Answer:
The lava flow is approximately 351.925 million years.
Explanation:
We find that atoms of Uranium-235 decays and turns into Lead-207 and that half-life of the former one is [tex]703.8\times 10^{6}[/tex] years. The initial amount of Uranium-235 is the sum of current atoms of Uranium-235 and Lead-207. The decay of isotopes is modelled by the following ordinary differential equation:
[tex]\frac{dn}{dt} = -\frac{n}{\tau}[/tex] (Eq. 1)
Where:
[tex]\frac{dn}{dt}[/tex] - Rate of change of the amount of Uranium-235 atoms, measured in atoms per year.
[tex]n[/tex] - Current amount of Uranium-235 atoms, measured in atoms.
[tex]\tau[/tex] - Time constant, measured in years.
The solution of this differential equation is described below:
[tex]n = n_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (Eq. 2)
Where:
[tex]n_{o}[/tex] - Initial amount of Uranium-235 atoms, measured in atoms.
[tex]t[/tex] - Time, measured in years.
In addition, we can calculate the time constant in terms of the half-life:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (Eq. 3)
If we know that [tex]n_{o} = 1000\,atoms[/tex], [tex]n = 707\,atoms[/tex] and [tex]t_{1/2} = 703.8\times 10^{6}\,yr[/tex], then the age of the lava flow is:
From (Eq. 2):
[tex]t = -\tau \cdot \ln \frac{n}{n_{o}}[/tex]
By (Eq. 3);
[tex]\tau = \frac{703.8\times 10^{6}\,yr}{\ln 2}[/tex]
[tex]\tau \approx 1.015\times 10^{9}\,yr[/tex]
[tex]t = -(1.015\times 10^{9})\cdot \ln \frac{707\,atoms}{1000\,atoms}[/tex]
[tex]t \approx 351.925\times 10^{6}\,yr[/tex]
The lava flow is approximately 351.925 million years.
