The half-life for the radioactive decay of U-238 is 4.5 billion years

QUESTION

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration.If a sample of U-238 initially contained 1.31018 atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms will it contain today?Please explain how you¦
Let t denote the time in billions of years, and let t = 0 denote œnow. Let A(t) denote that amount of U238 left after t billion years. Then a form of the half-life equation that I find more intuituve than the form using œe is A(t) = A(0) (1/2)^(t/h) where h is the half-life (in billions of years here, so that the exponent is dimensionless). If 15% of the atoms have decayed, then 85% remain. 0.85A(0) = A(0) (1/2)^(t/4.5) 0.85 = (1/2)^(t/4.5) log(0.85) = log[(1/2)^(t/4.5)] = (t/4.5) log(1/2) [log(0.85) ] / [log(1/2)] = t/4.5 t = 4.5 [log(0.85) ] / [log(0.50)] 1.055 billion years B) Now we need to¦

edefine t=0 to be the beginning of the universe (it doesnt matter which time we choose to be denoted by t = 0 ” its only time differences that matter). We have A(0) = 1.3 10^18 and want A(13.8): A(13.8) = (1.3 10^18) (1/2)^(13.8/4.5) Ill let you do the arithmetic. Note that 13.8 billion years is just over 3 half-lives, so the answer should be a bit less than (1/2) = 1/8 of the original number.

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