Class 12 Physics Nuclei Radioactive Decay

• Radioactive decay is a phenomenon in which unstable nucleus decay to form stable nuclei.
• There are 3 types of radioactive decay by which a unstable nuclei becomes stable nuclei:-
• Alpha decay:-
• In alpha decay α - particle (helium nucleus) is emitted.
• Beta decay:-
• In beta decay electrons or positrons are emitted.
• If electronsare emitted then it is known as β- decay; and when positronsare emitted then the decay is known as β+
• Gamma decay:-
• In gamma decay high energy photons(γ rays) are emitted.
• Law of radioactive decay states that the number of nuclei undergoing decay per unit time is ∝to total number of nuclei in the sample.
• Consider a sample of radioactive nuclei (Uranium). Thesample of uranium will have several uranium nuclei and they will undergo radioactive decay to convert into relatively stable lighter nuclei.
• That means the number of uranium nuclei will gradually keep on decreasing with time.
• For example:-
• If in a sample there are 10 uranium nuclei and in another 100 uranium nuclei the rate of decay will be slower where there are 10 uranium nuclei as compared to 100 uranium nuclei.

Generalized expression for law of radioactive decay:-

Suppose there are ‘N’ active nuclei at any instant of time (‘t’).

In small interval of time ‘dt’ =’dN’ (number of nuclei decay).

Rate of decay (dN/dt) ∝N (number of nuclei which are present in the sample).

(dN/dt) = -(λ N)

• Whereλ = decay constant and (-ive) sign means the number of nuclei is gradually decreasing with time.
• => (dN) =-λ dt, integrating both the sides,
• =>N0N∫(dN)/N = -λ0t∫ N dt
• Where N0=number of nuclei at t=0 and N= number of nuclei after time t.
• [ln N]N0N = -λ[t]0t
• (ln N0 – ln N) =-λt
• =>ln(N/N0) = -λt
• => (N/N0)=e- λt
• =>N = N0 e-λt always N0> N.
• Radioactive law is also known as exponential decay law as number of nuclei decreases exponentially which can be seen from the graph if plotted between N and t.
• It is followed only by radioactive nuclei.

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