# Rate of Radioactive Decay Worked Example Problem

Worked Chemistry Problems

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/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

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