Reaction Quotient (Q) Calculator

What is a reaction quotient calculator?

A reaction quotient calculator finds the value of Q, a number that compares the amount of product to the amount of reactant in a chemical reaction at any moment. It works for reactions with up to three reactants and three products. Enter each species' stoichiometric coefficient (its number in the balanced equation) and molar concentration, and the calculator returns Q along with the formula and the substituted values.

What is the reaction quotient (Q)?

The reaction quotient is the ratio of product concentrations to reactant concentrations, with each concentration raised to the power of its coefficient in the balanced equation. It can be worked out at any point during a reaction, not just at equilibrium. Comparing Q with the equilibrium constant K shows which way the reaction is heading.

Reaction quotient formula

For a general reaction

$aA + bB \rightarrow cC + dD$

the reaction quotient is

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

Here [A], [B], [C], and [D] are molar concentrations (mol/L), and a, b, c, and d are the whole-number coefficients from the balanced equation. Pure solids and pure liquids are left out of the expression, because their relative concentrations equal 1. Only gases and dissolved species appear.

How to calculate the reaction quotient

1. Write the balanced chemical equation.
2. Note the coefficient of each reactant and product.
3. Find the concentration of each species, in mol/L, at the moment of interest.
4. Raise each concentration to the power of its coefficient.
5. Multiply the product terms together for the numerator, and multiply the reactant terms together for the denominator.
6. Divide the numerator by the denominator.

Example calculation

Take the reaction that forms ammonia:

$N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$

Suppose the concentrations are $[N_2] = 0.5\text{ M}$, $[H_2] = 0.2\text{ M}$, and $[NH_3] = 0.1\text{ M}$.

$Q = \frac{[NH_3]^2}{[N_2]^1[H_2]^3} = \frac{(0.1)^2}{(0.5)(0.2)^3} = \frac{0.01}{0.004} = 2.5$

Q equals 2.5 at this moment in the reaction.

Q versus the equilibrium constant (K)

Q and K use the same formula, but K is the value Q takes at equilibrium, at a fixed temperature. Comparing the two predicts which way a reaction will shift:

• Q < K: the reaction moves forward, making more product.
• Q = K: the reaction is at equilibrium; concentrations stop changing.
• Q > K: the reaction moves in reverse, making more reactant.

Zero and undefined values of Q

If any product concentration is zero, the numerator is zero, so Q equals 0 (unless a reactant concentration is also zero, in which case the calculator shows ∞). This is typical near the start of a reaction, before much product has formed.

If any reactant concentration is zero, the denominator is zero and Q is mathematically undefined. The calculator flags this case with a warning and shows the result as ∞, since the ratio grows without bound as the reactant concentration approaches zero.

Uses of the reaction quotient

Chemists use Q to predict which direction a reaction will shift when concentrations change, an idea connected to Le Chatelier's principle: adding more reactant lowers Q below K, so the reaction shifts forward until Q rises back to K. Q also appears in the Nernst equation, $E = E^\circ - \frac{RT}{nF}\ln Q$, which relates the voltage of an electrochemical cell to the concentrations of the species involved, and in the relation between free energy and reaction mixture composition, $\Delta G = \Delta G^\circ + RT\ln Q$.

History

Norwegian chemists Cato Guldberg and Peter Waage first presented the law of mass action in 1864, in a lecture to the Academy of Sciences in Christiania (Oslo). The law relates the equilibrium state of a reaction to the amounts of the reacting substances, and it led to the equilibrium constant K. J. Willard Gibbs later connected chemical equilibrium to free energy in On the Equilibrium of Heterogeneous Substances (1876 and 1878), which gave Q its place in thermodynamics: a reaction proceeds in the direction that lowers the system's free energy, and Q measures how far the system is from the free-energy minimum represented by K.

How to use the calculator

1. Choose the number of reactants (1 to 3) and products (1 to 3).
2. For each species, enter its coefficient from the balanced equation. Coefficients must be whole numbers of 1 or more.
3. Enter each species' concentration in mol/L. Concentrations must be zero or a positive number.
4. Read the value of Q, along with the formula and the substituted numbers, in the results section.

Frequently asked questions

What is the difference between Q and K? They are calculated with the same formula. Q can be found at any point in a reaction; K is the value of Q once the reaction reaches equilibrium at a given temperature.

Can Q be zero? Yes. If any product has a concentration of zero, the numerator of the Q expression is zero, so Q is zero.

Can Q be undefined? Yes. If any reactant has a concentration of zero, the denominator is zero, which makes Q undefined in strict mathematical terms. The calculator shows this case as ∞ and displays a warning.

Does temperature affect Q? Temperature does not enter the Q formula directly, but it changes K, and it can change the concentrations used in Q. Compare Q with a value of K measured at the same temperature.

Do solids and liquids appear in the Q expression? No. Pure solids and pure liquids are left out, because their concentrations stay constant during the reaction. Only dissolved and gaseous species are included.

Why are concentrations raised to a power? The power matches each species' coefficient in the balanced equation, following the law of mass action. In the reaction 2A → B, doubling [A] raises the reactant term by a factor of four (2²), reflecting that two moles of A react for every mole of B formed.

References

1. OpenStax. Chemistry 2e, section 13.2. Equilibrium Constants — Q from the balanced equation with coefficients as exponents; K ≡ Q at equilibrium; liquids and solids have relative concentration 1 and are omitted; Q < K proceeds forward, Q > K in reverse.
2. OpenStax. Chemistry 2e, section 17.4. Potential, Free Energy, and Equilibrium — the Nernst equation, E = E° − (RT/nF) ln Q.
3. OpenStax. Chemistry 2e, section 16.4. Free Energy — ΔG = ΔG° + RT ln Q.
4. Waage, P. and Guldberg, C. M., trans. H. I. Abrash. Studies Concerning Affinity (hosted by the University of Notre Dame). Studies Concerning Affinity (PDF) — Guldberg and Waage's first presentation of the law of mass action, in 1864, to the Academy of Sciences in Christiania.
5. MacTutor History of Mathematics, University of St Andrews. Josiah Willard Gibbs. Gibbs biography — On the Equilibrium of Heterogeneous Substances, published in two parts in 1876 and 1878.