Reaction Half-life (Lecture 4)

I think the most important thing you should remember is what does reaction half-life mean conceptually, which pretty much follows from the meaning of words:

the half-life of a reaction is the time required for the reactant concentration to decrease to one-half its initial value.

After that, if you know the integrated rate laws (the integrated form will be given during exams), you just need to use the definition to do some algebra.

Zero-Order Reactions

Recall, that the integrated rate law is:

$$[A{]}_t=[A{]}_0-kt\text{\hspace{1em}(1)}$$

We’re interested in time $t_{1/2}$ when $[A{]}_{t_{1/2}}=0.5[A{]}_0$. Let’s substitute this into (1):

$$0.5[A{]}_0=[A{]}_0-k{t}_{1/2}⟹t_{1/2}=\frac{[A{]}_0}{2k}\text{\hspace{1em}(2)}$$

First-Order Reactions

The same drill. Recalling the integrated form:

$$\ln \frac{[A{]}_0}{[A{]}_t}=kt\text{\hspace{1em}(3)}$$

We’re interested in time $t_{1/2}$ when $[A{]}_{t_{1/2}}=0.5[A{]}_0$. Simple substitution leads to:

$$\ln 2=k{t}_{1/2}⟹t_{1/2}=\frac{\ln 2}{k}\text{\hspace{1em}(4)}$$

This actually turns out to be a very convenient form for a half-life. If we take the exponential form of the integrated rate law:

$$[A{]}_t=[A{]}_0{e}^{-kt}$$

and we substitute $k$ with $\ln 2/t_{1/2}$:

$$[A{]}_t=[A{]}_0{e}^{-\ln 2\times t/t_{1/2}}$$

We can take $-1$ and put it inside of the logarithm by raising $2$ to its power, i.e. $-\ln 2=\ln 2^{-1}=\ln 1/2$. Then, by definition of logarithm, we can simplify $e^{\ln 1/2}=1/2$. As a result:

$$[A{]}_t=[A{]}_0{\left(\frac{1}{2}\right)}^{t/t_{1/2}}$$

Which is convenient whenever we want to find concentrations after $t$ close to some integer number of half-lives. For example, $[A{]}_t$ after $2$, $3$, $4$ half-lives is $1/2^2=1/4$, $1/2^3=1/8$, $1/2^4=1/16$ of the initial concentration.

Second-Order Reactions

See if you can prove, using the same logic as above, that:

$$t_{1/2}=\frac{1}{k[A{]}_0}\text{\hspace{1em}(5)}$$

Some Meta-Notes

Note that you can state many corollaries from equations (2), (4), and (5), such as whether the half-life depends on the initial concentration, or whether it depends directly or inversely (as you see on Slide 29). If you were to ask me “what is the half-life of some reaction with order N”, I’d only be able to tell you result (4). If you were to ask me to give you half-life for 0-order or 2nd-order reactions, I’d start from the integrated rate laws (which I happen to remember just because I saw them so many times, but also I can integrate whenever I’m not confident) and do the derivations as shown on this page.

Personally, I don’t think there’s any benefit in trying to memorize information which can be directly and immediately inferred from some equation, which I can derive. Looking at equation (4) you can always and correctly make the conclusion that the half-life is independent of the initial concentration, but from the latter statement alone you’d never be able to reconstruct the equation (4).