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Half-Life is the time taken for the number of atoms in a radioactive sample to fall to half of the original value. Since the mass of radioactive material is proportional to the number of radioactive atoms, the mass of radioactive material also falls to half of the original value in one half-life If the amount of radioactive sample that decays is measured at various times, and a graph is plotted, it can readily be seen that the time taken for the activity of the substance to halve is constant. The half-life of radioactive substances varies widely from isotope to isotope and is independent of external conditions like chemical reactions, pressure, and temperature. In fact there is only one factor that affects the rate of decay, and that is the number of atoms left un-decayed. The graph of radioactive decay is
an example of what might be termed as exponential decay. This is a
situation where the rate of decay is proportional to the amount of
radioactive material remaining un-decayed. If the number of radioactive
atoms at any instant is N, then the rate at which these atoms are decaying
is given by dN/dt, so we can write:  where l is called the decay constant for the radioactive isotope under consideration. The solution to
this equation is:  This equation describes the number of un-decayed nuclei, N, left at time, t, in terms of the number of un-decayed nuclei, N0, at time, t0, and the decay constant, ?. Note that the larger the decay constant, the more rapid the decay. Half-lives vary over a very
wide range, for example the half-life of polonium-212 is 3 × 10-7 seconds,
and the half-life of uranium-236 is 4.5 × 109 years, while there are many
other values between these two extremes.  Where Nn = amount/mass of radioactive sample remaining; No= initial mass/amount of sample x = No. of half-life intervals. The half-life interval, x, can further be determined by using this formula: [1.1]  Where x = half-life interval; w= total time taken for sample to decay; T= half-life Worked Examples Q.1:If 10g of a radioactive sample with half-life of 2hrs is left to decay in a chamber. Calculate the amount of the radioactive sample that would be left after 10 hours. Solution {First, find the half-life interval, x,} X =? w (total time taken) = 10 hrs T (half-life) = 2 hrs Therefore X = 10 / 2 = 5 {We now have the half-life interval as 5. The amount remaining can now be calculated for, using the formula in [1.0]. Thus: Nn = 10 / 25 = 10 / 32 =0.3125g Q. 2:When 4g of a radioactive sample was left in a chamber for 12 hours, 3.5g decayed. Calculate the half-life of the sample. Solution Nn (Amount remaining) = (4.0 - 3.5)g = 0.5g No = 4g Therefore Nn = No / 2x 0.5 = 4 / 2x 2x = 4 / 0.5 2x = 8 2x = 23 X = 3 Using the formula in [1.1] w = 12 hours; x = 3 T = w / x = 12 / 3 T = 4 hrs The half-life is 4 hrs
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