Q19 CQ - College Physics (Urone)

Question

Does the number of radioactive nuclei in a sample decrease to exactly half its original value in one half-life? Explain in terms of the statistical nature of radioactive decay.

Step-by-Step Solution

Answer

As a result, in one half life, the starting number of nuclides decays by exactly half of the total quantity.

Step 1: Define Radioactivity

The spontaneous emission of radiation in the form of particles or high-energy photons as a result of a nuclear process is known as radioactivity.

Step 2: Explanation

A radioactive substance's half life is defined as the time it takes for the original number of nuclei to decay to half. We know that the formula for the number of nuclei as a function of time is:

 

\({\rm{N  =   }}{{\rm{N}}_{\rm{0}}}{{\rm{e}}^{{\rm{ - \lambda t}}}}.....................{\rm{(1)}}\)

Also, the following is the relationship between the decay constant and the half life:

\({\rm{\lambda   =   }}\frac{{{\rm{0}}{\rm{.693}}}}{{{{\rm{t}}_{{\rm{1/2}}}}}}....................{\rm{(2)}}\)

Now, putting the value of equation two in equation one.

So, we get:

\(\begin{align}N{\rm{ }} &= {\rm{ }}{N_0}{e^{ - 0.693}}\\ &= {\rm{ }}0.5{\rm{ }}{N_0}\end{align}\)

Then, in one half life the original number of nuclide gets decayed by exactly half of the total amount.

Therefore,as a result, the initial number of nuclides decays by exactly half the entire quantity in one half life.