumption.45

When students analyze these statements they can see that the first statement is false if "abundance" means "sufficient to continue currently accepted patterns of growth of rates consumption for as long as one or two human lifetimes."  An evaluation of the second and third statements show that they are falsely reassuring because they suggest the length of time our resources will last under the special condition of no growth of the rates of use of these resources.  The condition of no growth in these rates is absolutely contrary to the precepts of our national worship of growth.  It is completely misleading to introduce the results of "no growth:" unless one is advocating "no growth."

If it is true that our natural gas reserves will last 45 yr at present rates of consumption ( R / r0  =  45 yr), then Eq. (6) shows that this amount of gas would last only 23.6 yr at an annual growth rate of  5 % / yr, and only 17 yr at an annual growth rate of  10 % / yr.  When the third statement is analyzed one sees that the given figure of 160 x 109  barrels of reserves is roughly 60 % larger than Hubbert's estimate.  This amount would last 49 yr if oil was produced at the 1970 rate of  3.3 x 109  barrels / yr, held constant with no growth.  However, our domestic consumption is now roughly twice the rate of domestic production, so this amount of oil would satisfy domestic needs for only about 25 yr if there was no growth in these domestic needs.  If   R / r0  =  25 yr, then Eq. (6) shows that this amount of oil would last only 16.2 yr if production grew 5 % / yr and only 12.5 yr if it grew  10 % / yr.

We can conclude that the author is probably advocating growth in the rate at which we use fossil fuels from the following imprecise statement, "The fact is that we must produce more energy."  Therefore the author's statements about the life expectancy of resources at current rates of use are irrelevant.  When they are offered as reassurance of the lack of severity of our energy problem they are dangerously and irresponsibly misleading.

Students should be able to evaluate the same author's statement about coal, "At least 220 billion tons of immediately recoverable coal –awaits mining in the United States."  This "could supply our energy needs for several centuries."  Students can see that the size of the coal reserves given by the author is significantly smaller than either of the two estimates given by Hubbert.  They can see that it is imprecise and meaningless to suggest how long a resource will last if one says nothing about the rate of growth of production.  In addition to encouraging our students to carry out their responsibility to analyze what they read, we must encourage them to recognize the callous (and probably careless) inhumanity of a prominent person who is perhaps in his fifties,45  offering reassurance to younger readers to the effect, "don't worry, we have enough petroleum to last into the next century," The writer is saying that "There is no need for you to worry, for there is enough petroleum for the rest of my life."  Can we accept the urgings of those who advocate unending expansion and growth in the rates of consumption of our fossil fuel resources and who say "Why worry, we have enough to last into the next century."

We must give our students an appreciation of the critical urgency of evaluating the vague, imprecise, and meaningless statements that characterize so much of the public debate on the energy problem.  The great benefits of the free press place on each individual the awesome responsibility of evaluating the things that he or she reads.  Students of science and engineering have special responsibilities in the energy debate because the problems are quantitative and therefore many of the questions can be evaluated by simple analysis.

Students must be alert not only to the writings in the popular press but to the writings in college textbooks.  In the bookstore of a school of engineering I purchased a book that was listed for one of the courses, possibly in political science.  Here are a few interesting statements from the book:46

    Our population is not growing too rapidly, but much too slowly... To approach the problem ("the population scare") from the standpoint of numbers per se is to get the whole thing hopelessly backward... Our coal supply alone, for example, is sufficient to power our economy for anywhere for 300 to 900 years –depending on the uses to which it is put– while gas and oil and coal together are obviously good for many centuries... So whatever the long-term outlook for these energy sources, it is obvious (that) natural shortage cannot account for the present energy crunch.

Dr. Hubbert, speaking recently, noted that we do not have an energy crisis, we have an energy shortage.  He then observed that the energy shortage has produced a cultural crisis. (See Fig. 7.)

We must emphasize to our students that they have a very special role in our society, a role that follows directly from their analytical abilities.  It is their responsibility (and ours) to become the great humanists.

