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# Rate Equations

Chemical reactions are processes in which a set of reactants are converted into products as a result of changes to their structures. These structural changes can happen at different speeds, similar to how race cars can travel at different speeds. Just like how it's important to understand how the speed of a race car can be affected, understanding how the speed…

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# Rate Equations

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Chemical reactions are processes in which a set of reactants are converted into products as a result of changes to their structures. These structural changes can happen at different speeds, similar to how race cars can travel at different speeds. Just like how it's important to understand how the speed of a race car can be affected, understanding how the speed of chemical changes can be affected is an important part of physical chemistry.

The rate equation is an expression that links the rate of a reaction to the concentration of the species involved.

• We will be looking at the rate equation.
• We'll see what it tells us about the rate of a reaction.
• Finally, we'll explore the rate constant and reaction orders, and we'll briefly touch on methods used to determine the rate equation.

## Rate equation chemistry

The rate of a reaction is how quickly a reaction occurs. But what does that mean and what does it tell us? Well, one way of looking at it is to think about how much product is made in a period of time, which will depend on how much reactant is used up. In essence, we can say that the rate of a reaction is the speed at which reactants are converted into products.

The rate of reaction is the change in concentration of reactants or products over time. It is typically measured in mol dm-3 s-1.

### Measuring rate of reaction

To calculate the rate of a reaction, we need to measure the change in the amount of reactant/product from the start of the reaction to the end.

We define the rate of a reaction as a measure of how much product is formed, or how much reactant is used, over a period of time.

We can measure this by observing things like colour change, pH change, volume of gas produced, or change in mass of a solid reactant. You should then be able to convert your data values into figures for concentration. This data is plotted on a line graph, with time on the x-axis and concentration on the y-axis. From the graph, we can find a value for the rate of reaction by working out the line's gradient. We either calculate an overall rate of reaction or an instantaneous rate of reaction. Both use the following equation:

$\mathrm{rate}\mathrm{of}\mathrm{reaction}=\frac{\mathrm{change}\mathrm{in}\mathrm{concentration}}{\mathrm{time}\mathrm{taken}}$

#### Overall rate of reaction

Calculating an overall rate of reaction is fairly straightforward. You divide the overall change in concentration of a reactant or product by the time taken. For a graph of concentration against time, this means dividing the change in y-values by the change in x-values. Here's an example.

Calculate the overall rate of reaction for the following graph.

Fig. 1 - A concentration-time graph used to calculate rate of reaction

To find the overall rate of reaction, we divide the change in concentration by the time taken.

It doesn't matter whether you measure the concentration of a product or reactant - both will give you a valid answer.

Fig. 2 - A concentration-time graph used to calculate rate of reaction

Here, the concentration starts at 40 mol dm-3 and ends at 8 mol dm-3. This is a change of 40 - 8 = 32 mol dm-3. The reaction takes 200 seconds. The rate of reaction is therefore $\frac{32}{200}=0.16\mathrm{mol}{\mathrm{dm}}^{-3}{\mathrm{s}}^{-1}$.

#### Instantaneous rate of reaction

Sometimes, finding an overall rate of reaction isn't that useful. You might instead want to know how the rate of reaction changes over time. To do this, you calculate instantaneous rates of reaction. This involves drawing a tangent to the curve at a particular point and finding its gradient. Again, this is given by the change in concentration divided by the time taken - in other words, the change in y-values divided by the change in x-values.

Calculate the instantaneous rate of reaction for the following graph at 60 seconds.

Fig. 3 - A concentration-time graph used to calculate rate of reaction

We first need to find the 60-second mark on the curve. We draw a tangent to the curve at this point.

Remember that a tangent is a straight line that just touches the curve at a specified point.

Next, we calculate the gradient of this tangent by dividing the change in concentration by the time taken. You do this by turning the tangent into a right-angled triangle.

Fig. 4 - A concentration-time graph used to calculate rate of reaction

Here, we can see from our right-angled triangle that the concentration starts at 22 mol dm-3 and ends at 6 mol dm-3. This is an overall change of 16 mol dm-3. This change in concentration takes place between 20 and 120 seconds, meaning it takes 100 seconds in total. The instantaneous rate of reaction is therefore $\frac{16}{100}=0.16\mathrm{mol}{\mathrm{dm}}^{-3}{\mathrm{s}}^{-1}$

## Rate of reaction equation

Let's look at something different: the rate equation. The rate equation in chemistry is a formula that we can use to find the rate of a reaction using the concentration of species involved in the reaction. Here's what it looks like:

Fig. 5 - Rate equation

At first glance it certainly looks confusing, but once you understand what's going on it isn't all that bad.

