Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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# A Radioactive Isotope is Being Produced at a Constant Rate Dn/Dt = R in an Experiment. the Isotope Has a Half-life T1/2. Show that After a Time T >> T1/2 - Physics

#### Question

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t1/2. Show that after a time t >> t1/2 the number of active nuclei will become constant. Find the value of this constant.

#### Solution

Given:
Half life period of isotope = t1/2

Disintegration constant, lambda = 0.693/t_"1/2"

Rate of Radio active decay (R) is given by

R = "dN"/"dt"

We are to show that after time t >> t_"1/2"the number of active nuclei is constant.

("dN"/"dt")_"present" = R = ("dN"/"dt")_"decay"

therefore R = ("dN"/"dt")_"decay"

Rate of radioactive decay, R = lambdaN

Here, λ = Radioactive decay constant
N = Constant number

R = 0.693/t_"1/2" xx N

⇒Rt_"1/2" = 0.693 N

⇒N = (Rt_"1/2")/0.693

This value of N should be constant.

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Solution A Radioactive Isotope is Being Produced at a Constant Rate Dn/Dt = R in an Experiment. the Isotope Has a Half-life T1/2. Show that After a Time T >> T1/2 Concept: Radioactivity - Law of Radioactive Decay.
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