Note added in proof: 

Two incredible misrepresentations of the life expectancy of U.S. coal reserves have been called to my attention recently.  Time (April 17, 1978, p.74) said:

    Beneath the pit heads of Appalachia and the Ohio Valley, and under the sprawling strip mines of the West, lie coal seams rich enough to meet the country's power needs for centuries, no matter how much energy consumption may grow." (emphasis added)

In reply to my letter correcting this, Time justified their statement by saying that they were using the Citibank estimate of U.S. coal reserves which is larger than the estimate used by Hubbert.

A beautiful booklet, "Energy and Economic Independence" (Energy Fuels Corporation of Denver, Denver, 1976) said: "As reported by Forbes magazine, the United States holds 437 billion tons of known (coal) reserves.  That is equivalent to 1.8 trillion barrels of oil in British Thermal units, or enough energy to keep 100 million large electric generating plants going for the next 800 years or so." (emphasis added)   This is an accurate quotation from Forbes, the respected business magazine (December 15, 1975, p.28).  Long division is all that is needed to show that 437 x 109 tons of coal would supply our 1976 production of  0.665 x 109 tons per year for only 657 years, and we probably have fewer than 500 large electric generating plants in the U.S. today.  This booklet concluded, "Your understanding of the facts about "energy and economic independence' issue is of great importance."

A very thoughtful comment on fusion was made to me recently by a person who observed that it might prove to be the worst thing that ever happened to us if we succeed in using nuclear fusion to generate electrical energy because this success would lead us to conclude that we could continue the unrestrained growth in our annual energy consumption to the point (in a relatively few doubling times) where our energy production from the unlimited fusion resource was an appreciable fraction of the solar power input to the earth.  This could have catastrophic consequences.

Richard Stout, columnist for the New Republic, noted (Time, March 27, 1978, p.83) that in America, "We consume one third of all the energy, one third of the food and enjoy one half of the world's income.  Can a disparity like this last?  I think that much of the news in the next 50 years is going to turn on whether we yield to the inevitable graciously or vindictively."


When a quantity such as the rate  r( t )  of consumption of a resource grows a fixed percent per year, the growth is exponential:                                                                                        

r ( t )   =   r0  e k t  =   r0  2 t / T2                                                (1)

where  r0  is the current rate of consumption at  t  =  0, e is the base of natural logarithms, k  is the fractional growth per year, and  t  is the time in years.  The growing quantity will increase to twice its initial size in the doubling time T2where:
T2 (yr)  =  (ln 2) / k  »  70 / P                                                     (2)

and where  P, the percent growth per year, is 100k.  The total consumption of a resource between the present ( =  0) and a future time T is:

C  =  {T  to  0}      r(t) dt                                                              (3)

The consumption in a steady period of growth is:

C  =  r0 {T  to  0}    e kt dt =  ( r0 / k ) ( e kt - 1 )                        (4)

If the known size of the resource is R tons, then we can determine the exponential expiration time (EET) by finding the time  Te  at which the total consumption C is equal to R:

R  =  ( r0 / k ) ( e kTe - 1 )                                                             (5)

We may solve this for the exponential expiration time Te where:

EET  =  Te  =  ( 1 / k )  ln ( k R / r0   + 1 )                                    (6)

This equation is valid for all positive values of  k  and for those negative values of  for which the argument of the logarithm is positive.


1. This paper is based on a series of articles, "The Exponential Function" which is appearing in The Physics Teacher:
(a) Vol. 14, p. 393 (Oct. 1976); (b) Vol. 14, p. 485 (Nov. 1976);

(c) Vol. 15, p. 37 (Jan. 1977); (d) Vol. 15, p. 98 (Mar. 1977);
(e) Vol. 15, p. 225 (Apr. 1977); (f) Vol. 16, p. 23 (Jan. 1978);
(g) Vol. 16, p. 92 (Feb. 1978); (h) Vol. 16, p. 158 (Mar. 1978).

