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/t1/2
where k is the rate and t1/2 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
