^{226}_{88}Ra, 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:

log_{10} X_{0}/X = kt/2.30

where X_{0} 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:

log_{10} X_{0}/X = (4.28 x 10^{-4}/year)/2.30 x 100 years = 0.0186

Taking antilogs: X_{0}/X = 1/1.044 = 0.958 = **95.8% of the isotope remains**