# Meaning of Partial Half-Life for Two Parallel Reactions

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 $$\ce{C}$$.

• 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.) – 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:

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

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

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

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.
• 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. – porphyrin Sep 5 at 16:12