Particles in a nucleus are bound by the strong force. A reasonable model of this

Question

Particles in a nucleus are bound by the strong force. A reasonable model of this binding would be a barrier potential of finite energy. Quantum mechanics allows a small, nonzero probability for a particle to "tunnel" through a potential barrier. Working with this rough model, we can say that, for a given unstable nucleus, there is a constant probability of some particle, say an alpha particle, tunneling through the barrier and being emitted as radiation. Since this probability is constant, depending only on the particular nuclide involved, you would expect the number of decays per second to be proportional to the number of nuclei present. Since the number of decays is the same as the amount by which the total number of nuclei (of the particular type being considered) is reduced, you can construct the relationship dN(t)dt=N(t), where N(t) is the number of nuclei at a time t. The constant of proportionality is called the decay constant. If N(t)=Cet, where C is some constant, what is dN(t)dt? Express your answer in terms of C, , and t.

Explanation / Answer

N(t) = Ce-t

dN(t)/dt = -Ce-t --- (answer)

but  Ce-t = N (t)

Therefore, dN(t)/dt = -Ce-t = - N(t)

So, dN(t) / dt = - N(t)

This is how the relationship is constructed.

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