Category: Chemical Engineering Math, Differential Equations
"Published in Newark, California, USA"
How old is a bottle of wine if the tritium 3H content is 45% of a new wine? The half-life of tritium is 12.5 years.
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 tritium at time t
t = decaying time of tritium
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 = x0 and t = 0 at the start, then the value of C is
Hence, the particular solution of the working equation is
If x = ½ x0 at t = 12.5 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 x = 0.45 x0 as stated in the problem, therefore, the age of a bottle of wine that contains tritium is
Take the natural logarithm on both sides of the equation,w e have
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 = x0 and t = 0 at the start, then the value of C is
Hence, the particular solution of the working equation is
If x = ½ x0 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
