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We have a second order irreversible reaction with 2 reactants (A and B, order 1 for both) so that the initial concentrations are different. I've read that the half-life doesn't make sense in this context but I can't fully understand it.

$$\ce{A +B ->[$k$]P}$$

$$-\frac{\mathrm d[\ce{A}]}{\mathrm dt} =-\frac{\mathrm d[\ce{B}]}{\mathrm dt} = \frac{\mathrm d[\ce{P}]}{\mathrm dt} = k[\ce{A}][\ce{B}]$$

Any explanation?

orthocresol
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Jorge Bonifaz
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2 Answers2

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Because half life is only a well defined constant for a first order reaction

Half life for simple, first order reactions is the length of time it takes for half the substance to disappear. The classic case being radioactive decay. The rate of the reaction depends only on the amount of the single substance present but the length of time for the amount of substance to half stays the same whatever the initial amount.

For more complex reactions involving more than one substance, the reactions will not be first order and the rate of the reaction depends on the amounts of two (or more) reactants. One can calculate how long it would take for half of one of the components to disappear, and you could call that a half-life, but it would be a fairly useless number. In non first order reactions that half life would depend on the specific concentrations of all the components. And it would be different for all the combinations of the starting concentrations. And it would differ over time because those concentrations change over time.

So, for first order reactions, half life is a well-defined constant but for more complex reactions it is a number which changes depending on the concentrations and changes over time. So, in those reactions, it is a totally useless number and it would rarely be helpful to calculate it or use it. That is why it is said that it doesn't make sense.

matt_black
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Before I actually answer the question, let's address the elephant in the room.

What is half-life?

The half life is a term that denotes the amount of time it takes for the amount of a substance to be halved. A more formal definition would be as follows.

In a chemical reaction, the half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value.

Now we've got that out of the way, let's actually talk about the question you've asked.

We have a second order irreversible reaction with 2 reactants (A and B, order 1 for both) so that the initial concentrations are different. I've read that the half-life doesn't make sense in this context.

Half-life doesn't make sense in this context. Why? It's because you don't have enough context. Go back to the "formal" definition. The half-life of a species is what is defined. This isn't the same as the half-life of a reaction (Because this term doesn't exist for anything that has more than one reactant in the mix).

However, you could ask the following question.

  1. What is the half-life of A (or B) in said reaction?

This has more context as you've given the person enough info to actually corelate the question and definition of what they know to be half life. Can you calculate it theoretically with the data given? That's a different question, whose answer I believe is no - for the same reason, you don't have enough information.

Safdar Faisal
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  • Ok, thanks, now I understand that is specific to one species. When you say you can't calculate it is because you don't have enough information or because is not calculable anyways? – Jorge Bonifaz Aug 23 '22 at 12:45
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    "whose answer I believe is no" - That's not (entirely) correct; if you know the reaction stoichiometry (in the question, $\ce{A + B -> P}$) then that means that there is only one unknown, because $[\ce{A}] = [\ce{A}]_0 - x$ and $[\ce{B}] = [\ce{B}]_0 - x$. The initial concentrations $[\ce{A}]_0$ and $[\ce{B}]_0$ are constants, and the only unknown is $x$. Now, I don't know whether there's a closed-form expression for the half-lives of $\ce{A}$ and $\ce{B}$ (you'd have to do some calculus, and I'm too lazy to figure that out). But you can definitely at least calculate it numerically. – orthocresol Aug 23 '22 at 13:00
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    @orthocresol, as it happens there is an algebraic solution, in the case $A+B\rightleftharpoons P$ the lifetime $\tau$ is given by $1/\tau = k_1([A_e]+[B_e])+k_{-1}$ where the subscript 'e' indicates the equilibrium concentration – porphyrin Aug 24 '22 at 07:17