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# If the half-life of radium is$7$ days. Its decay constant in a day will beA. $3 \times 10^{ - 13}$ /dayB. $9 \times 10^{ - 3}$ /day C. $1 \times 10^{ - 3}$ /dayD .$6 \times 10^{ - 6}$ /day

Last updated date: 22nd Jul 2024
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Hint: Decay constant is also known as disintegration constant. It is ratio of number of radioactive atoms disintegrating in a short interval of time to the number of atoms of same kind present at the starting. It is represented by the symbol$\lambda$. Decay constant unit is$\delta ^{ - 1}$. It is the inverse of decay time.
The half-life$( T_{1/2} )$ for a radioisotope is the time taken to decay nuclei of a radioactive sample to half of its initial number of nuclei. The half-life of radioactive decay is inversely proportional to the decay constant.

Formula Used:In this solution we will be using the following formulae, $\begin{array}{l}\lambda =\dfrac{0.693}{T_{1/2}} \end{array}$
where $( T_{1/2} )$ is the half-life of radioactive atoms and$( \lambda )$is the decay constant.
Decay constant$( \lambda )$ is the fraction of total nuclei that will decay per unit of time $( t )$.

Complete step by step solution:Radioactive elements are made up of atoms. The emitting radiation transforms radioactive atoms into another chemical element. It may be stable or maybe radioactive. Then it undergoes further decay.
The time interval required for one-half of the atomic nuclei of a radioactive sample decay. Half-life, in radioactivity, is the interval of time required for one-half of the atomic nuclei of a radioactive sample to decay.
The first-order reaction in the half-life is a constant. It is related to the rate constant for the reaction. Where $( T_{1/2} )$ is half-life and $( \lambda )$ is the decay constant. It is given that the half-life of given substances is $7$ .
Now, comes the solution;$\lambda =\dfrac{0.693}{T_{1/2}}$
Where half-life $( T_{1/2} )$is given that i$7$ . We can get $\lambda =\dfrac{0.693}{77}$ $=$$0.00$$=$$9 \times 10^{ - 3}$ /day
That is the decay of the day will be $9 \times 10^{ - 3}$

option B is the correct answer

Note: The radioactive decay law and the half-life formula are used to find the age of organic material, which is known as radioactive dating. By knowing the half-life of radium i $7$ days, the decay constant can be calculated directly by using the decay relation.