The Reaction Quotient Q and Predicting Shift

The reaction quotient Q has the same form as the equilibrium constant but uses the concentrations at any moment, and comparing Q with K shows in which direction a reaction mixture will move: Q < K shifts toward products, Q > K shifts toward reactants and Q = K means equilibrium. In brief: for H2 + I2 ⇌ 2HI with K = 54.3, a mixture with [H2] = 0.0200 M, [I2] = 0.0100 M and [HI] = 0.0500 M has Q = 12.5, so it moves toward more HI.

Knowing the equilibrium constant is not enough to say what happens in a given mixture; one must also know whether the mixture is already at equilibrium. The reaction quotient answers this question. This article defines Q, gives the rule for the direction of the shift, explains its link to Gibbs energy and works through examples for gas-phase and aqueous reactions.

What is the reaction quotient?

For aA + bB ⇌ cC + dD, the reaction quotient is:

Q = [C]c[D]d / ([A]a[B]b)

The expression is written like Kc, but the concentrations are those present at the moment of interest, not necessarily at equilibrium. For gases, Qp uses partial pressures. Pure solids and liquids are left out, as in K. At the start of a reaction with only reactants, Q = 0. As the reaction proceeds, Q changes until it reaches the value of K, and from that moment it stays constant.

How do you predict the direction of the shift?

  1. Write the equilibrium expression and calculate Q from the given concentrations.
  2. Compare Q with K at the same temperature.
  3. If Q < K, there are too many reactants (or too few products), and the net reaction goes forward, from left to right.
  4. If Q > K, there are too many products, and the net reaction goes in reverse.
  5. If Q = K, the system is at equilibrium and no net change occurs.

Summary of the comparison

Predicting the direction of a reaction
Comparison Meaning Net change Concentration of products
Q < KRatio of products to reactants too smallForward reactionIncreases
Q = KAt equilibriumNoneConstant
Q > KRatio of products to reactants too largeReverse reactionDecreases

Worked examples

Example 1: a mixture that moves forward

For H2(g) + I2(g) ⇌ 2HI(g), Kc = 54.3 at 430 °C. A mixture has [H2] = 0.0200 M, [I2] = 0.0100 M and [HI] = 0.0500 M.

  1. Q = [HI]2 / ([H2][I2]) = (0.0500)2 / (0.0200 × 0.0100) = 0.00250 / 0.000200 = 12.5.
  2. Q (12.5) < K (54.3), so the reaction proceeds in the forward direction and more HI forms.

Example 2: a mixture that moves backward

For the same reaction, a mixture has [H2] = [I2] = 0.0050 M and [HI] = 0.0800 M. Q = (0.0800)2 / (0.0050 × 0.0050) = 0.00640 / 2.5 × 10−5 = 256. Since Q > K, HI decomposes in the reverse direction, and the concentrations of H2 and I2 rise.

Example 3: ammonia synthesis

For N2 + 3H2 ⇌ 2NH3, Kc = 0.50 at a particular temperature. A mixture contains 0.10 M N2, 0.20 M H2 and 0.050 M NH3. Q = (0.050)2 / (0.10 × 0.203) = 0.0025 / (0.10 × 0.0080) = 0.0025 / 0.00080 = 3.1. Since Q > K, the reaction proceeds in reverse and NH3 decomposes.

Example 4: only reactants present

If only N2O4 is present at the start of N2O4 ⇌ 2NO2, then [NO2] = 0, Q = 0 and Q < K for any K > 0. The reaction must go forward.

Example 5: a heterogeneous equilibrium

For CaCO3(s) ⇌ CaO(s) + CO2(g), Q = P(CO2), since the solids are not included. If the pressure of CO2 over the solid is lower than the equilibrium value Kp, the carbonate decomposes until the pressure reaches Kp; if it is higher, CO2 combines with CaO until the pressure falls to Kp.

Reaction quotient values of two mixtures compared with K on a logarithmic axis 1 10 100 1000 Q = 12.5 K = 54.3 Q = 256 net forward net reverse Reaction quotient Q (logarithmic scale)
For H2 + I2 ⇌ 2HI with Kc = 54.3 (long bar), the mixture of Example 1 has Q = 12.5, which is less than K, so the net reaction goes forward. The mixture of Example 2 has Q = 256, which is greater than K, so the net reaction goes in reverse. Either way, Q moves toward K.

How is Q related to free energy?

The Gibbs energy change for a reaction mixture is ΔG = ΔG° + RT ln Q. Since ΔG° = −RT ln K, this becomes:

ΔG = RT ln (Q/K)

If Q < K, the logarithm is negative, ΔG < 0 and the forward reaction is spontaneous. If Q > K, ΔG > 0 and the reverse reaction is spontaneous. At equilibrium Q = K and ΔG = 0. This is the thermodynamic basis for the rule.

Where is Q used?

Where does the concept come from?

The idea of comparing the actual composition with the equilibrium composition developed from the work of van ‘t Hoff and Gibbs in the 1880s and 1890s. Van ‘t Hoff’s isotherm, ΔG = ΔG° + RT ln Q, connected the driving force of a reaction to the deviation from equilibrium. The symbol Q for the reaction quotient became standard in the 20th century.

Which mistakes are common?