An early version of this paper was presented at the Third Annual UMR-MEC Conference on Energy, held at the University of Missouri at Rolla, Oct. 12-14, 1976, and appears in the volume of the Proceedings of the Conference.  The early version, or minor revisions of it have been published in Not Man Apart  published by Friends of the Earth: July / Aug. 1977, Vol. 7, No. 14 pp. 12-13; The Vermillion Flycatcher (Tucson, Arizona Audubon Society, December 1977);  The Colorado Business Review (Grad. Sch. of Business Admin. of the University of Colorado, Jan / Feb 1978).
2. Newsweek, Dec. 6, 1976.
3. A. A. Bartlett; Civil Engineering, Dec. 1969,  p. 71.
4. Time, April 25, 1977, p. 27.
5. W. Von Engelhardt, J. Goguel, M. King Hubbert, J. E. Prentice, R.A. Price, and R. Trumpy; Environmental Geology, Vol. 1, 193-206 (1976).
6. A. A. Bartlett, Proceedings of the Third Annual UMR-MEC Conference on Energy,  University of Missouri at Rolla, Missouri, October 12-14, 1976, p. 10.
7. U.S. Energy Resources, a Review as of 1972, a background paper prepared at the request of the Hon. Henry M. Jackson, Chairman of the Committee on Interior and Insular Affairs of the United States Senate, pursuant to Senate Resolution 45:  M. King Hubbert, A National Fuels and Energy Policy Study, Serial 93-40 (92-75) Part 1 (U.S. GPO, Washington, D.C., 1973), $2.35, 267 pages.  This document is an invaluable source of data on consumption rates and trends in consumption, for both the U.S.A. and the world.  In it Hubbert also sets forth the simple calculus of his methods of analysis.  He does not confine his attention solely to exponential growth.  He predicts that the rate of rise and subsequent fall of consumption of a resource will follow a symmetrical curve that looks like the normal error curve.  Several figures in this paper are redrawn from Hubbert's paper.

