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 $\ce{C}$.

  • $\begingroup$ 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

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:


Analogically the similar for the other reaction:

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


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.

| improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.