• k is the rate constant.
• The letters A and B in the rate equation are used to represent species involved in the reaction. These could be reactants or catalysts.
• The square brackets around the letters represent concentration. So, [A] is used to show the concentration of species A.
• The letters m and n represent the order of the reaction with respect to a certain species. They show the power that the concentration of that species is raised to in the rate equation. Overall, [A]m represents the concentration of A, raised to the power of m. This means that A has the order m.

### The rate constant

k is the rate constant. It is used in the rate equation to link the concentrations of certain species to the rate of that reaction. The value of k changes depending on the reaction and reaction conditions. However, k is always constant for a certain reaction at a particular temperature. If you were to carry out the exact same reaction at different temperatures, k would change, but if you carried it out at the same temperature, k would stay the same: after all, it is a constant!

To learn more about how the rate constant relates to temperature, read The Arrhenius Equation. And if you want to find out how to calculate the rate constant, alongside its units, head over to Determining Rate Constant.

### Orders of reaction

In chemical reactions, reactants and catalysts (if there are any) have an order of reaction. The sum of the individual orders of species in a reaction equals the overall order of the equation.

In the rate equation, the order of a reaction with respect to a species is shown using a power. For example, in the rate equation we looked at above, the order of A is represented by the letter m. The order of a reaction with respect to a species tells us how the concentration of that particular species affects the reaction rate. Some species don't affect the rate whatsoever, while other species affect it dramatically.

Any non-negative number can be an order, and species can also have fractional orders like 5/2. But for the purpose of your exams, you only need to know about zero, first and second-order reactants.

#### Zero-order reactants

The concentration of a zero-order reactant doesn't affect the rate of reaction. If you double its concentration, the rate stays the same. This is because 20 = 1. Because they have no effect on the rate of reaction, zero-order reactants don't appear in the rate equation.

#### First-order reactants

The concentration of first-order reactants is directly proportional to the rate of reaction. If you double the concentration of a first-order reactant, the rate of reaction also doubles. This is because 21 = 2.

If a reactant is first-order, it appears in the rate equation raised to the power of 1. However, we don't tend to write the number 1 because raising something to the power of 1 has no effect on its value. You'll see first-order reactants in the rate equation as [A], where A represents the species.

#### Second-order reactants

The concentration of second-order reactants has an exponential effect on the rate of reaction. Doubling the concentration of a second-order reactant causes the rate of reaction to quadruple. This is because 22 = 4.

If a reactant is second-order then we put it in the rate equation raised to the power of 2; in other words, squared. You'll see it in the rate equation as [A]2.

#### Order of a reaction

The overall order of a reaction is the sum of all the individual reactant orders. Remember that the order of a reactant is the power that it is raised to in the rate equation. If you're ever asked to find the overall order of reaction, simply add together all of the powers present in the equation and you'll reach your final answer.

Understanding orders of reaction is a bit tricky, so let's look at an example to help you understand.

A reaction has the chemical equation and rate equation shown below.

$\mathrm{A}+\mathrm{B}+\mathrm{C}\to \mathrm{D}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{rate}=\mathrm{k}\left[\mathrm{A}\right]{\left[\mathrm{B}\right]}^{2}$

Describe the effect of doubling the concentrations of A, B, and C.

First of all, looking at the rate equation, we can see that the only species present are A and B. C does not appear at all. It must therefore be zero-order. Hence, doubling the concentration of C will have no effect on the rate of reaction.

On the other hand, A does appear in the rate equation. It looks like [A] isn't raised to any power, but as we learnt above, [A] is the same as saying [A]1. A is therefore first-order. Doubling the concentration of A will cause the rate of reaction to double, because 21 = 2.

B also appears in the rate equation. It is raised to the power of 2, meaning that it is second-order. Doubling the concentration of B will cause the rate of reaction to quadruple, because 22 = 4.

## Determining the rate equation

There are a few different methods we can use to determine the rate equation for a reaction. The basic principles come down to determining the species involved in the rate equation and then finding each of their orders. The main methods for doing this are:

• The initial rates method.
• Using rate-concentration graphs.
• Finding first-order reactants from their half-life.
• Inspecting the reaction mechanism.

We cover these methods in much more detail in Determining Reaction Order, but we'll explore them briefly now.

### Initial rates

The initial rates method involves measuring the rate of the same reaction over several experiments, each with different starting concentrations of a particular reactant. This method allows us to see numerically how the concentration of the reactant affects the rate of the reaction. We do this for each reactant, and can use the information to determine the reactant's order.

### Rate-concentration graphs

Earlier in the article, we looked at how you use graphs showing concentration of a species against time to calculate rate of reaction at a specific instant. You can then take the values for instantaneous rate of reaction and plot them against concentration to make a rate-concentration graph. These take specific shapes, depending on the order of the species involved.

