Question

Plot a graph showing the variation of undecayed nuclei $$N$$ versus time $$t$$. From the graph, find out how one can determine the half-life and average life of the radioactive nuclei.

Answer 1

Law of Radioactivity defines that the number of Nuclei undergoing number of Nuclei present in the sample at that Instant.

Since from the graph; we have

$$N=N_{ - }e^{-\lambda t }$$ ....1

where $$\lambda $$=Disintegration constant

For $$\therefore $$ For $$T_{1/2}$$ is the time at $$N-\frac{1}{2}N_D$$

$$\Rightarrow$$ $$t=T_{1/2}$$

$$\frac{N_0}{2}$$=$$N_0e^{-\lambda\frac{T}{2}}$$

$$\Rightarrow$$ $$\boxed { T_{ 1/2 }=\frac { Ln2 }{ \lambda } =\frac { 0.693 }{ \lambda } } $$

And for Mean life -we have to sum it over the whole Range for

$$N(t)=N_0e^{-\lambda t}$$

for number of nuclei which decay in time t to t t $$\triangle$$;

$$N(t)\triangle t=\lambda N_{ 0 }e^{ -\lambda \, t}\triangle t$$

For Integration it over the Range $$T=0\, to\,\infty$$

$$\tau$$=$$\frac { \lambda N_{ 0 }\iint _{ 0 }^{ \infty }{ te^{ -\lambda t }dt } }{ N_ 0 }$$

$$=\lambda \int ^{ \infty }_0{ t } e^{ \lambda t }dt$$

$$\boxed { \tau =\frac { 1 }{ \lambda } } $$

Mean-life