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# Representations of Equilibrium

It is easy to get bogged down with numbers when working with dynamic equilibria. A different way of looking at them is through visual representations. They can help you identify the point of equilibrium and work out the value of the equilibrium constant in what is perhaps a more intuitive way.

• We'll define equilibrium and the equilibrium constant before looking at graphical and particulate ways of representing equilibrium.
• You'll be able to practice not only identifying the point of equilibrium, but also working out the value of the equilibrium constant from such diagrams.

## What is an equilibrium state?

Take a "reversible reaction". This is a reaction in which the reactants react to form the products, as you'd expect, but the products also react again to form the reactants. For example, in a reaction, A and B react to form C and D, which then react to form A and B once more. At first, one reaction might happen more than the other. This causes the concentrations of A, B, C, and D to change. But if you leave the reaction in a closed system, eventually the rate of the forward reaction equals the rate of the backward reaction, and the concentrations of all of the species settle down. This is a point of dynamic equilibrium.

A dynamic equilibrium is the state of a reversible reaction in which the rate of the forward reaction equals the rate of the backward reaction and the concentrations of reactants and products remain constant.

Check out "Dynamic Equilibrium" for a more in-depth look at dynamic equilibria.

## Characteristics of equilibrium constant

We've learned that in a state of dynamic equilibrium, the concentrations of reactants and products in a closed system don't change. This means that the ratio between them doesn't change, either. We represent this ratio using the equilibrium constant, Keq.

The equilibrium constant, Keq, is a value that tells us the relative amounts of reactants and products in a system at equilibrium.

We use the equilibrium constant Kc to represent the relative concentrations of gaseous or aqueous species at equilibrium, and the equilibrium constant Kp to represent the relative partial pressures of gaseous species at equilibrium.

## Equilibrium Constant Expression

The values of Kc and Kp are worked out using the following expressions:

$$K_c=\frac{[C]_{eq}^c[D]_{eq}^d}{[A]_{eq}^a[B]_{eq}^b}$$

$$K_p=\frac{(P_C)_{eq}^c(P_D)_{eq}^d}{(P_A)_{eq}^a(P_B)_{eq}^b}$$

Not sure what these expressions mean? "Equilibrium Constant" has got you covered!

## Visual representations of equilibrium

So, equilibrium - a state in which the rate of the forward reaction equals the rate of the backward reaction, and the concentrations of reactants and products don't change. We can represent an equilibrium in different ways:

• Graphical representations.
• Particulate representations.

We'll look at both of these types of representations, as well as consider how to interpret them. Let's start with graphical representations.

## Graphical representations of equilibrium

The first way of representing an equilibrium is by using a concentration-time graph. We plot concentration on the y-axis and time on the x-axis, showing how the concentrations of all of the different species involved in the reaction change as the reaction progresses. Here's an example:

Fig.1-A concentration-time graph for a system of reversible reactions.

Notice a few things:

• The change in concentration of the products is proportional to the change in concentration of the reactants. This makes sense - after all, the reactants react to form the products!
• The change in concentration is initially fast, but after a period of time, it slows.
• The concentrations of both products and reactants eventually level out.
• However, the concentrations of products and reactants aren't necessarily the same as each other.

Let's now consider what we can learn from concentration-time graphs.

### Interpreting graphical representations

In the introduction, we said how we can use representations of equilibrium to work out the point at which the reaction reaches equilibrium, and the value of the equilibrium constant. Here's how that works for concentration-time graphs.

#### Point of equilibrium

At equilibrium, the overall concentrations of reactants and products remain constant. We can see this on a concentration-time graph. It is the point at which the lines showing the concentration of reactants and products become horizontal - their gradient is 0, and so concentration doesn't change.

Fig.2-A concentration-time graph for a system of reversible reactions. Here, the point at which equilibrium is reached is labeled.

#### Value of equilibrium constant

We can also calculate the value of the equilibrium constant from concentration-time graphs. To do this, we use the balanced chemical equation to write an expression for Keq. We then identify the point of equilibrium and substitute the concentrations of reactants and products at this point (or any point after) into the expression.

34 M H2 and 20 M Cl2 are left in a closed system to reach dynamic equilibrium. Using the following concentration-time graph, calculate:

1. The time at which equilibrium is reached.
2. The value of Kc for this reaction.

Fig.3-A concentration-time graph for the reversible reaction between hydrogen and chlorine.

To find the time at which equilibrium is reached, we look for when the concentrations of reactants and products level out. Here, that is at the 20-second mark. Therefore, equilibrium is reached at 20 seconds.

To calculate the equilibrium constant Kc, we first need to find an expression for it. This is done using the balanced chemical equation:

$$K_c=\frac{[HCl]_{eq}^2}{[H_2]_{eq}[Cl_2]_{eq}}$$

We then substitute in the equilibrium concentrations of each species. This is their concentration at any point at or beyond 20 seconds:

$$[H_2]_{eq}=24\,M\,\,\,[Cl_2]_{eq}=10\,M\,\,\,[HCl]_{eq}=20\,M$$

$$K_c=\frac{(20)^2}{(24)(10)}=1.67$$

You can also use rate-time graphs to indicate an equilibrium. These plot the rate of the forward and backward reaction on the y-axis against time on the x-axis. At equilibrium, the rate of the forward reaction equals the rate of the backward reaction, and so the two rates converge onto one horizontal line. Here's an example:

Fig.4-A rate-time graph for a system of reversible reactions

## Equilibrium Particle Representation

Another way of representing an equilibrium is with a particle diagram. Here, we use different colored particles to represent the relative amounts of reactants and products in a system. Here's an example:

Fig.5-A particle diagram for a system of reversible reactions.

