Determination of Rate Equations & the Rate Determining Step

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Determination of Rate Equations & the Rate Determining Step

Rate equations are determined experimentally by investigating the effect of changing concentrations and then deducing the order.

Using Initial Rate to Calculate Reaction Variables

Rate equations have a general form:

$\text{Rate} =k[X]^n$

To work out the value of n graphically, we need to take logarithms of both sides of the equation to convert it to the form $y = mx + c$.

$\text{log Rate}=\text{log k}+\text{n log [X]}$

A graph of this equation would produce a straight line, where the $\text{y-intercept}$ is equal to $\text{log k}$ allowing us to calculate the rate constant $\text{K}$.

$\text{K}=10^{\text{ y-intercept}}$

The gradient of the line is equal to the order of reaction, $\text{n}$, with respect to X.

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The Rate Determining Step

The slowest step in a reaction is known as the rate-determining step (RDS). It is rate-determining because it controls the overall rate of a reaction.

The rate equation contains species up to and including the rate-determining step. Therefore, we can use the rate equation to determine the rate-determining step. Take the reaction

$\text{A} + 3\text{B} + 2\text{C} \rarr\text{D}$

A suggested mechanism for this reaction is

Step 1: $\text{A} + 2\text{B}\rarr\text{X}$
Step 2: $\text{B} + 2\text{C}\rarr\text{Y}$
Step 3: $\text{X} + \text{Y}\rarr\text{D}$

The rate equation for the reaction is

$\text{Rate} = \text{k[A][B]}^2$

The rate equation tells us that there is $1$ mole of A and $2$ moles of B up to and including the rate-determining step. Step 1 contains both $1$ mole of A, $2$ moles of B, Step 2 contains only $1$ mole of B and none of A, and Step 3 contains no moles of either species.

Step 3 cannot be the RDS as it does not contain either of the species contained in the rate equation. step 2 contains $1$ mole of B, but taking both steps 1 and 2 together gives $3$ moles of B in total. As the rate equation contains only $2$ moles of B step 3 can’t be the RDS. Step 1 contains the correct number of both moles according to the rate equation and so must be the rate determining step.

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Determining Reaction Orders From Experimental Data.

By carrying out a series of experiments using different concentrations of reactants, we can determine the rate equation of a reaction. To do this we need to compare the effects of the different concentrations on the initial rate.

 Experiment Initial Concentration of X /$\text{mol dm}^{-3}$ Initial Concentration of Y /$\text{mol dm}^{-3}$ Initial Rate /$\text{mol dm}^{-3}\text{ s}^{-1}$ 1 $0.12$ $0.26$ $2.10\times10^{-4}$ 2 $0.36$ $0.26$ $1.98\times10^{-3}$ 3 $0.72$ $0.13$ $3.78\times10^{-3}$

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We can start by comparing the rates of experiments 1 and 2. Although the concentration of Y remains constant, the initial rate changes. That immediately tells us that the order of X is greater than $0$ and is part of the rate equation.

From experiments 1 to 2, the concentration of X triples while the initial rate increases by a factor of $9$. This means that the order of X must be $2$ (you can work out the amount the initial rate increases if you divide the rates).

$\text{Rate} = \text{k[X]}^2$

We can now look at experiments 2 and 3 to find the effect of Y. Firstly, we have to consider the effect of X. Between experiments 2 and 3, the concentration of X doubles. We know that X is second order so from experiments 2 to 3, the initial rate would be expected to quadruple from $1.8\times10^{-3}\text{ mol dm}^{-3}\text{ s}^{-1}$ to $7.56\times10^{-3}\text{ mol dm}^{-3}\text{ s}^{-1}$.

Since that is not the initial rate of the reaction, Y must have an order greater than $0$. From experiments 2 to 3, the concentration of Y halves, meanwhile, the rate is half the expected value of $7.56\times10^{-3}\text{ mol dm}^{-3}\text{ s}^{-1}$, so Y must be first order.

$\text{Rate} = \text{k[X]}^2\text{[Y]}$

To calculate the value for $\text{k}$, the rate equation needs to be rearranged to make $\text{k}$ the subject, and then the known values can be substituted into the equation.

For example, if we use the rate equation above, $\text{Rate} = k [X]^2[Y]$, we can find $\text{k}$ using the equation $k=\frac{\text{Rate}}{[X]^2[Y]}$.

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Initial Rate Experiments – The Iodine Clock

The initial rate of a reaction can be calculated by carrying out a clock reaction. In clock reaction, a system is designed so that it can be monitored over a set period of time or until one product has been produced in a specific concentration.

Method
In this common clock reaction, hydrogen peroxide $(\text{H}_2\text{O}_2)$, iodide ions $(\text{I}^-)$, thiosulfate ions $(\text{S}_2\text{O}_3^{2-})$ and starch are mixed together. The following reactions take place:

$\text{H}_2\text{O}_{\text{2(l)}} + 2\text{H}^+_{\text{(aq)}} + 2\text{I}^{-}_{\text{(aq)}} \rarr \text{I}_{\text{2 (aq)}}$

$2\text{S}_2\text{O}_3^{2-}\text{}_{\text{(aq)}}+\text{I}_{\text{2(aq)}}\rarr2\text{I}^-_{\text{(aq)}}+\text{S}_4\text{O}_6^{2-}\text{}_{\text{(aq)}}$

1. Fill a burette $(50\text{ cm}^3)$ with potassium iodide solution. Rinse the $50\text{ cm}^3$ burette with potassium iodide beforehand to improve accuracy.
2. Put $10.0\text{ cm}^3$ of hydrogen peroxide into a clean and dry measuring cylinder.
3. Use a measuring cylinder to add $25\text{ cm}^3$ of sulfuric acid to a second  $250\text{ cm}^3$ beaker. Then, add $20\text{ cm}^3$ of deionised water into the beaker.
4. Use a plastic dropping pipette to add about $1\text{ cm}^3$ of starch solution to the $250\text{ cm}^3$ beaker.
5. Use the burette to add $5.0\text{ cm}^3$ of potassium iodide solution to the mixture in the $250\text{ cm}^3$ beaker.
6. Finally add $5.0\text{ cm}^3$ of sodium thiosulfate solution to the mixture in the $250\text{ cm}^3$ beaker.
7. Stir the mixture in the $250\text{ cm}^3$ beaker and pour the $10.0\text{ cm}^3$ hydrogen peroxide solution from the measuring cylinder into the $250\text{ cm}^3$ beaker and immediately start the timer.
8. Stop the timer when the mixture in the $250\text{ cm}^3$ beaker turns blue-black and record the time.
9. Repeat steps 1-8 at least two more times, changing the potassium iodide concentration each time. This is to allow the order of reaction to be determined.
10. Plot a graph of initial rate versus concentration to determine the order.
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Determination of Rate Equations & the Rate Determining Step Example Questions

$\text{Rate} = \text{k[NO}_2\text{]}^2$.

$\text{NO}_2$ appears twice in the rate-determining step while $\text{CO}$ does not appear at all.

Gold Standard Education

\begin{aligned}\text{k}&=\frac{2.5\times10^{-3}}{0.15\times0.03}\\ &=0.555\end{aligned}

Units:$\text{ mol}^{-1}\text{dm}^{3}\text{ s}^{-1}$

Order with respect to X: $2$ or $\text{[X]}^2$
Order with respect to Y: $0$ or $\text{[Y]}^0$