For small quantities of radioactive material, the rate of…
For small quantities of radioactive material, the rate of decay is directly proportional to the amount of radioactive material present. Find the mass of radioactive material as a function of time.
Explanation of Method
Translate the physical law into a linear first-order ODE and solve it by separation of variables, then express the result in terms of the half-life.
Step 1: Set up and solve the ODE
With $m(t)$ the mass and $k>0$ the decay constant, "rate of decay proportional to mass" means $\dfrac{dm}{dt}=-km$. Separating and integrating,
\[\frac{dm}{m}=-k\,dt \quad\Longrightarrow\quad \ln m=-kt+C \quad\Longrightarrow\quad \boxed{\,m(t)=m_0e^{-kt},}\]
where $m_0=m(0)$ is the initial mass.
Step 2: Half-life form
The half-life $t_{1/2}$ satisfies $m_0e^{-kt_{1/2}}=m_0/2$, so $t_{1/2}=\dfrac{\ln2}{k}$, and the law can equivalently be written
\[m(t)=m_0\left(\frac12\right)^{t/t_{1/2}}.\]
Result
Final answer:
\[\boxed{\,m(t)=m_0e^{-kt}=m_0\left(\dfrac12\right)^{t/t_{1/2}}.}\]