# 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 $\left[A\right]$. 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 $A$?

Jun 14, 2017

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$\textcolor{w h i t e}{0 , 2 , \frac{1}{2} , 1 , 0}$

1)

Well, consider the following half-lives:

${t}_{\text{1/2}} = \frac{{\left[A\right]}_{0}}{2 k}$ (zero order)

${t}_{\text{1/2}} = \frac{\ln 2}{k}$ (first order)

${t}_{\text{1/2}} = \frac{1}{k {\left[A\right]}_{0}}$ (second order)

${t}_{\text{1/2}} = \frac{3}{2 k {\left[A\right]}_{0}^{2}}$ (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...

$r \left(t\right) = k {\left[A\right]}^{n}$

Knowing that doubling the concentration leads to quadrupling the rate, i.e. $\textcolor{red}{2} \left[A\right] \to \textcolor{red}{4} r \left(t\right)$, we have that

$\textcolor{red}{4} \cdot r \left(t\right) = k {\left(\textcolor{red}{2} \left[A\right]\right)}^{n} = {\textcolor{red}{2}}^{n} \cdot k {\left[A\right]}^{n}$

Thus, we have that ${2}^{n} = 4$. What must $\boldsymbol{n}$ be? The reaction order with respect to $A$ is of this $n$th order.

3)

Same process as $\left(2\right)$. Now we claim that:

$\textcolor{red}{1.41} \cdot r \left(t\right) = k {\left(\textcolor{red}{2} \left[A\right]\right)}^{n} = {\textcolor{red}{2}}^{n} \cdot k {\left[A\right]}^{n}$

But $1.41 \approx \sqrt{2}$. Hence, we have that $\sqrt{2} = {2}^{n}$. What is $\boldsymbol{n}$ 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 ${\left[A\right]}_{0}$? Hint: it's not a prime number.

5)

This says that the rate of reaction does not depend on the current $\left[A\right]$. 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:

$r \left(t\right) = k {\left[A\right]}^{0} = k$