Zerfallsgesetz Beispiel Essay

Aufgrund der $\alpha$- und $\beta$-Zerfälle findet ja eine Umwandlung der Kerne des Ausgangsnuklids statt. Das bedeutet ja zunächst auf qualitativer Ebene, dass die Anzahl der Kerne des Ausgangsnuklids mit der Zeit abnimmt.

Wir wollen nun diese radioaktiven Kernzerfälle mathematisch genauer beschreiben. Entscheidend ist dabei die Frage, wie viele Kerne eines Ausgangsnuklids nach einer Zeit $t$ übrig bleiben.

Zum Zerfallsgesetz: Anzahl $N$ der Kerne eines Nukilds in Abhängigkeit von der Zeit $t$

Zerfallsgesetz

Der radioaktive Zerfall ist ein stochastischer (zufallsbedingter) Prozess, weil man nicht vorhersagen kann, wann genau jeder einzelne Kern zerfällt. Für eine große Anzahl von Kernen lässt sich aber mit statistischen Mitteln ein Gesetz gewinnen, welches den radioaktiven Zerfall exakt beschreibt.

Hier klicken zum Ausklappen

Ist $N_0$ die Anzahl der Kerne des Ausgangsnuklids, so beträgt die Anzahl der Kerne dieses Nuklids nach einer Zeit $t$

$N(t)=N_0\cdot e^{-\lambda t}$

$\lambda$ heißt dabei Zerfallskonstante des entsprechenden Nuklids.

Halbwertszeit $T_{1/2}$

Hier klicken zum Ausklappen

Die Halbwertszeit ist diejenige Zeit, nach der die Hälfte der Kerne des Ausgangsnuklid zerfallen ist.

Mit Hilfe des Zerfallsgesetzes kann man die Halbwertszeit $T_{1/2}$ allein durch die Zerfallskonstante $\lambda$ darstellen.

$N(T_{1/2})=\frac{N_0}{2} \quad \Rightarrow \quad N_0\cdot e^{-\lambda T_{1/2}}=\frac{N_0}{2}\quad \Rightarrow \quad e^{-\lambda T_{1/2}}=\frac{1}{2} $

Invertiert man nun die letzte Gleichung durch Bildung des Logarithmus, erhält man

$-\lambda\cdot T_{1/2}=\ln{\frac{1}{2}}=\ln 1-\ln 2=-\ln 2$

$\Rightarrow T_{1/2}=\frac{\ln 2}{\lambda}$

Hier klicken zum Ausklappen

Zwischen Halbwertszeit $T_{1/2}$ und Zerfallskonstante $\lambda$ eines bestimmten Nuklids besteht der Zusammenhang

$T_{1/2}=\frac{\ln 2}{\lambda}$

Hier klicken zum Ausklappen

Zerfallskonstante für Cäsium-137

Als Beispiel wollen wir die Zerfallskonstante für das radioaktive Cs-137 bestimmen.

In einer Messung wird festgestellt, dass nach etwa nach 60 Jahren 3/4 des Nuklids zerfallen ist.

Wie groß ist dann die Zerfallskonstante $\lambda$?

Antwort:

Durch Umformung des Zerfallsgesetzes bekommt man $\frac{\ln({\frac{N_0}{N(t)}})}{t}=\lambda$. Es ist $N(60 a)=\frac{1}{4}N_0$. (a steht für Jahre)

$\Rightarrow \lambda=\frac{\ln 4}{60a}=0,023 a^{-1}$

Die Halbwertszeit ist gemäss obiger Formel $T_{1/2}=\frac{\ln 2}{\lambda}=\frac{\ln 2}{0,023 a^{-1}}\approx 30 a$ (ca. 30 Jahre)

Aktivität

Es ist im Rahmen des Zerfalls eines radioaktiven Nuklids auch wichtig zu wissen, wie stark er strahlt (Anzahl der Zerfälle pro Zeiteinheit). Die Anzahl der Zerfälle pro Zeiteinheit lässt sich zunächst so schreiben

$-\frac{\Delta N}{\Delta t}$.

Darin ist $-\Delta N=-(N(t+\Delta t)-N(t))$ die Anzahl der zerfallenen Kerne innerhalb eines Zeitintervalls $\Delta t$.

Aus der Analysis sollte man wissen, dass solche Differentialquotienten benutzt werden, um die Ableitung bzw. die momentane Änderungsrate zu definieren.

Hier klicken zum Ausklappen

Die Aktivität $A$ eines radioaktiven Nuklids ist definiert als die Anzahl der Zerfälle pro Zeiteinheit.

$A(t):=-\dot N(t)$

$\dot N(t)$ ist die Ableitung von $N(t)$ nach der Zeit $t$.

Differenzieren wir nun entsprechend der Definition, so bekommen wir

$A(t)=-\dot N(t)=N_0\cdot \lambda \cdot e^{-\lambda t}=\lambda \cdot N(t)$.

Die Einheit ist $[A]=1 Bq=1 s^{-1}$ und heißt Becquerel (Bq).

Man beachte, dass Nachweisgeräte wie das Geiger-Müller-Zählrohr gerade die Anzahl der Zerfälle pro Sekunde, also die Aktivität eines spezifischen Nuklids, messen.

Nuklidkarte

Video: Das Zerfallsgesetz

Video: Das Zerfallsgesetz

@wwp if there are any veterans who are applying to college & need help with their admission essays, it will be our honor to help (1/2)

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