Calculateur Half-Life

Calculer la demi-vie et la quantité d'une substance étape par étape

Cette calculatrice permet de calculer la demi-vie, la quantité initiale, la quantité restante et le temps, avec les étapes indiquées.

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: $N(t)=\frac{125 \sqrt[3]{2}}{64}\approx 2.46078330057592$ .