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प्रश्न
State the law of radioactive decay. hence derive the relation N = Noe-λt . Represent it graphically.
उत्तर
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)`
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