Answer:
The wood sample has an age of approximately 22800 years.
Explanation:
The Becquerel ([tex]Bq[/tex]) is a SI unit which describes radioactive activity related to decay of radioactive isotopes, which is equivalent to [tex]\frac{1}{s}[/tex]. The decay of radioactive isotope is described by the following ordinary differential equation:
[tex]\frac{dN}{dt} = -\frac{N}{\tau}[/tex] (Eq. 1)
Where:
[tex]\frac{dN}{dt}[/tex] - Disintegration rate, measured in [tex]\frac{1}{s}[/tex].
[tex]N[/tex] - Amount of remaining radioactive nuclei, dimensionless.
[tex]\tau[/tex] - Time constant, measured in seconds.
By integration the solution of this differential equation is obtained:
[tex]\int {\frac{dN}{N} } = -\frac{t}{\tau}\int dt[/tex]
[tex]\ln N = -\frac{t}{\tau} + C[/tex]
[tex]N(t) = N_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (Eq. 2)
Let [tex]N_{1}[/tex] and [tex]N_{2}[/tex] different disintegration rates for Carbon-14 samples, so that:
[tex]N_{1} = N_{o} \cdot e^{-\frac{t_{1}}{\tau} }[/tex] (Eq. 3)
[tex]N_{2} = N_{o}\cdot e^{-\frac{t_{2}}{\tau} }[/tex] (Eq. 4)
If we divide (Eq. 4) by (Eq. 3), then:
[tex]\frac{N_{2}}{N_{1}} = \frac{N_{o}\cdot e^{-\frac{t_{2}}{\tau} }}{N_{o}\cdot e^{-\frac{t_{1}}{\tau} }}[/tex]
[tex]\frac{N_{2}}{N_{1}} = e^{-\frac{1}{\tau}\cdot (t_{2}-t_{1}) }[/tex] (Eq. 5)
If [tex]\Delta t = t_{2}-t_{1}[/tex], we proceed to clear that variable:
[tex]\ln \frac{N_{2}}{N_{1}} = -\frac{1}{\tau}\cdot \Delta t[/tex]
[tex]\Delta t = -\tau\cdot \ln \frac{N_{2}}{N_{1}}[/tex] (Eq. 6)
Time constant is also a function of half-life ([tex]t_{1/2}[/tex]), measured in seconds:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex]
If [tex]t_{1/2} = 1.798\times 10^{11}\,s[/tex], [tex]N_{1} = 14\,\frac{1}{s}[/tex] and [tex]N_{2} = 0.875\,\frac{1}{s}[/tex], the age of the wood sample is:
[tex]\tau = \frac{1.798\times 10^{11}\,s}{\ln 2}[/tex]
[tex]\tau = 2.594\times 10^{11}\,s[/tex]
[tex]\Delta t = -(2.594\times 10^{11}\,s)\cdot \ln \left(\frac{0.875\,\frac{1}{s} }{14\,\frac{1}{s} } \right)[/tex]
[tex]\Delta t \approx 7.192\times 10^{11}\,s[/tex]
[tex]\Delta t \approx 22805.682\,yr[/tex]
The wood sample has an age of approximately 22800 years.
