Answer:
Differential equation: [tex]\frac{dA}{dt} = k\cdot A(t)[/tex], [tex]k < 0[/tex], [tex]k = -\frac{1}{\tau}[/tex], [tex]\tau > 0[/tex].
Solution: [tex]A(t) = A_{o}\cdot e^{-\frac{t}{\tau} }[/tex]
Step-by-step explanation:
Let suppose that rate of change of the amount of Carbon-14 atoms are directly proportional to the current amount of Carbon-14 atoms. That is:
[tex]\frac{dA}{dt}\propto A(t)[/tex]
[tex]\frac{dA}{dt} = k\cdot A(t)[/tex]
Where:
[tex]A (t)[/tex] - Amount of Carbon-14 atoms, dimensionless.
[tex]k[/tex] - Proportionality constant, measured in [tex]\frac{1}{yr}[/tex].
[tex]k[/tex] must be negative as death rate is constant and birth rate is zero. ([tex]k< 0[/tex]). Dimensionally, we can rewritte this constant as following:
[tex]k = -\frac{1}{\tau}[/tex]
Where [tex]\tau[/tex] is the time constant ([tex]\tau > 0[/tex]), measured in years.
We can find the solution of the ordinary differential equation by separating each variable:
[tex]\frac{dA}{dt} = -\frac{1}{\tau}\cdot A[/tex]
[tex]\int {\frac{dA}{A} } = -\frac{1}{\tau} \int \, dt[/tex]
[tex]\ln A(t) = -\frac{t}{\tau} + C[/tex]
[tex]A(t) = e^{-\frac{t}{\tau}+C }[/tex]
[tex]A(t) = e^{C}\cdot e^{-\frac{t}{\tau} }[/tex]
[tex]A(t) = A_{o}\cdot e^{-\frac{t}{\tau} }[/tex]
Where [tex]A_{o}[/tex] is the initial amount of atoms of Carbon-14.
