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Halbwertszeit-Rechner

Berechnen Sie die Halbwertszeit und die Menge einer Substanz Schritt für Schritt

Dieser Rechner berechnet die Halbwertszeit, die Anfangsmenge, die verbleibende Menge und die Zeit, wobei die Schritte angezeigt werden.

There are units of mass of a substance with a half-life of units of time. In units of time, there will remain units of mass of the substance.

Enter any three values.

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