Question

(a) 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.

(b) The air in some caves includes a significant amount of radon gas, which can lead to lung cancer if breathed over a prolonged time. In British caves, the air in the cave with the greatest amount of the gas has an activity per volume of $1.55×10_{5}Bq/m_{3}$ . Suppose that you spend two full days exploring (and sleeping in) that cave. Approximately how many $_{222}Rn$ atoms would you take in and out of your lungs during your two-day stay? The radionuclide $_{222}Rn$ in radon gas has a half-life of 3.82 days. You need to estimate your lung capacity and average breathing rate.Open in App

Updated on : 2022-09-05

Solution

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(a) 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

$N=N_{−}e_{−λt}$ ....1

where $λ$=Disintegration constant

For $∴$ For $T_{1/2}$ is the time at $N−21 N_{D}$

$⇒$ $t=T_{1/2}$

$2N_{0} $=$N_{0}e_{−λ2T}$

$⇒$ $T_{1/2}=λLn2 =λ0.693 $

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

$N(t)=N_{0}e_{−λt}$

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

$N(t)△t=λN_{0}e_{−λt}△t$

For Integration it over the Range $T=0to∞$

$τ$=$N_{0}λN_{0}∬_{0}te_{−λt}dt $

$=λ∫_{0}te_{λt}dt$

$τ=λ1 $

Mean-life

(b) The equation for the activity is given by:

$R=λN$

Here, R is the activity, N is the number of nuclei and $λ$ is the decay constant. The equation for the decay constant is given by,

$λ=T_{1/2}ln2 $

Here, $T_{1/2}$ is the half - life

Thus, $R=T_{1/2}ln2 N$

$N=ln2RT_{1/2} $

Dividing by volume, $VN =VR ln2T_{1/2} $

Substituting the value,

$=VN =(1.55×10_{5}Bq/m_{3})ln23.82×24×3600s $

$=7.38×10_{10}$ $atoms/m_{3}$

Thus, these are $7.38×10_{10}$ atoms $/m_{3}Rn$ atms per unit volume in the cave.

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