E09 Applications of the equilibrium law : a change in concentration of one component

It is obvious, that such a simple fundamental law has many interesting and useful applications:

1. If the equilibrium constant of a specific reaction at particular temperature is known, one can predict (calculate) for known reactant concentrations the concentrations applying at equilibrium. The degree of conversion can also be calculated, but not the equilibration time.
2. If the initial composition of an equilibrium reaction mixture and the equilibrium concentration of one of the components are known, then the other equilibrium concentrations can be calculated and also the K_c-value of the reaction at that particular
   temperature.
3. The equilibrium law can be used in industrial chemical reactions to increase the yield of a reaction which does not go to completion by varying the initial concentrations of the reactants.
   These application possibilities will be discussed at greater length later in this text.

We could even agree that a change in temperature changes the equilibrium law itself: for each reaction temperature a different equilibrium constant applies. Its influence clearly merits separate discussion.

The only other factor, apart from reaction temperature,which can change the equilibrium mixture is a change in concentration of one of the components or all of the components at the same time.

In illustration E09a, the effect of a change in the NO₂- concentration on the NO₂-dimerization reaction is shown. The concentrations of the reactant, NO₂, and product, N₂O₄ are plotted as a function of time. The initial situation corresponds with the equilibrium of experiment A in the table on illustration E06.

The initial equilibrium concentrations of NO₂ and N₂O₄are 0.011 mol/L and 0.027 mol/L respectively. The brown band shows a doubling of NO₂-concentrations at time t₁, by adding NO₂ from outside the closed system.

Between time t₁ and the achievement of a new equilibrium mixture at time t₂ there is no equilibrium, the NO₂- and N₂O₄-concentrations in the reaction mixture changing as shown in illustration E09a until new equilibrium concentrations are achieved. The new equili-brium concentrations for NO₂ and N₂O₄ are 0.012 mol/L and 0.032 mol/L respectively.

After the concentration change, the concentrations of the components vary in the closed system.
This can seem strange at first : a rapid doubling in NO₂- concentration leading to a strong increase in N₂O₄- concentration (N.B. this specific case may not be freely generalized).

These sometimes unexpected changes is concentration can be explained in terms of the equilibrium law. In illustration E09b, use is made of the term reaction quotient Q_c, to help explain this. The reactant quotient of an equilibrium reaction Q_c is analogous to K_c, differing in the use of the actual concentrations of the components present at a given time, rather than the equilibrium concentrations of the components.

In the case of the NO₂-dimerization reaction, the appropriate expressions for Q_c and K_c are:

At time t₁, a doubling of the NO₂-concentration yields a Q_c value of about 55 i.e. (0.027)/(0.0022)². This as a quarter of the K_c-value for this equilibrium at the same temperature. Thus Q_c < K_c. According to the equilibrium law, Q_c  will increase until Q_c equals K_c i.e. 222. It is this change in Q_c with time, which can be followed as the green band in illustration E09b.

To attain the K_c value, the numerator has to increase by an amount x to (0.027 + x). The denominator has to decrease correspondingly by 2x to (0.011 - 2x), as 2 molecules of NO₂ are consumed for each molecule of N₂O₄ formed.

The new equilibrium concentrations, indicated by [ ]’_e, can be calculated using the equilibrium law and the K_c value by solving the equation:

In the following table, quantititive results are summarized, which are useful in the interpretation of illustrations E09a and E09b.

The new equilibrium concentrations 0.012 mol/L for NO₂ and 0.032 mol/L for N₂O₄, when inserted in the expression for K_c give a K_c value of 222.
The result of this intervention, an be explained qualitatively as follows:

“If change occurs in the concentration of a component in an equilibrium mixture, the system will tend to adjust itself so as to annul, as far as possible, the effect of that change”.

This rule of thumb is known as “Le Chatelier’s Principle” or the principle of “least stress”.

BEWARE

The disturbance in a chemical equilibrium is not completely counteracted, as appears it to be in the case of phase-equilibria. The NO₂-concentration increase is counteracted as much as possible, the initial 0.022 mol/L being reduced to 0.012 mol/L. However, this is still higher than the initial value of 0.011 mol/L. The disturbance has an interesting result, the position of the equilibrium being shifted to the right, although the degree of conversion has decreased. For the sake of completeness, it should be repeated that a closed system at chemical equilibrium is being considered in which the reaction temperature remains constant. That this disturbance is almost completely counteracted by adaptions in the reaction mixture is thanks to the fairly large value of K_c , in this case. In the case of reactions with much smaller K_c values (e.g. weak acids) the amount of counteraction is much smaller.