The Reaction Quotient and the Equilibrium Constant
Gary L. Bertrand         gbert@umr.edu
Professor Emeritus of Chemistry         University of Missouri-Rolla

LeChatelier's Principle tells us how a chemical system will respond when a change is imposed on it.  However, there is a much clearer way to determine these changes in a quantitative manner, and to determine whether or not a reaction is at equilibrium.  The fundamental quantity which does this is called The Reaction Quotient.

You will see that the Reaction Quotient will eventually look just like something you may have called the Equilibrium Constant.  Even though they look alike, the two are very different.

For a reaction such as

2 A + B C + 2D
the reaction quotient (Q) is written: The reaction quotient contains the activities ( C, D) of the products in the numerator and the activity of each product is raised to a power corresponding to its stoichiometric constant - thus the activity of product C is raised to the power of "1" and the activity of the product D is raised to the power of "2".  The reaction quotient also contains the activities ( A, B) of the reactants in the denominator with the activity of each reactant raised to a power corresponding to its stoichiometric constant - thus the activity of reactant A is raised to the power of "2" and the activity of reactant B is raised to the power of "1".

At this point, we have introduced two things that you've probably never heard of before - the reaction quotient and activities.  Follow along for a while, and maybe these will start to make sense.

Activities are closely related to concentration, but concentrations may be expressed in many ways - weight fraction, mole fraction, molarity, molality, etc - and in the case of gases, we use often the partial pressure in place of concentration.  The standard variables that Chemists use for concentration are:

 Physical State: Concentration Variable: Gas (g) Partial Pressure (PA ,bar) or Molarity ([A], moles/liter of solution) Pure Solid (s) 1 Pure Liquid (l) 1 Solvent (l) Mole Fraction (XA) Solute (aq) Molarity ([A], moles/liter), or Molality (mA, moles/kG of solvent)

The activity is defined as a dimensionless quantity and is equated to the concentration variable multiplied by an activity coefficient (f or g).  Most concentration variables have mathematical dimensions, so the activity coefficient must have dimensions which cancel out those of the concentration.

The concentration units to be used for activity depend on the physical state of the component, so the reaction that we are considering must include these physical states:

2A(g) + B(l) C(s) + 2D(aq)

For a gas, the activity may be written in terms of the partial pressure in atmospheres or bars, or in terms of molarity:

gas: A= PAfA   or A= [A]gA   .

For a dissolved compound [(aq) if the solvent is water] the concentration variable is molarity (moles/liter of solution), but for some special applications (electrochemistry, chemical thermodynamics) molality (moles/kilogram of solvent) is used:

solute: D = [D]gD.

For a pure liquid or a pure solid, the concentration variable is "1", provided that some of the material is present (otherwise it's activity is zero):

pure liquid or solid: B = (1)gB C = (1)gC.

There are cases in which a component such as water (H2O) may play the role of solvent as well as that of a reacting component.  The physical state is shown as (l), but a solvent cannot be a pure liquid.  In this case, the concentration variable is the mole fraction (XB):

solvent: B = XBgB.

Activities and activity coefficients have been designed so that activity coefficients may be approximated as "1" for many of the situations of interest to Chemists.  These situations are called "ideal" as in "ideal gas", "ideal solution", or "ideal dilute solution":

Ideal gas or real gas at low pressure: A = PA   or A = [A]   .

Solute at low concentration: D = [D]  .

Pure liquid or solid at 1 atmosphere (or 1 bar) pressure: B = 1 C = 1 .

Solvent with dilute solutes: B = XB .

Writing the Reaction Quotient:

The first step is to inspect the reaction as written, and write the reaction coefficient in terms of activities:

2A(g) + B(l) C(s) + 2D(aq) Concentration units may then be substituted for activities for an approximation of Q:

Q = (1)[D]2/PA2(1)  = [D]2/PA2    ,

or (using concentration units for the gas)

Q = (1)[D(aq)]2/[A(g)]2(1)  = [D(aq)]2/[A(g)]2    .

As was pointed out earlier, the Reaction Quotient is written exactly the same way we write an equilibrium constant.  Actually, it's a great deal more than an equilibrium constant.

A chemical reaction comes to equilibrium, irrespective of the amounts of the components that we start with.  If we introduce a gas A and an immiscible liquid B into a container along with water to serve as a solvent for D, there is a tendency for them to react and form C and D.  However, there may be kinetic barriers which prevent this reaction.  Kinetic barriers can usually be overcome with a suitable catalyst and/or an increase in temperature.

Before any reaction occurs, we can calculate the value of Q for this reaction.  The concentration of component D is zero, and the partial pressure (or concentration) of A is greater than zero, so the value of Q is zero.

As the reaction proceeds, the concentration of D increases and the partial pressure (or concentration) of A decreases, so the value of Q increases.  Eventually, the reaction will slow down and come to equilibrium, with a specific value for Q when the reaction is at equilibrium.  While the components B and D do not appear in our calculation of Q, they must also be in the container for the reaction to reach equilibrium.  Reactant A cannot react without the presence of B, and it isn't possible to form Product D without also forming product C.

When the reaction comes to equilibrium, the value of Q is equal to the equilibrium constant K.

If we start with only the solid compound C and an aqueous solution of component D in the container, the concentration of component A is zero.  Before any reaction occurs, the value of Q is infinite.

The reaction will begin to occur, increasing the concentration (or partial pressure) of A and the amount of B while the concentration of D and the amount of C decreases.  The reaction quotient Q decreases as the reaction proceeds toward equilibrium.

When the reaction comes to equilibrium, the value of Q is again equal to the equilibrium constant K.

Irrespective of the amounts of the various components that one starts with, the reaction will come to equilibrium with Q equal to K, provided that all solids or liquids involved in the reaction are present.

This gives us a way to predict which way the reaction will go (whether it will produce more products or more reactants) if we know the concentrations or partial pressures of the reacting components so that we can calculate Q, AND we must know the equilibrium constant, K.

If Q is less than K, more products will be formed.  If Q is greater than K, more reactants will be formed.  If Q= K, no changes will occur.

If a reaction is at equilibrium (Q = K) and something is done to disturb the system (adding something to the container, changing the volume of the container, changing the temperature), this may change the relationship between Q and K.

The value of K may depend on the temperature, but not on the amounts of the compounds nor on the pressure.  The value of Q depends only on partial pressures and concentrations.

If there is a change in either Q or K, the reaction will go in the direction which will re-establish the condition Q = K.

Therefore, if the disturbance causes Q to become less than K, more products will be formed.  If the disturbance causes Q to become greater than K, more reactants will be formed.  If the disturbance does not change either Q or K, no changes will occur.

If a reaction is endothermic as it comes to equilibrium, an increase in temperature will increase the value of K.  If the reaction is exothermic as it comes to equilibrium, an increase in temperature will decrease the value of K.  Temperature has no effect on the equilibrium constant for an athermal reaction.