Question

Help with these kinetics questions about reaction order with respect to one reactant `A`??

`1)` The decay of `A` proceeds in a way such that the half-life is directly proportional to `[A]`. What is the order with respect to `A`?
`2)` If doubling the concentration of reactant `A` quadruples the rate, then what would the order with respect to `A` be?
`3)` If doubling the concentration of `A` leads to a reaction rate that is `1.41` times as fast, what is the order with respect to `A`?
`4)` Two initial concentrations of `A` were tested to determine the half-life, but both half-lives were equal even though the two concentrations were different. What is the order with respect to `A`?
`5)` If the reaction rate is constant, what is the order with respect to `A`?

Answer 1

Highlight to reveal answers once you've figured it out.

`color(white)(0, 2, 1/2, 1, 0)`

---

`1)`

Well, consider the following half-lives:

`t_"1/2" = ([A]_0)/(2k)` (zero order)

`t_"1/2" = (ln2)/k` (first order)

`t_"1/2" = 1/(k[A]_0)` (second order)

`t_"1/2" = 3/(2k[A]_0^2)` (third order)

In all of these, only the zero order half-life is directly proportional to the concentration of `A`. Hence, the reaction is zero order with respect to `A`.

`2)`

You can figure out a lot from the rate law...

`r(t) = k[A]^n`

Knowing that doubling the concentration leads to quadrupling the rate, i.e. `color(red)(2)[A] -> color(red)(4)r(t)`, we have that

`color(red)(4)cdotr(t) = k(color(red)(2)[A])^n = color(red)(2)^n cdot k[A]^n`

Thus, we have that `2^n = 4`. What must `bbn` be? The reaction order with respect to `A` is of this `n`th order.

`3)`

Same process as `(2)`. Now we claim that:

`color(red)(1.41)cdotr(t) = k(color(red)(2)[A])^n = color(red)(2)^n cdot k[A]^n`

But `1.41 ~~ sqrt2`. Hence, we have that `sqrt2 = 2^n`. What is `bbn` this time? (What is `2` raised to that correlates with a square root?) The reaction is of this `n`th order with respect to `A`.

`4)`

It discusses the first and second half-lives and claims that they are equal. This just says that the half-life (of THIS order) does not depend on what concentration you start at.

Look above. Which order half-life does not depend on `[A]_0`? Hint: it's not a prime number.

`5)`

This says that the rate of reaction does not depend on the current `[A]`. That is, `A` has no influence on the rate. That can only be the case if `A` is zero order, i.e. the rate depends only on the rate constant:

`r(t) = k[A]^0 = k`