Reaction Quotient

If you haven't eaten anything for a while, your blood glucose levels might drop. Your body responds by releasing glucagon, a hormone that causes your liver to break down glycogen. This increases your blood glucose levels. On the other hand, if you've just eaten a large meal, your blood glucose levels might increase. This time your body responds by releasing insulin, a hormone that causes your cells to take up glucose and store it as glycogen. The system works in an equilibrium. Its overall aim is to keep your blood glucose levels constant, at a fixed point.

However, sometimes our body isn't quite at an equilibrium. There might be too much glucose in our blood, or perhaps not enough. The reaction quotient is a handy way of looking at reversible reactions that haven't yet reached equilibrium.

• This article is about the reaction quotient, Q, in chemistry.
• We'll define the reaction quotient and look at its expression before seeing how it differs from the equilibrium constant, K_eq.
• We'll then go through an example of calculating the reaction quotient.
• Finally, we'll take a deep dive into how the reaction quotient relates to Gibbs free energy.

What is the Reaction Quotient?

If you've read the articles "Dynamic Equilibrium" and "Reversible Reactions", you'll know that if you leave a reversible reaction in a closed system for enough time, it will eventually reach a point of dynamic equilibrium. At this point, the rate of the forward reaction equals the rate of the backward reaction and the relative amounts of products and reactants don't change. Provided you keep the temperature the same, the position of the equilibrium doesn't change either.

It doesn't matter whether you start with lots of the reactants or lots of the products - as long as the temperature remains constant, you'll always end up with fixed relative amounts of each. This is analogous to your body always trying to bring your blood sugar levels back to a fixed point.

We can express the ratio between the relative amounts of products and reactants using the equilibrium constant, K_eq. Because the position of an equilibrium is always the same at a certain temperature, K_eq is always the same too. At equilibrium, the value of K_eq is constant.

However, reactions may take a while to get to equilibrium. What if we want to compare the relative amounts of reactants and products in a system that still isn't quite there? For this, we use the reaction quotient.

You should be familiar with the different types of K_eq. They measure the amounts of substances in different systems of reversible reactions at equilibrium in different ways. For example, K_c measures the concentration of aqueous or gaseous species in an equilibrium, whilst K_p measures the partial pressure of gaseous species in an equilibrium. Likewise, we can also get different types of the reaction quotient. In this article, we'll focus on just two of them:

• Q_c is similar to K_c. It measures the concentration of aqueous or gaseous speciesin a system at a particular moment.
• Q_p is similar to K_p. It measures the partial pressure of gaseous speciesin a system at a particular moment.

Let's now move on to look at the expressions for Q_c and Q_p.

Reaction Quotient Expression

The expressions for the reaction quotients Q_c and Q_p are very similar to the respective expressions for K_c and K_p. But whilst K_c and K_p take measurements at equilibrium, Q_c and Q_p take measurements at any one time - not necessarily at equilibrium.

Q_c Expression

Take the reaction \(aA + bB \rightleftharpoons cC + dD\). Here, the capital letters represent species whilst the lowercase letters represent their coefficients in the balanced chemical equation. For the above reaction, Q_c looks a little something like this:

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

Here's what that all means:

• Square brackets show the concentration of a species at a given moment. Therefore, [A] means the concentration of species A.

• The superscript lowercase letters are exponents, based on the coefficients of species in the balanced chemical equation. Therefore, [A]^a means the concentration of species A, raised to the power of the number of moles of A in the balanced equation.

• Overall, the numerator represents the concentrations of the products, raised to the power of their coefficients, and then multiplied together. The denominator represents the concentrations of the reactants, raised to the power of their coefficients, and then multiplied together. To find Q_c, you simply divide the numerator by the denominator.

Notice how similar this expression is to the expression for K_c. The only difference is that K_c uses equilibrium concentrations, whilst Q_c uses concentrations at any given moment:

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

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

Q_p Expression

Let's take the reaction again. But this time, instead of measuring concentration, let's measure the partial pressure of each species. This is the pressure it would exert on the system if it occupied the same volume on its own. To compare the ratio of partial pressures of gases in a system, we use Q_p. Here's the expression:

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

Let's break that down:

Once again, notice how similar this is to the expression for K_p. The only difference is that K_p uses equilibrium partial pressures, whilst Q_p uses partial pressures at any given moment:

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

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

Reaction Quotient Units

Q takes the same units as K_eq - which, as you might remember, doesn't have any units. Both K_eq and Q are unitless.

Difference Between the Equilibrium Constant and the Reaction Quotient

Before we go any further, let's consolidate our learning by providing a summary of the differences between the equilibrium constant and the reaction quotient. We'll further break it down into K_c, K_p, Q_c and Q_p:

Fig.1-A table comparing the equilibrium constant and the reaction quotient

Reaction Quotient Example

Before we finish, let's have a go at calculating the reaction quotient for a particular reaction at a given moment. In the article "Using the Reaction Quotient", we'll then compare this to the reaction's equilibrium constant and see what it tells us about the reaction.

That's the end of this article. By now you should understand what we mean by the reaction quotient and be able to explain the difference between the equilibrium constant and the reaction quotient. You should also be able to derive an expression for the reaction quotient based on a system of reversible reactions then use your expression to calculate the reaction quotient.