Radioactive materials decay by a first-order process. The rate constant for the decay of an isotope whose half-life is 7.69 years is​

k = 0.0901 yrs⁻¹

Explanation:

1st order decay is defined by the equation A = A₀e^-k·t. If A = A₀/2 and t represents the 1st half-life of the isotope then ...  

t(1/2) = ln2/k = 0.693/k => k = 0.693/t(1/2) = 0.693/7.69yrs = 0.0901 yrs⁻¹

radioactive decay occurs in unstable atomic nuclei – that is, ones that don't have enough binding energy to hold

the nucleus together due to an excess of either protons or neutrons.