the rubidium isotope 87Rb is a beta emitter with a half-life of 4.9E10 y that decays into 87Sr. It is used to determine the age of rocks and fossils. Rocks containing the fossils of early animals contain a ration of 87Sr to 87Rb of 0.090. Assuming that there was no 87Sr present when the rocks were formed, calculate the age of these fossils. Answer in units of y.
Assume we start with 100 atoms Rb87 @ t=0 when the fossil just started to form. Then after some time, y will have disintegrated and formed Sr87. Thus 100-y will be the number of Rb87 atoms left.
(Sr87/Rb87)=0.090 from the problem.
y/(100-y)=0.090
Solve for y and 100-y
k = 0.693/t1/2= ??
ln(No/N) = kt
No = 100
N = 100 - y from above.
k = from above
solve for t, the age of the fossil.
To determine the age of the fossils, we can use the decay equation and solve for time (t).
Given:
- Half-life of 87Rb isotope: t₁/₂ = 4.9 × 10¹⁰ years
- Ratio of 87Sr to 87Rb in rocks with fossils: 0.090
- Assuming we start with 100 atoms of 87Rb at t = 0, so initially 100 - y atoms remain (where y is the number of decays that occurred)
First, let's solve for y by using the given ratio:
87Sr / 87Rb = 0.090
y / (100 - y) = 0.090
Solving this equation for y will give us the number of decay events that occurred:
y = 9.0(100 - y)
y = 900 - 9y
10y = 900
y = 90
Now we have the number of decays (y), and we can substitute it into the decay equation:
k = 0.693 / t₁/₂ = 0.693 / 4.9 × 10¹⁰
k = 1.415 × 10⁻¹² (decay constant)
Using the equation for radioactive decay, ln(No / N) = kt, we substitute the initial number of atoms (No = 100) and the number of remaining atoms (N = 100 - y = 100 - 90 = 10):
ln(100 / 10) = 1.415 × 10⁻¹² × t
Simplifying the equation:
ln(10) = 1.415 × 10⁻¹² × t
t = ln(10) / (1.415 × 10⁻¹²)
Calculating this expression gives us the age of the fossils:
t ≈ 2.173 × 10¹² years
Therefore, the age of the fossils is approximately 2.173 × 10¹² years.