• A horizontal straight line shows that the rate of reaction is unaffected by the concentration of the species. The species is therefore zero-order.
• A sloping straight line through the origin shows that the rate is directly proportional to the concentration of the species. The species is therefore first-order.
• A curved line through the origin shows that the rate is exponentially proportional to the concentration of the species. The species is second-order or higher.

Fig. 6 - Rate-concentration graphs for reactants with different orders

### Half-life equations

The half-life, ${\mathrm{t}}_{1/2}$, of a reactant is the time it takes for the concentration of that reactant to become half of what it was where you started measuring from. There's an interesting feature of first-order reactants: they have a constant half-life. This means that it takes the same amount of time to get from, say, a concentration of 1.0 to a concentration of 0.5 mol dm-3, as it does to get from a concentration of 0.8 to 0.4 mol dm-3. In both cases, the concentration has halved.

You can measure half-life using concentration-time graphs. Pick any point on the graph and look at the concentration for that time value. Then, see how long it takes to halve the concentration. Repeat this again to find multiple half-lives for a species. If all of the half-lives are the same, the species is first-order.

Fig. 7 - A concentration-time graph used to show half-life

#### Half-life and rate constant

The half-life of a first-order reactant relates to the rate constant, k, using the following equation:

$\mathrm{k}=\frac{\mathrm{ln}\left(2\right)}{{\mathrm{t}}_{1/2}}$

This means that once you know the half-life of a first-order reactant, you can easily find k.

### Reaction mechanism

Reactions can have mechanisms with one step or multiple steps. Each step happens at a different speed, and the rate of reaction is determined by the slowest step. We call the slowest step in a reaction the rate-determining step, and it gives us an idea of what the rate equation is likely to look like. This is because the rate equation is only made up of reacting species found in the steps up to and including the rate-determining step. The number of moles of each species relates to its order.

If you know the reaction mechanism and the rate-determining step of a reaction, you can predict the rate equation!

## Rate Equations - Key takeaways

• The rate of a chemical reaction is the change in concentration of reactants or products over time.

• Rate of reaction can be represented by a rate equation. Rate equations are composed of a rate constant (k), and reactant concentrations raised to the power of their respective order.

• Rate constants are constant for a particular reaction at a certain temperature.

• The order of a reaction with respect to a species tells us how the rate of reaction depends on the concentration of that species.

• The concentration of zero-order reactants has no effect on the rate of reaction.

• The concentration of first-order reactants is directly proportional to the rate of reaction.

• The concentration of second-order reactants has an exponential effect on the rate of reaction.

• The rate equation can be determined using the initial rates method, by identifying the shapes of graphs, by calculating half-lives, and by inspecting the reaction mechanism.

To calculate the rate equation, you need to find out the order of reaction with respect to each species involved in the reaction. You also need to find the rate constant, k. You can do this experimentally. Once you've formed a rate equation, you can substitute in known concentration values and find the rate of reaction at a particular instant.

Rate equations are written in the form rate = k [A]m [B]n. The rate constant, k, is a value that is always constant for a particular reaction at a particular temperature. [A] represents the concentration of A, whilst the letter m represents the order of the reaction with respect to A. Overall, [A]m means the concentration of A, raised to the power of m. To write a rate equation, you work out the rate constant and the orders of reaction with respect to each species involved, and write them in the form given above.

The rate equation tells us the rate of a reaction. This means that it tells us the rate of change of reactant or product concentration during a reaction. So, by calculating the rate equation, you can find the rate of change.

You can find the rate of a reaction by using the rate equation. The rate equation is a formula that tells us the rate of any reaction from the concentration of its reactants.

Some factors that affect the rate of a reaction include reactant concentration, surface area, temperature, and activation energy.

## Rate Equations Quiz - Teste dein Wissen

Question

Why is the Arrhenius equation useful in chemistry?

It allows us to relate the temperature of a reaction with its rate

Show question

Question

What does the letter A represent in the Arrhenius equation?

A is the Arrhenius constant

Show question

Question

What information about a reaction can the value of ex give us in the Arrhenius equation?

the number of reacting particles that have enough energy to react

Show question

Question

Rearrange the Arrhenius equation into its logarithmic form.

ln(k) = ln(A) - Ea/RT

Show question

Question

Given a graph showing an Arrhenius plot for a chemical reaction, how could you use the plotted data to determine activation energy and the rate constant for that reaction?

The activation energy would be equal to the gradient of the line.
The rate constant would be equal to the y-intercept.

Show question

Question

When drawing an Arrhenius plot, what would you label your axis?

The x-axis would be 1/T

The y-axis would be ln(K)

Show question

Question

What does the Arrhenius equation show us about the effect of temperature on the rate of a reaction?

The rate of a reaction will increase if the temperature at which the reaction occurs increases.

Show question

Question

What is a rate equation?

A mathematical equation that links the rate of reaction with the concentration of species involved in the reaction.

