Question

Obtain an expression for the decay law of radioactivity. Hence show that the activity A(t) =λN_O e^-λt.  

Answer 1

Expression for a decay law of radioactivity:

If ‘N(t)’ is the number of parent nuclei present at any instant ‘t’, ‘dN’ is the number of nuclei disintegrated in a short interval of time ‘dt’, then,
dN ∝ −N(t) dt
dN = −λ N(t) dt   

where λ is known as decay constant or disintegration constant. The negative sign indicates the disintegration of atoms.

Integrating both sides of the equation,

`int_{"N"_0}^{"N"_(("t"))} "dN"/("N"("t")) = int_0^"t" -lambda  "dt"`

where, N₀ is number of parent atoms at time t = 0.

∴ log_e = `("N"("t"))/("N"_0) = -lambda"t"`

∴ N(t) = `"N"_0 "e"-lambda"t"`

Expression for activity:

1. The rate of decay, i.e., the number of decays per unit time −`("dN"("t"))/"dt"`, is called as activity A(t).
2. It is given as,
   A(t) = −`("dN"("t"))/"dt"`
   = λN(t)
   = λN₀e^−λt 
   At t = 0, the activity is A₀ = λN₀. 
   ∴ A(t) = A₀e^−λt