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 August 7th, 2008, 02:45 PM #1 Newbie   Joined: Aug 2008 Posts: 1 Thanks: 0 Half-life problem. Please help :) Guys, Could you please help me with this problem? "What is the half-life of a radioactive substance that has a decay rate of 25% per decade? Hint: Remember that the Rule of 70 does not give a good approximation to half-life when the rate is larger than 15%. We must use the actual formula here." Thanks a bunch..
 August 7th, 2008, 03:06 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Half-life problem. Please help :) Hi The total amount of radioactive material remaining in a decay is always given by $Ke^{-\alpha t}$ where K is the starting amount and $\alpha$ is some positive constant related to the rate of decay. If we measure t in years, then since after 10 years there will be 75% of the starting amount, we have the relationship $\frac34K=Ke^{-10\alpha}\\ \qquad\frac43=e^{10\alpha},$ so $\alpha=\frac1{10}\log\frac43.$ To work out the halflife, we need to find the value of t - call this $\lambda$ - that corresponds to the time where the amount remaining is half the initial amount. This can be represented by the formula $Ke^{-\alpha \lambda}=\frac12K,$ so $e^{\alpha \lambda}=2\\ \quad\qquad \lambda=\frac1\alpha\log2,$ or $\lambda=10\frac{\log2}{\log\frac43}.$
 August 8th, 2008, 09:53 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,747 Thanks: 2133 If the half-life is t years, (3/4)^(t/10) = 1/2, so t = 10ln(1/2)/ln(3/4) = 24.1 to 3 significant figures.

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