Answer :
Final answer:
The exponential decay function for a substance with a known half-life can be used to calculate the amount of the substance remaining after a given time. In the case of Mercury-197 with a half-life of 3 days and an initial amount of 50 grams, approximately 0.60 grams will remain after 20 days.
Explanation:
The question deals with the concept of half-life in relation to the decay of the isotope Mercury-197. The half-life of a substance is the time required for half the atoms in a sample to decay. For Mercury-197, this time is given as three days.
For decay of this type, we often use an exponential decay function, usually in the form of N(t) = No * e^-λt, where No represents the initial amount of the substance, e is the base of the natural logarithm (approximately 2.71828), λ is the decay constant, and t is the time elapsed. However, since we know the half-life, we can use the fact that during one half-life, the substance decays to half of its original amount to replace λ with ln(2)/T (where T represents the half-life), giving us the formula N(t) = No * e^-(ln(2)/T)t.
For the Mercury-197 example with an initial amount No of 50 grams and a half-life T of 3 days, we want to find the amount N leftover after t = 20 days. Plugging these values into the formula gives: N(20) = 50 * e^-(ln(2)/3)*20. This equation results in approximately 0.60 grams of Mercury-197 remaining after 20 days.
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