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$^{64}_{29}\ce{Cu}$(" half life" =$12.8$ "hours" ) decay by $\beta^{-}$- emission (38%), $\beta^+$- emission(19%), and electron capture (43%). Write the decay products and calculate partial half lives for each of the decay processes. Source: JEE Advanced

Soltn: Half life of beta-= $\frac{}{}=\frac{12.8}{.38}$, beta+ =$ \frac{12.8}{.19}$ and electron capture$ \frac{12.8}{.43}$

From googling I find, that the term partial half life is popularly known as branching fraction:

the branching fraction (or branching ratio) for a decay is the fraction of particles which decay by an individual decay mode with respect to the total number of particles which decay

Source

Question: I don't get why the half life of each individual path is the total half life divided by the fraction of nuclei decayed that way. I foudnd this related post but there it involves a the actual reaction being given.

I also found this wiki article on half life, but I am not sure how to use it to derive this.

tryst with freedom
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  • Deriving the relation of the half life, partial half lifes and probability percentage of the particular decay path is not complicated algebraic exercise. – Poutnik Aug 07 '21 at 15:45
  • What equations should one begin with? @Poutnik I'll try for a self answer then – tryst with freedom Aug 07 '21 at 15:48
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    Equations for the reaction kinetic of the first order - all radioactive decays follow it. – Poutnik Aug 07 '21 at 15:50
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    The solution given does seem strange but maybe something common in study of radioactive decay. The Cu decays with a single half life and each product appears with the same half life, 12.8 hours even though the rate constants for each product can be different. If the rate constants are (in order as above) $k_1, k_2, k_3$ then $\beta^-$ appears with fraction (branching ratio) $k_1/(k_1+k_2+k_3)=0.38$ and so on and the sum is the rate constant for decay of the Cu which is easily related to the half life. I suspect partial half lives means get k's for each product and convert to half lives. – porphyrin Aug 07 '21 at 17:05
  • Note that "partial half lifes" is not very useful concept, it is better to stay with the overall half life, decay constants and decay mode probability. – Poutnik Aug 07 '21 at 17:53
  • How is the percent of product equal to fraction of rate constant over total ?? @poryphrin Secondly if the equation is true, how would I find rate constants anyways – tryst with freedom Aug 07 '21 at 17:59
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    It is easier to use rate constants than half lives. You have three equations and three unknowns, $k_1,k_2,k_3$ you know the sum of these given by $\ln(2)/t_{1/2}$ and each ratio, 0.38 etc. – porphyrin Aug 07 '21 at 20:25

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