Decay Formula

Decay Formula

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. The decay law calculates the number of undecayed nuclei in a given radioactive substance.

Decay Formula – 

Formula for Half-Life in Exponential Decay –

\[\large N(t)=N_{0}\left ( \frac{1}{2}^{\frac{t}{t_{\frac{1}{2}}}} \right )\]

\[\large N(t)=N_{0}e^{\frac{-t}{\tau }}\]

\[\large N(t)=N_{0}e^-\lambda t\]

\(\begin{array}{l}N_{0}\end{array} \)
is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
N(t) is the quantity that still remains and has not yet decayed after a time t,
\(\begin{array}{l}t_\frac{1}{2}\end{array} \)
is the half-life of the decaying quantity,
\(\begin{array}{l}\tau \end{array} \)
is a positive number called the mean lifetime of the decaying quantity,

\(\begin{array}{l}\lambda\end{array} \)
is a positive number called the decay constant of the decaying quantity.

The three parameters

\(\begin{array}{l}t_\frac{1}{2}\end{array} \)
\(\begin{array}{l}\tau \end{array} \)
, and
\(\begin{array}{l}\lambda\end{array} \)
are all directly related:

\[\large t_{\frac{1}{2}}=\frac{\ln (2)}{\lambda}=\tau  \ln(2)\]

Decay constant (

\(\begin{array}{l}\lambda\end{array} \)
) gives the ratio of number of radioactive atoms decayed to the initial number of atoms, which is

\[\LARGE \lambda=\frac{0.693}{t_{\frac{1}{2}}}\]

Decay Law is used to find the decay rate of a radioactive element.

Here are few Radioactive Isotopes and their half-life:

1) As per decay rate of

\(\begin{array}{l}10^{-24}\end{array} \)

\(\begin{array}{l}\large Hydrogen-7 =23\end{array} \)
\(\begin{array}{l}\large Hydrogen-5 =80\end{array} \)
\(\begin{array}{l}\large Hydrogen-4 =139\end{array} \)
\(\begin{array}{l}\large Hydrogen-10 =200\end{array} \)
\(\begin{array}{l}\large Hydrogen-6 =290\end{array} \)
\(\begin{array}{l}\large Lithium-5 =304\end{array} \)
\(\begin{array}{l}\large Lithium-4 =756\end{array} \)
\(\begin{array}{l}\large Boron-7 =350\end{array} \)
\(\begin{array}{l}\large Helium-5 =760\end{array} \)


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