Show question

Question

What does the letter k represent in the rate equation?

The rate constant

Show question

Question

What do the letters m and n represent in the rate equation?

The order of the reaction with respect to a certain species.

Show question

Question

Give four ways of determining the rate equation.

• Initial rates method
• Rate-concentration graphs
• Half-life
• Reaction mechanism

Show question

Question

When using the initial rates method to determine the rate equation, you _____.

Keep the external conditions the same.

Show question

Question

How can we use half-lives to help determine the rate equation?

We can use half-lives to identify first-order reactants. First-order reactants have a constant half-life. You can see this by plotting a concentration-time graph.

Show question

Question

What is half-life?

The half-life (t1/2) of a species is the time it takes for half of the species to be used in the reaction. In other words, it is the time it takes for its concentration to halve.

Show question

Question

True or false? Second-order reactants have a constant half-life.

False

Show question

Question

Only species in _____ feature in the rate equation.

The steps up to and including the rate-determining step

Show question

Question

Which step in a chemical reaction is the rate-determining step?

The slowest step

Show question

Question

What is rate of reaction?

Change in concentration of reactants or products over time.

Show question

Question

What are the units for rate of reaction?

mol dm-3 s-1

Show question

Question

Give three ways of measuring rate of reaction.

• pH change
• Change in mass
• Volume of gas produced
• Colour change

Show question

Question

What does the letter k represent in the rate equation?

The rate constant

Show question

Question

What does [A] represent in the rate equation?

The concentration of species A, typically in mol dm-3

Show question

Question

What do the letters m and n represent in the rate equation?

The order of reaction with respect to a particular species. They show the power that the species is raised to in the rate equation.

Show question

Question

rate = k [A]2 [B]

What is the order of reaction with respect to A?

2

Show question

Question

rate = k [A]2 [B]

What is the overall order of the reaction?

3

Show question

Question

'k' is constant at different temperatures. True or false?

False

Show question

Question

What are the units of the rate constant k?

It depends

Show question

Question

rate = k [A] [B]

What are the units of k for this reaction?

mol-1 dm3 s-1

Show question

Question

How does the concentration of zero-order species affect the rate of reaction?

It has no effect on the rate of reaction.

Show question

Question

How does the concentration of first-order species affect the rate of reaction?

It is directly proportional to the rate of reaction.

Show question

Question

How does the concentration of second-order species affect the rate of reaction?

It has an exponential effect on the rate of reaction.

Show question

Question

The concentration of a second-order species doubles. Predict the effect on the rate of reaction.

Show question

Question

The concentration of a first-order species increases by a factor of 3. Predict the effect on the rate of reaction.

The rate of reaction increases by a factor of 3.

Show question

Question

Which of these do not appear in the Arrhenius equation?

ΔT

Show question

Question

What are the units of Ea in the Arrhenius equation?

J mol-1

Show question

Question

What are the units of k in the Arrhenius equation?

mol-1 dm3 s-1

Show question

Question

What does k represent in the Arrhenius equation?

The rate constant

Show question

Question

What does T represent in the Arrhenius equation and what are its units?

Temperature, K

Show question

Question

According to the Arrhenius equation, increasing the temperature _____ the rate of reaction.

Increases

Show question

Question

According to the Arrhenius equation, decreasing the activation energy _____ the rate of reaction.

Increases

Show question

Question

What is the significance of the y-value at the point where x = 0 on an Arrhenius plot?

The y-value at the point where x = 0 is equal to ln(A)

Show question

Question

What is the Arrhenius equation?

The Arrhenius equation is a mathematical formula that relates the rate constant of a reaction with the activation energy and temperature of that reaction.

Show question

Question

What goes on the y-axis on an Arrhenius plot?

ln(k)

Show question

Question

What information does the y-coordinate of the point at which x = 0 give you?

ln(A)

Show question

Question

A reaction takes place at 600K. The Arrhenius constant equals 3.8 x 1011 s-1 and the activation energy of this reaction is 240 kJ mol-1. Calculate the rate constant, k.

4.73 x 10-10 s-1

Show question

Question

A reaction takes place at 650K. The Arrhenius constant equals 5.617 x 1012 mol-1 dm3 s-1 and the rate constant equals 0.53 mol-1 dm3 s-1. Calculate the activation energy, giving your answer to three significant figures.

162000 J mol-1 (162 kJ mol-1)

Show question

Question

True or false? For reactions with an activation energy of around 50 kJ mol-1, increasing the temperature from 290K to 200K will triple the rate constant.

False

Show question

Question

What is the rate constant?

The rate constant is a proportionality constant that links the concentrations of certain species to the rate of a chemical reaction.

Show question

Question

What is the symbol for the rate constant?

k

Show question

Question

Which of the following statements is always true about the rate constant?

0

Show question

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