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So we can substitute this value in for ?y?, and then simplify the decay formula. But regardless of the substance, when we’re looking at half life, we know thatīecause ?y? is the amount of substance that remains as the substance decays, and because ?C? is the amount of substance we started with originally, when the substance has decayed to half of its original amount, ?y? will be equivalent to ?C/2?. Because every substance decays at a different rate, each substance will have a different half life. Since substances decay at different rates, ?k? will vary depending on the substance.Įvery decaying substance has its own half life, because half life is the amount of time required for exactly half of our original substance to decay, leaving exactly half of what we started with. Hence the formula to calculate the half-life of a substance is: t1 2 0.693 t 1 2 0.693. Here is called the disintegration or decay constant. The natural logarithm (ln) is a mathematical function that is the inverse to the exponential (e) function. Mathematically, the half life can be written in terms of the decay rate: Half-life - ln(2) / k. Where ?C? is the amount of a substance that we’re starting with, ?k? is the decay constant, and ?y? is the amount of the substance we have remaining after time ?t?. The formula for the half-life is obtained by dividing 0.693 by the constant. The radioactive half-life is defined as the amount of time taken to reduce the number of nuclei by 50 percent.
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In 24,110 years, youd still have 50 pounds left. Given the basic exponential growth/decay equation h(t) abt, half-life can be found by. So, for now, we’ll just state that the basic equation for exponential decay is Half-life means that, if you have 100 pounds of Plutonium-239. Mathematics for Public and Occupational Health Professionals.
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We won’t work through how to prove these formulas, because in addition to derivatives, we also use integrals to build them, and we won’t learn about integrals until later in calculus.