Rate of Radioactive Decay Worked Example Problem

You can use the equation of the rate of radioactive decay to find how much of an isotope is left after a specified length of time. Here is an example of how to set up and work the problem.

Problem

²²⁶₈₈Ra, a common isotope of radium, has a half-life of 1620 years. Knowing this, calculate the first order rate constant for the decay of radium-226 and the fraction of a sample of this isotope remaining after 100 years.

Solution

The rate of radioactive decay is expressed by the relationship:

k = 0.693/t_1/2

where k is the rate and t_1/2 is the half-life.

Plugging in the half-life given in the problem:

k = 0.693/1620 years = 4.28 x 10^-4/year

Radioactive decay is a first order rate reaction, so the expression for the rate is:

log₁₀ X₀/X = kt/2.30

where X₀ is the quantity of radioactive substance at zero time (when the counting process starts) and X is the quantity remaining after time t. k is the first order rate constant, a characteristic of the isotope that is decaying. Plugging in the values:

log₁₀ X₀/X = (4.28 x 10^-4/year)/2.30 x 100 years = 0.0186

Taking antilogs: X₀/X = 1/1.044 = 0.958 = 95.8% of the isotope remains