8. L. Ruedisili and M. Firebaugh, Perspectives on Energy, (Oxford University Press, New York,  1975).
9. M. King Hubbert, Resources and Man, National Academy of Sciences and National Research Council, (Freeman, San Francisco, 1969), Chapter 8.
10.  M. King Hubbert, "Energy Resources of the Earth," Scientific American, Sept. 1971, p. 60. Reprinted as a book (Freeman, San Francisco, 1971).
11. M. Iona, Physics Teacher, Vol. 15, p. 324 (1977).
12. Emile Benoit, "The Coming Age of Shortages," Bulletin of Atomic Scientists, January 1976, p. 7.  Benoit attributes his information to David Pimintel et al., "Food Production and the Energy Crisis," Science, Vol. 182, p. 448 (Nov. 2, 1973).  This article is the first of three by Benoit (Bulletin of Atomic Scientists, Jan., Feb., Mar., 1976,).   These are one of the best presentations I have read of coming problems of food, fuels, and resources.
13. Newsweek, Jan. 31, 1977.
14. "Factors Affecting the Use of Coal in Present and Future Energy Markets" a background paper prepared by The Congressional Research Service at the request of Sen. Henry M. Jackson, Chairman of the Committee on Interior and Insular Affairs of the United States Senate pursuant to Senate Resolution 45, a National Fuels and Energy Policy Study Serial No. 93-9 (92-44) (U.S. GPO, Washington, D.C., 1973) pp. 41, 42, 15.
15. "The Energy Crisis" a booklet by the U.S. Energy Research and Development Agency  (ERDA) Oak Ridge, Tennessee, no date, p. 3. (1975 or 1976).
16. Associated Press story "Energy Head Stresses Coal Reserves," in the Boulder Daily  Camera, July 5, 1975.
17. "America's Coal: A Gold Mine of Energy," Exxon Corporation two-page full-color ad in Newsweek, 1975.
18. "They're trying to tell us something.  We're foolish not to listen," American Electric Power Company, Inc.  Two-page ad in Newsweek, 1975.
19. "The call to greater energy independence" American Electric Power Company, Inc., ad in Newsweek, Nov. 3, 1975.
20. "An open letter on energy to those who are still employed." American Electric Power Company, Inc., ad in Newsweek, Jan. 12, 1976.
21. W. H. Miernyk, Journal of Energy and Development, Vol. 1, No. 2, p. 223 (1976).
22. "The Whale Oil, Chicken, and Energy Syndrome," an address before the Economic Club of Detroit by Walter B. Wriston, Chairman, First National City Corporation, Feb. 25, 1974.
23. "The Transitional Storm, Part I, An Explanation," by the Edison Electric Institute for the Electric Companies, in  Broadcasting, July 26, 1976.
24. Charles O. Frush, "Moral Basis for Mineral Resource Use and Development Policy"  The Mines Magazine, Colorado School of Mines, March 1973, p. 20.
25. J. C. Fisher, "Physics and the Energy Problem," Physics Today, American Institute of Physics, New York, 1974.
26. "Opening Remarks, UMR-MEC Conference on Energy," R. L. Bisplinghoff, Proceedings of the Conference, Oct. 7-9, 1975, University of Missouri at Rolla.
27. Washington Star, Feb. 12, 1977.
28. L. G. Hauser, "Creating the Electric Energy Economy," Proceedings of the Second Annual UMR-MEC Conference on Energy,  October 7-9, 1975, p. 3., University of Missouri at Rolla.
29. Gil Bailey, "Conservation–Development Proposed As Solution," Washington Bureau of the Boulder Daily Camera, March 13, 1973.
30. Time, May 19, 1975, p. 55.
31. D. Brower, Not Man Alone, Vol. 6, No. 20, Nov. 1976; Friends of the Earth, 529 Commercial, San Francisco.
32. C. C. Garvin, Jr., Chairman of the Exxon Corporation; Full page ad in the Rocky Mountain News, July 23, 1976.
33. G. Pazik, in a special editorial feature, "Our Petroleum Predicament," in Fishing Facts ("The magazine for today's freshwater fisherman"), Northwoods Publishing Co., P.O. Box 609, Menomonee Falls, WI 53051.  Nov. 1976.  Reprints are available at $0.30 each from the publisher.  This is an excellent summary of the present situation and of the way we got into our petroleum predicament.
34. The Arizona Republic, Feb. 8, 1978.
35. "Conservation is like Cholesterol" an ad copyrighted 1976 by the Mobil Oil Corporation.
36. Boulder Daily Camera, April 4, 1977.
37. Boulder Daily Camera, May 16, 1977.
38. U.S. News & World Report, July 25, 1977, P. 8.
39. Boulder Daily Camera, July 10, 1977.
40. Amory Lovins, "Energy Strategy, the Road Not Taken," Foreign Affairs, Oct. 1976.  This material is now available as a book, Soft Energy Paths; Toward a Durable Peace Ballinger, Cambridge, MA, 1977).  It is said that this book "could very well be the most important book on energy policy of this decade."
41. W. L. Rogers, Special Assistant to the Secretary of the Interior, quoted in the Denver Post, Nov. 19, 1976.
42. Time, April 4, 1977, p. 63.
43. Technology Review, Dec. 1976, p. 21, reprinted in the second edition of Ref. 8.
44. Robert H. Romer, Energy–An Introduction to Physics (Freeman, San Francisco, 1976), pp. 594-597.  In addition to making energy the central theme of an introductory text, this book has 18 appendices (61 pages) of data ranging from  "Units and conversion factors" to the "History of energy production and consumption in the world and in the United States" to "Exponential growth" to "Consumer prices of common sources of energy."  The book is at once a text and a valuable source of reference data.
45. Melvin Laird, "The Energy Crisis: Made in U.S.A." Reader's Digest, Sept. 1977, P. 56.
46. M. Stanton Evans, Clear and Present Dangers, (Harcourt Brace Jovanovich, New York, 1975).


