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Exponential Decay Problems, 2

__Category__: Chemical Engineering Math, Differential Equations

"Published in Vacaville, California, USA"
The half-life of Sr-90 is 29 years. What fraction of the atoms in a sample of Sr-90 would remain in 100 years later?
__Solution__:
From the description of a given problem, it is about exponential decay problem. The rate of change of a substance is directly proportional to the negative of its substance present. The working equation can be expressed as follows
where
x = amount of Sr-90 at time t
t = decaying time of Sr-90
k = proportional constant for decaying
By separation of variables, transfer x to the left side of the equation and dt to the right side of the equation as follows
Integrate on both sides of the equation, we have
Take the inverse natural logarithm on both sides of the equation, we have
If x = x_{0} and t = 0 at the start, then the value of C is
Hence, the particular solution of the working equation is
If x = ½ x_{0} at t = 29 years, then the value of k is

Take natural logarithm on both sides of the equation, we have
Hence, the complete working equation of the above equation is

If t = 100 years, therefore, the fraction of Sr-90 in a sample is