Forgotten Fundamentals of the Energy Crisis - Part 4

by Prof. Al Bartlett

IV. Exponential growth in a finite environment

Bacteria grow by division so that 1 bacterium becomes 2, the 2 divide to give 4, the 4 divide to give 8, etc. Consider a hypothetical strain of bacteria for which this division time is 1 minute. The number of bacteria thus grows exponentially with a doubling time of 1 minute. One bacterium is put in a bottle at 11:00 a.m. and it is observed that the bottle is full of bacteria at 12:00 noon. Here is a simple example of exponential growth in a finite environment. This is mathematically identical to the case of the exponentially growing consumption of our finite resources of fossil fuels. Keep this in mind as you ponder three questions about the bacteria:

(1) When was the bottle half-full? Answer: 11:59 a.m.!

(2) If you were an average bacterium in the bottle, at what time would you first realize that you were running out of space?

Answer: There is no unique answer to this question, so let's ask, "At 11:55 a.m., when the bottle is only 3% filled (1 / 32) and is 97% open space (just yearning for development) would you perceive that there was a problem?" Some years ago someone wrote a letter to a Boulder newspaper to say that there was no problem with population growth in Boulder Valley. The reason given was that there was 15 times as much open space as had already been developed. When one thinks of the bacteria in the bottle one sees that the time in Boulder Valley was 4 min before noon! See Table II.

Table II.
The last minutes in the bottle.

11:54 a.m.

1/64 full (1.5%)

63/64 empty

11:55 a.m.

1/32 full (3%)

31/32 empty

11:56 a.m.

1/16 full (6%)

15/16 empty

11:57 a.m.

1/8  full (12%)

7/8   empty

11:58 a.m.

1/4  full (25%)

3/4   empty

11:59 a.m.

1/2  full (50%)

1/2   empty

12:00 noon

full (100%)

0% empty

Suppose that at 11:58 a.m. some farsighted bacteria realize that they are running out of space and consequently, with a great expenditure of effort and funds, they launch a search for new bottles. They look offshore on the outer continental shelf and in the Arctic, and at 11:59 a.m. they discover three new empty bottles. Great sighs of relief come from all the worried bacteria, because this magnificent discovery is three times the number of bottles that had hitherto been known. The discovery quadruples the total space resource known to the bacteria. Surely this will solve the problem so that the bacteria can be self-sufficient in space. The bacterial "Project Independence" must now have achieved its goal.

(3) How long can the bacterial growth continue if the total space resources are quadrupled?

Answer: Two more doubling times (minutes)! See Table III.

Table III.
The effect of the discovery of three new bottles.

11:58 a.m.

Bottle  No. 1 is one quarter full.

11:59 a.m.

Bottle  No. 1 is half-full.

12:00 noon

Bottle  No. 1 is full.

12:01 p.m.

Bottles No. 1 and 2 are both full.

12:02 p.m.

Bottles No. 1, 2, 3, 4 are all full.

Quadrupling the resource extends the life of the resource by only two doubling times!  When consumption grows exponentially, enormous increases in resources are consumed in a very short time!

James Schlesinger, Secretary of Energy in President Carter's Cabinet recently noted that in the energy crisis "we have a classic case of exponential growth against a finite source."4

Reprinted with permission from Bartlett, A., American Journal of Physics, 46(9), 876, 1978. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Association of Physics Teachers. Copyright 1978, the American Association of Physics Teachers.
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