Question

The half-life of titanium-51 is approximately 5.76 minutesDetermine a so that A(t) = Aoa' describes the amount of titanium-51 left after t minutes, where A, is the amount at time t = 0. Round to six decimal places.

Answer 1

The half-life of titanium-51 is used to find the decay constant 'a' for its exponential decay formula, and the concept of half-life is also applied to determine the remaining amount of titanium-44 after a given time period.

Step-by-step explanation:

The half-life of a radioactive isotope is the time required for half of the initial amount to decay. For titanium-51 with a half-life of 5.76 minutes,

we can determine the constant a in the exponential decay formula A(t) = Ao * a^t. Since half the substance remains after one half-life, we have 0.5 = a^5.76. Taking the natural logarithm of both sides gives ln(0.5) = 5.76 * ln(a), and solving for a would provide the decay constant.

To find how much titanium-44 remains after multiple half-lives, for example, 240 years when the half-life is 60 years, we would use the formula: Amount remaining = initial amount * (1/2)^(elapsed time/half-life).

Substituting the given values yields 0.600 * (1/2)^(240/60) = 0.600 * (1/2)^4 = 0.600 * 1/16 = 0.0375 grams of titanium-44 remaining.