Note the following:

• Each particle represents a fixed number of moles of a species. For example, in this diagram, a particle could represent 0.1 moles.
• The volume of the container should be given. For example, the volume of this diagram could be 1 liter. Using the relative amounts and the volume of the system, we can work out the concentration of each species.
• We might just be given one diagram, showing the system at equilibrium, or perhaps a series of diagrams, showing how the relative amounts of products and reactants change over time.
• Like with concentration-time graphs, the concentrations initially change rapidly, but then level off.

Here's how we interpret particle diagrams.

### Interpreting particulate representations

Particulate representations of equilibrium can be used much like concentration-time graphs, in order to find the point at which equilibrium is reached and the value of the equilibrium constant.

#### Point of equilibrium

At equilibrium, the concentrations of reactants and products remain constant - once a system has reached equilibrium, the relative amounts of products and reactants don't change. When it comes to particulate representations, we look for the diagram in which the numbers of each type of particle stop changing.

Fig.6-A series of particle diagrams for a system of reversible reactional

For example, in diagrams C and D, the relative amounts of reactants and products remain constant. This means that the system reached equilibrium at point C.

#### Value of equilibrium constant

We can also use particle diagrams to work out the equilibrium constant Kc. We first find an expression for the equilibrium constant using the balanced chemical equation. We then calculate the equilibrium concentrations of each species using the number of particles and the overall volume of the system. Finally, we substitute them into the equilibrium concentration expression to get our final answer.

Using the following particle diagram, calculate:

1. The time at which equilibrium is reached.
2. The value of Kc for this reaction.

Note that here, each particle represents 1 mole, and the system has a volume of 1 liter. Time is given in seconds.

Fig.7-A further example of series of particle diagrams for a system of reversible reactions

To find the point at which the system reaches equilibrium, we look for the diagram in which the numbers of each particle stop changing. In this representation, you can see that the number of orange particles - representing N2O4 - decreases from t = 0 to t = 20, whilst the number of purple particles - representing NO2 - increases. However, their numbers don't change from t = 20 to t = 25. Therefore, we reach equilibrium at the 20 second mark.

Next, to calculate the equilibrium constant Kc, we first need to write an expression for it. Once again, this is done using the balanced chemical equation:

$$K_c=\frac{[NO_2]_{eq}^2}{[N_2O_4]_{eq}}$$

We then calculate the equilibrium concentration of each species. We worked out above that equilibrium is reached at t = 20. At this point, there is 1 particle of N2O4 and 8 particles of NO2. The question told us that each particle represents 1 mole and that the volume of the system is 1 liter. We can therefore work out their equilibrium concentrations:

$$[N_2O_4]_{eq}=\frac{1}{1}=1\,M$$

$$[NO_2]_{eq}=\frac{8}{1}=8\,M$$

All we need to do now is substitute these values into the equilibrium constant expression, and we'll get our final answer:

$$K_c=\frac{(8)^2}{1}=64$$

That's it for this article. You should now understand the different ways of representing an equilibrium, using concentration-time graphs and particle diagrams. You should also be able to interpret these diagrams to find not only the point at which equilibrium is reached but also the value of the equilibrium constant.

## Representations of Equilibrium - Key takeaways

• At a state of equilibrium, the rate of the forward reaction equals the rate of the backward reaction and the concentrations of products and reactants remain constant.
• We can use the equilibrium constant Keq to represent the ratio between the relative amounts of products and reactants ina system at equilibrium.
• The equilibrium constant Kc measures the relative concentrations of gaseous or aqueous species at equilibrium.
• The equilibrium constant Kp measures the relative partial pressures of gaseous species at equilibrium.
• We can use both graphical and particulate representationsto show an equilibrium:
• Concentration-time graphs show how the concentration of products and reactants change throughout a reaction.
• Particle diagrams describe the relative amounts of products and reactants either just at equilibrium or throughout the reaction.
• From equilibrium representations, we can work out the point at which the system reaches equilibrium and the value of the equilibrium constant Kc.

Equilibrium expressions are used to find the equilibrium constant, Keq. They describe the ratio of products to reactants in a system at equilibrium.

Examples of equilibria include the reaction between hydrogen and iodine to form hydrogen iodide: H2(g) + I2(g) ⇌ 2HI(g). Another example is the reaction between nitrogen and hydrogen to form ammonia: N2(g) + 3H2(g) ⇌ 2NH3(g)

At equilibrium, the rate of the forward reaction equals the rate of the backward reaction and the concentrations of reactants and products remain constant. Head over to "Dynamic Equilibrium" for more.

The three factors that affect an equilibrium are temperature, pressure and concentration. You can find out more about these factors and the effect that they have on an equilibrium in the article "Le Chatelier's Principle".

An system at equilibrium is represented by combining two reactions with two half-headed arrows, ⇌. The arrows show that the first reaction is the reverse of the second - the initial reactants react to form the products, which can then react to form the reactants again.

## Final Representations of Equilibrium Quiz

Question

What is a state of equilibrium?

The state of a reversible reaction in which the rate of the forward reaction equals the rate of the backward reaction and the concentrations of reactants and products remain constant.

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Question

What is the equilibrium constant?

The equilibrium constant, Keq, is a value that tells us the relative amounts of reactants and products in a system at equilibrium.

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Question

Which of the following are true about Kc?

It uses concentration at equilibrium.

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Question

Which of the following are true about Kp?

It only involves gaseous species.

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Question

Concentration-time graphs show _____.

Concentration on the y-axis and time on the x-axis.

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Question

How do you identify the point at which equilibrium is reached from a concentration-time graph?

The concentrations of reactants and products level off.

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Question

True or false? On a concentration-time graph, the lines showing the concentrations of reactants and products always converge at equilibrium.

False

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Question

How do you identify the point at which equilibrium is reached from a particle diagram?

The number of particles of each species remains constant.

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