# Growth and Decay

Begin with one bacterium capable of dividing each hour. After 10 hours there will be 1,024 bacteria (2^10). That's an example of geometric, or exponential, growth. If the colony is confined to a small jar of food, and half that food is converted into those 1,024 bacteria in 10 hours, you can imagine their leader declaring: "Do not concern yourselves. There remains as much food in this jar as we have consumed during our entire past history of 10 generations." His audience will be unpleasantly surprised when the food is gone after just one additional hour, at which point there are 2,048 hungry bacteria.

A useful relationship between percentage growth rate, or annual interest rate, and the doubling time is the 70 percent rule: (Growth Rate) x (Doubling Time) = 70 percent. For example, a 7 percent annual interest rate would double your money in 10 years. The math is related to the natural logarithm of 2, equal to 0.693, approximated by 70 percent.

Exponential decay, the mirror image of growth, is pertinent to nuclear technology, nuclear medicine and geologic dating. Decay Rate multiplied by Half-life = 0.693. For example, Tritium (Hydrogen-3), illuminating your watch dial, has a half-life of 12 years, so the probability of any one atom decaying within any one year is 0.693/12 yr = 0.058/year. Half of the Earth's original inventory of Uranium-238, with a half-life of 4.5 billion years (the age of the Earth) has been converted into Lead-206. The original inventory of fissionable Uranium-235, with a half life of 0.7 billion years, is 99 percent decayed into Lead-207. Modern uranium is only 0.7 percent U-235 and needs to be enriched for use in weapons or power stations. Among its many fission products is Iodine-131, with a half-life of eight days. This isotope accumulates in the thyroid, causes thyroid cancer, and is used in treating thyroid cancer as well as hyperthyroidism. Patients receiving iodine radiotherapy should stay away from children.

Geologic history is quantified by a dozen independent radiometric dating methods in addition to the U-Pb methods. The one creationists love to criticize is Carbon-14 dating. C-14, a trace constituent in the atmosphere and hence plants, is produced from atmospheric nitrogen by cosmic rays and decays back into N-14. Because daughter nitrogen cannot be distinguished from abundant background nitrogen, radiocarbon dating must be calibrated by measuring the C-14 remaining in wood of known age, determined by counting tree rings. Anyway, C-14, with a half-life of 5,730 years, is barely detectable in objects older than 45,000 years. The Earth is 100,000 times older than that.

As an existential exercise, plot on my diagram the percent of food remaining in the bacterial colony during the last three hours of its existence.

#### Don Garlick

Bio:
Don Garlick is a geology professor retired from Humboldt State University. He invites any questions relating to North Coast science, and if he cannot answer it he will find an expert who can. E-mail dorsgarlick@yahoo.com.

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