Question

State the law of radioactive decay. hence derive the relation N = Noe^-λt . Represent it graphically.

Answer 1

Radioactive decay law :
The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at that instant.
Proof:

• Let ‘N’ be the number of nuclei present at any instant ‘t’.
• Let 'dN’ be the number of nucleic that disintegrated in short interval of time ‘dt’.
• According to law

`(-"dN")/"dt" ∝ "N"`

`therefore ("dN")/"dt"= -λ "N"`........................(1)

Where λ - decayconst

`therefore("dN")/"N"=-λ"dt"`

`therefore \text{integrating both sides}`

`∫ ("dN")/("N")=∫-λ  "dt"`

` \[\ell\]"N"= λ t+C `................................(2)

Where C → integration consta

∴ at  t = 0 , N =N₀

∴ \[\ell\] nN₀ = C                   .............(3)

`therefore ` Frome equation (2) and (3)

`l n N = -λ t +`\[\ell\] `nN_0`

`λ t = In\ N - In\ N_0`

-λT= In `(N/N_0)`

`N/N_0= e^-(λt)`

`therefore N=N_0e^-(λt)`