This paper has been rewritten and reprinted many times in the 20 years since it was first published.

The paper was enlarged and was published in: Mineral & Energy Resources, Colorado School of Mines, Golden, Colorado ;Part I, Vol. 22, Sept. 1979, pp. 1-46; Part II, Vol. 22, Nov. 1979, pp. 1-9; Part III, Vol. 23, Jan. 1980, pp. 1-10.

The enlarged version was also published in the Journal of Geological Education, Vol. 28, Jan. 1980, pp. 4-35.

The paper was rewritten as a chapter in the book, Perspectives on Energy  by L.C. Ruedisili and M.W. Firebaugh, Third Edition, Oxford University Press, New York City, 1982.

The paper was reprinted in New Trends in Physics Teaching, Vol. IV, 1984, pp. 20-37 by the United Nations Educational Scientific and Cultural Organization in Paris, France.

Short versions of this paper have been printed as essays in introductory physics textbooks by Halliday & Resnick, Serway, and Tipler.  Other authors of physics texts have written chapters or sections in their texts using these applications of exponential arithmetic.

The paper has been reprinted in full or abridged in over  30  different publications or proceedings, and was translated into Spanish for publication in Mexico.

I adapted the paper to data on energy in Canada, and it was published as  "Forgotten Fundamentals of the Energy Crisis: A Canadian Perspective," by the Industrial Energy Division of the Ministry of Energy, Mines, and Resources of the Federal Government of Canada, Ottawa, Canada, May 1986.

This paper was listed as one of ten "memorable papers" for the year 1978 that was included in a list of  "Memorable papers from the American Journal of Physics, 1933-1990"  R.H. Romer, American Journal of Physics, Vol. 59, March 1991, p. 205.

The paper was included in the "Physics Teachers' CD-ROM Toolkit" published by the University of Nebraska, 1993.

About the Author

Albert A. Bartlett is Professor Emeritus of Physics at the University of Colorado in Boulder.   A distinguished and dedicated educator and widely published author,  Dr. Bartlett recently received the first George Gamow Memorial Lecture Award in recognition of his "most significant contribution to the public's understanding of science."   Besides an  illustrious career as a physicist, educator, and activist for stopping population growth, Dr. Bartlett is perhaps best known for his lecture, "Arithmetic, Population, and Energy: The Forgotten Fundamentals of the Energy Crisis," which has been given over a thousand times, including repeat performances to members of Congress at the U.S. Capitol.  NPG is honored to have Dr. Bartlett as one of our Board of Advisors.

Video copies of Dr. Bartlett's lecture are available from University of Colorado Television; Academic Media Services; Campus Box 379; Boulder, CO 80309-0379; (303) 492-1857.

Negative Population Growth is indeed privileged to be able to reprint "Forgotten Fundamentals of the Energy Crisis" on the 20th anniversary of its first publication in the American Journal of Physics in September of 1978. Since then, it has been reprinted in full or abridged in over 30 different publications or proceedings, including translation into Spanish for publication in Mexico. The special importance of this article as a resource for educators is reflected in its use in several introductory physics textbooks, in the "Physics Teachers' CD-ROM Toolkit" published by the University of Nebraska, and in the United Nations Educational and Cultural Organization's journal New Trends in Physics Teaching.

Reprinted with permission from The American Journal of Physics,  Volume 46, September 1978, pages 876 to 888.  Copyright 1978 American Association of Physics Teachers.

Return to Section I of II. Go to "Author: Albert Bartlett"

Regarding exponential growth, also see on the MFS website "The Pond Lilly Parable".
* Courtesy of NPG, Negative Population Growth
See original at < http://www.npg.org/specialreports/bartlett_index.htm >.


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