Theory of Radioactive decay

The “k∆t” term here is equivalent to the in the dice analogy.

So in the real radioactive decay, the number of atoms decaying in an interval of time ∆t is given by:

        

                ∆N = -k∆t.N

Rearranging this, we see that,

                Rate of decay = -kN

This rate of decay is going to vary with time, and so in order to find the rate of decay at “an instant”, ∆t must be small so that N will not change much within it.

What are the units of k?

        

Well, if –kN has units of atoms/sec and N = number of atoms at start then k must have units of  or.

The exponential function – exponential changes

“All exponential changes arise whenever the rate of change of something or other is directly proportional to how much of that something there presently happens to be.”

Exponential growth:

• Sales of pop records/fashion fads etc.
• The spread of an epidemic in a community.

Exponential decay:  

• Charge on a capacitor in a RC circuit.
• Amputude v time for damped oscillations of a harmonic oscillator.
• Radioactive decay.

The equation of radioactive decay:

If k is the chance of decay in 1 second then k∆t is the chance of decay in ∆t seconds.

In an interval of time ∆t, the number of decayed atoms (∆N) is related to the number of atoms originally present (N) by the equation:

 ∆N = -k∆t.N

From this:

                 = -k∆t

Integrating both sides gives:

        

                 dN        =         dt

Where N          = number of atoms at time t=0

Where N            = number of atoms at time t=t

                =

         =        -kt

        

        N = N. e

Half life

This is the time taken for N to become.

i.e. the number of atoms remaining after time t to become equal to half of the original number of atoms present.

So, putting N =  in the equation N = N. e

We get:

        

         = N. e

        e

        

Taking log to the base e gives us:

        t =

The activity of a radioactive material

This is exactly the same thing as its rate of decay. Radioactive sources are marked with “strengths” or activities in curies 1 Curie = 3.7*10 disintegrations per second. OR nowadays in Bequerels 1Bq = 1 disintegration per second.

Since activity = rate of decay.

        A =

And since

        N = Ne

        

        A =

        

        A = -kNe

But

        N = Ne

        A = -kN

By the use of substitution we come get the following equation.

         A = Ae