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Consider two parallel reactions:

$$\ce{A -> B }\tag1$$ and $$\ce{A -> C}\tag2$$

What is the meaning of partial half life of the equation $(1)$ and that of equation $(2)$ with respect to above reactions?

It seems to me that it means the half life when one of these reactions are taking place individually, one-at-a-time, and not simultaneously. For example, partial half life of the equation $(1)$ is when half of $\ce{A}$ converts to only $\ce{B}$ and not to $\ce{C}$.

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  • $\begingroup$ As you have written it A must go to both B and C with rate constant $k_1+k_2$: A cannot choose and both B and C appear with rate constant $k_1+k_2$. If you know the rate constants then you can choose to define the half life but that is just an alias for the rate constant anyway so why bother ? (The yield to B is $k_1/(k_1+k_2)$ if that is what you are actually looking for.) $\endgroup$ – porphyrin Sep 5 at 14:06
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The partial reaction half-life relates to the reaction speed constant by the same way as for the "normal" reaction half-life.

If there are 2 parallel reactions of the first order, $\ce{A -> B}$ and $\ce{A -> C}$, and if there is the reaction rate for the former:

$$\frac{\mathrm{d}[\ce{B}]}{\mathrm{d}t}=k_{\ce{B}} \cdot [\ce{A}]$$

then the partial half-life w.r.t. $\ce{A -> B}$ is:

$$t_{1/2,\mathrm{B}}=\frac{\ln{2}}{k_{\ce{B}}}$$

Analogically the similar for the other reaction:

$$\frac{\mathrm{d}[\ce{C}]}{\mathrm{d}t}=k_{\ce{C}} \cdot [\ce{A}]$$

$$t_{1/2,\ce{C}}=\frac{\ln{2}}{k_{\ce{C}}}$$

The overall half-life w.r.t. $\ce{A -> X}$ is then

$$t_{1/2}=\frac{\ln{2}}{k_{\ce{B}}+k_{\ce{C}}}=\frac{1}{\frac 1{t_{1/2,\ce{B}}}+\frac 1{ t_{1/2,\ce{C}}}}$$

---- Responses to comments

The partial half-life is the extrapolated time after which all $\ce{A}$ would have decayed, if it had been decaying by the current and constant rate of given reaction and only by this reaction. But the main meaning is as a kind of reciprocal value of the reaction constant.

If we draw the chart of the partial reaction rate, then it's tangenta at $t=0$ will cross $x$-axis at the reaction partial half time. It is the same as if it was the only reaction. It does not say half of $\ce{A}$ decays in this partial half time.

If $\Delta t \ll \min{(t_{1/2})}$, then it does not matter it is just partial halftime, as we can neglect decay of the other parallel reactions.

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  • $\begingroup$ Does this term "partial half life" carry any actual physical meaning? $\endgroup$ – Tony Stark Sep 5 at 14:38
  • $\begingroup$ Sir, " if it was decaying by constant rate of given reaction and only by this reaction." -- This statement makes me think that we are talking in hypothetical sense. Are we? $\endgroup$ – Tony Stark Sep 5 at 14:49
  • $\begingroup$ The ratio of concentration and its (total or partial ) time rate is a very real parameter. See A update. $\endgroup$ – Poutnik Sep 5 at 15:01
  • $\begingroup$ Are you not assuming that the rate constant A to B is the same as that B to A and similarly for C in making your final half life $t_{1/2}$? This is not true unless by numerical accident. The reaction studied is A to B and C and not the reverse. $\endgroup$ – porphyrin Sep 5 at 16:12
  • $\begingroup$ Why should I ? Reversibility of reactions has not come into account at all. E.g. We can consider 2 different modes of radioactive nuclide decay. $\endgroup$ – Poutnik Sep 5 at 16:28

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