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Half-Life Calculator
Calculate the half-life, and quantity of a substance step by step
This calculator will calculate the half-life, initial quantity, quantity remained, and time, with steps shown.
Solution
Your input: find $$$N(t)$$$ in $$$N(t)=N_0e^{-\lambda t}$$$ given $$$N_0=250$$$, $$$t_h=15$$$, $$$t=100$$$.
$$$N(t)$$$ is the amount after the time $$$t$$$, $$$N_0$$$ is the initial amount, $$$t_h$$$ is the half-life.
First, find the constant $$$\lambda$$$ (also known as decay constant or decay rate).
We know that after half-life there will be twice less the initial quantity: $$$N\left(t_h\right)=\frac{N_0}{2}=N_0e^{-\lambda t_h}$$$.
Simplifying gives $$$\frac{1}{2}=e^{-\lambda t_h}$$$ or $$$\lambda=-\frac{\ln\left(\frac{1}{2}\right)}{t_h}$$$.
Plugging this into the initial equation, we obtain that $$$N(t)=N_0e^{\frac{\ln\left(\frac{1}{2}\right)}{t_h}t}$$$ or $$$N(t)=N_0\left(\frac{1}{2}\right)^{\frac{t}{t_h}}$$$.
Finally, just plug in the given values and find the unknown one.
From $$$N(t)=250\left(\frac{1}{2}\right)^{\frac{100}{15}}$$$, we have that $$$N(t)=\frac{125 \sqrt[3]{2}}{64}$$$.
Answer: $$$N(t)=\frac{125 \sqrt[3]{2}}{64}\approx 2.46078330057592$$$.