# Rate Equations

Chemical reactions are like a race where reactants turn into products due to changes in their structures. These changes can happen at different speeds, just like race cars. Understanding how the speed of chemical changes can be affected is important in physical chemistry. To do this, scientists use something called a rate equation. This equation shows how the rate of a reaction is linked to the concentration of the species involved. In this article, we will dive into the rate equation and explore what it tells us about the speed of chemical reactions. We will also talk about the rate constant and reaction orders, and briefly touch on the methods used to determine the rate equation. By the end of this article, you'll be a pro at understanding how chemical reactions work! Keywords: rate equations, chemical reactions, physical chemistry, rate constant, reaction orders, species concentration.

## Rate equation chemistry

The rate of a reaction is how quickly a reaction occurs and is typically measured in terms of the change in concentration of the reactants or products over a period of time. For example, the rate of a reaction can be determined by looking at how much product is made in a given period of time, which will depend on how much reactant is used up. This rate of reaction is typically measured in mol dm-3 s-1.

The rate of a reaction can tell us a lot about the reaction itself, such as the activation energy needed for the reaction to occur, the reaction mechanism, and the overall efficiency of the reaction. Knowing the rate of a reaction can also help us to better understand the underlying chemistry of the reaction, as well as to predict the rate of future reactions under similar conditions.

**Measuring rate of reaction**

To calculate the rate of a reaction, we need to measure the change in the amount of reactant or product over time. This can be done by observing changes in colour, pH, volume of gas produced, or changes in mass of a solid reactant. Once we have collected our data, we can plot it on a line graph, with time on the x-axis and concentration on the y-axis. From the graph, we can calculate the rate of reaction by finding the line's gradient.

There are two types of rates we can calculate: overall rate of reaction and instantaneous rate of reaction. To calculate the overall rate of reaction, we 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. For example, if we have a graph of concentration against time and the concentration changes from 1 mol/dm³ to 0.5 mol/dm³ over a period of 20 seconds, the overall rate of reaction would be:

Overall rate of reaction = (1 mol/dm³ - 0.5 mol/dm³) / (20 seconds) = 0.025 mol/dm³s.

By calculating the rate of reaction, we can better understand the chemistry of the reaction and predict the rate of future reactions under similar conditions.

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.

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. 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.

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.

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

## 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:

Understanding the rate equation may seem confusing at first, but it becomes clearer once you understand the symbols and their meaning. k is the rate constant, which is specific to a particular reaction and reaction conditions. The letters A and B in the rate equation represent species involved in the reaction, which could be reactants or catalysts. The square brackets around the letters represent concentration, so [A] means the concentration of species A. The letters m and n represent the order of the reaction with respect to a certain species. The power to which the concentration of a species is raised is called the order of the reaction with respect to that species. So, [A]m represents the concentration of A, raised to the power of m, which means that A has the order m.

The rate constant, k, 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, but it remains constant for a certain reaction at a particular temperature. If you carry out the same reaction at different temperatures, k will change. However, if you carry it out at the same temperature, k will stay the same.

To learn more about how the rate constant relates to temperature, you can read about the Arrhenius Equation. If you want to calculate the rate constant and learn about its units, you can head over to resources on Determining Rate Constant.

**Orders of reaction**

In chemical reactions, the order of reaction for reactants and catalysts determines how the concentration of those species affects the rate of reaction. The overall order of the reaction is the sum of the individual orders of species present in the reaction. The rate equation shows the order of the reaction with respect to each species using a power.

Zero-order reactants have no effect on the rate of reaction, so their concentration doesn't appear in the rate equation. First-order reactants have a direct proportionality between their concentration and the rate of reaction, so their concentration appears in the rate equation raised to the power of 1. Second-order reactants have an exponential relationship between their concentration and the rate of reaction, so their concentration appears in the rate equation raised to the power of 2.

To determine the overall order of a reaction, you simply add the individual orders of all species in the rate equation. For example, if the rate equation is rate = k[A]2[B], the overall order of reaction would be 2 + 1 = 3.

An example problem was given to help understand the effect of doubling the concentrations of species A, B, and C. Only A and B appeared in the rate equation, with A being first-order and B being second-order. Doubling the concentration of A would cause the rate of reaction to double, while doubling the concentration of B would cause the rate of reaction to quadruple. Doubling the concentration of C, which was zero-order, would have no effect on the rate of reaction.

Understanding the order of reaction is crucial for predicting the behavior of chemical reactions and determining reaction kinetics.

## 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.

**Half-life equations**

The half-life, , 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.

The half-life of a first-order reactant relates to the rate constant, k, using the following equation: This means that once you know the half-life of a first-order reactant, you can easily find k.

**Reaction mechanism**

In addition to understanding the order of reaction, it's important to know that reactions can have mechanisms with one or multiple steps. Each step in the mechanism occurs at a different rate, and the slowest step is known as the rate-determining step. This step determines the overall rate of the reaction, and the rate equation includes only the species involved in the steps up to and including the rate-determining step. The order of each species in the rate equation is determined by the stoichiometry of the reaction and the order of each species in the rate-determining step.

The rate equation includes a rate constant (k), which is specific to a particular reaction and reaction conditions. The concentration of each species involved in the reaction is raised to the power of its respective order. Zero-order reactants have no effect on the rate of reaction, while first-order reactants have a direct proportionality between their concentration and the rate of reaction. Second-order reactants have an exponential relationship between their concentration and the rate of reaction.

The rate equation can be determined through various methods such as the initial rates method, identifying the shapes of graphs, calculating half-lives, and inspecting the reaction mechanism. Knowing the reaction mechanism and rate-determining step can also help predict the rate equation. Overall, understanding the rate equation and reaction mechanisms is essential to predicting and controlling the behavior of chemical reactions.

## Rate Equations

**How do you calculate the rate equation?**

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.

**How do you write a rate equation?**

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.

**How do you find the rate of change from an equation?**

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.

**How do you find the rate of reaction from an equation?**

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.

**What factors affect the rate of reaction?**

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