Recall that radioactive/nuclear decay of isotopes follows first-order kinetics, and the integrated rate law for first-order reactions is:
[N]t = concentration at time t
k = decay constant
t = time
[N]0 = initial concentration.
Also, recall that half-life is the time needed for the amount of a reactant to decrease by 50% or one-half.
The half-life of a first-order reaction is given by:
We first need to calculate for the decay constant using the given half-life of 87Rb, 4.7 x 1010 years:
k = 1.4748x10-11 yr-1
Calculate the initial amount of 87Rb atoms present:
initial amount of 87Rb = final amount of 87Rb + amount of 87Sr
amu is approximately equal to molar mass (g/mol)
87Rb = 7.6012x1017 atoms
87Sr = 2.148x1016 atoms
initial amount of 87Rb = 7.6012x1017 atoms + 7.6012x1017 atoms
initial amount of 87Rb = 7.816x1017 atoms
Calculate the age of a rock sample:
[N]0 = 87Rb initial = 7.816x1017 atoms k = 1.4748x10-11 yr–1
[N]t = 87Rb final = 7.6012x1017 atoms t = ???
Solving for t:
Rubidium-87 decays by β-particle production to strontium-87 with a half-life of 4.7 x 1010 years. What is the age of a rock sample that contains 109.7 μg of 87Rb and 3.1 μg of 87Sr? Assume that no 87Sr was present when the rock was formed. The atomic masses for 87Rb and 87Sr are 86.90919 u and 86.90888 u, respectively.
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Our data indicates that this problem or a close variation was asked in Chemistry: An Atoms First Approach - Zumdahl Atoms 1st 2nd Edition. You can also practice Chemistry: An Atoms First Approach - Zumdahl Atoms 1st 2nd Edition practice problems.