D22.4 Reaction Quotient

If we start a reaction with only reactants present, the reaction must initially be spontaneous in the forward direction, even if only very small concentrations of products are present when equilibrium is reached. Similarly, starting a reaction with only products present requires that the reaction proceed in the reverse direction.

However, which direction does a reaction go when both reactants and products are present but the reaction is not at equilibrium? The reaction quotient can help us answer this question. For a generic solution phase reaction:

mA(aq) + nB(aq) ⇌ xC(aq) + yD(aq)

the reaction quotient (Q) is defined as:

Q = \dfrac{[\text{C}]_t^{\;x}[\text{D}]_t^{\;y}}{[\text{A}]_t^{\;m}[\text{B}]_t^{\;n}}

As was true of K, the numeric value of each concentration expressed in mol/L is entered into the Q expression. Concentrations are represented by “[…]t” where the subscript “t” emphasizes that Q for a reaction depends on the concentrations present at the time when Q is determined. This is usually not at equilibrium. For a gas-phase reaction, Q is expressed in terms of numeric values of partial pressures expressed in bar.

When only reactants are present, Q = 0. As the reaction proceeds, Q increases because product concentrations (or partial pressures) increase and reactant concentrations (or partial pressures) decrease. For an example, see panels (a) and (b) in the figure below.

Four graphs are shown. The y-axis on top left graph is labeled, “Concentration,” and the x-axis is labeled, “Time.” Three curves are plotted on graph. The first is labeled, “[ S O subscript 2 ];” this line starts high on the y-axis, ends midway down the y-axis, has a steep initial slope and a more gradual slope as it approaches the far right on the x-axis. The second curve on this graph is labeled, “[ O subscript 2 ];” this line mimics the first except that it starts and ends about fifty percent lower on the y-axis. The third curve is the inverse of the first in shape and is labeled, “[ S O subscript 3 ].” The y-axis on top right graph is labeled, “Concentration,” and the x-axis is labeled, “Time.” Three curves are plotted on graph b. The first is labeled, “[ S O subscript 2 ];” this line starts low on the y-axis, ends midway up the y-axis, has a steep initial slope and a more gradual slope as it approaches the far right on the x-axis. The second curve on this graph is labeled, “[ O subscript 2 ];” this line mimics the first except that it ends about fifty percent lower on the y-axis. The third curve is the inverse of the first in shape and is labeled, “[ S O subscript 3 ].” The y-axis on bottom left graph is labeled, “Reaction Quotient,” and the x-axis is labeled, “Time.” A single curve is plotted on graph c. This curve begins at the bottom of the y-axis and rises steeply up near the top of the y-axis, then levels off into a horizontal line. The top point of this line is labeled, “kc.”
Figure: Reaction Quotient. The changes in the partial pressures of reactants and products are depicted for the 2 SO2(g) + O2(g) ⇌ 2 SO3(g) reaction. (Left) The reaction starts with only reactants present. Graph (a) shows changes in partial pressures and graph (b) shows the change in Q as the reaction approaches equilibrium over time. Once equilibrium has been reached Q = K and Q stops changing. (Right) The reaction starts with only products present. Graph (c) shows changes in partial pressures and graph (d) shows the change in Q as the reaction approaches equilibrium over time.

When the reaction reaches equilibrium, Q no longer changes over time because the concentrations no longer change, and, at equilibrium, Q = K:

Q \text{(at equilibrium)} = \dfrac{[\text{C}]_e^{\;x}[\text{D}]_e^{\;y}}{[\text{A}]_e^{\;m}[\text{B}]_e^{\;n}} = K

The subscript “e” emphasizes that the concentrations are equilibrium concentrations.

A system that is not at equilibrium proceeds spontaneously in the direction that establishes equilibrium (Q changes until it equals K). Hence, we can predict directional shifts of a reaction by comparing Q to K: when Q < K, the reaction proceeds spontaneously in the forward direction (from left to right; to the product side); when Q > K, the reaction proceeds spontaneously in the reverse direction (from right to left; to the reactant side).

For example, for the water-gas shift reaction,

CO(g) + H2O(g) ⇌ CO2(g) + H2(g)          K (800 °C) = 0.64

different starting mixtures of CO, H2O, CO2, and H2 react (and the concentrations of reactants and products change) until the compositions reach the same value of Q; that is, until Q = K.

Figure: water-gas shift. Partial pressures of four different mixtures are shown before and after reaching equilibrium at 800 °C for the water-gas shift reaction: CO(g) + H2O(g) ⇌ CO2(g) + H2(g).

It is important to recognize that, for an equilibrium system with the same elemental composition, Q reaches the same equilibrium value (K) whether the reaction starts from all reactants, from all products, or from a mixture of both. In fact, one technique to determine whether a reaction is truly at equilibrium is to start with only reactants in one experiment and start with only products in another. If the same value of the reaction quotient is observed when the concentrations have stopped changing in both experiments, then it is highly likely that the system has reached equilibrium.

Exercise: Determining Which Direction a Reaction Goes

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