Finding the quantities at equilibrium

10 minAS Level7.1.8

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can set up a table of initial, change and equilibrium amounts, solve for a single unknown using $K_c$ or $K_p$ , and check your answer by substituting it back.

The question runs backwards

Until now you were given the equilibrium mixture and asked for the constant. This lesson reverses the direction: you are given the constant and asked for the mixture.

The method is still one unknown at a time. Call the amount that reacts $x$ , express everything else in terms of $x$ , then substitute.

The three-row table

For a reaction $a\text{A} \rightleftharpoons c\text{C}$ , write three rows:

	A	C
initial	$n_0$	$0$
change	$-ax$	$+cx$
equilibrium	$n_0 - ax$	$cx$

The change row always follows the coefficients. If 2 mol of a substance react, the change is $-2x$ , not $-x$ .

Then divide the equilibrium row by the volume (for $K_c$ ) or by the total moles (for $K_p$ ).

Worked example: Worked example: hydrogen and iodine

Hydrogen and iodine are mixed in a $1.00\ \text{dm}^3$ flask at $700\ \text{K}$ , $1.00\ \text{mol}$ of each:

$\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g) \qquad K_c = 64$

Calculate the equilibrium concentration of each gas.

Have a go first?

Recognising the friendly case

Cambridge questions are built so the algebra collapses. Two patterns cover nearly all of them.

Pattern 1 — a perfect square. If the expression is a ratio of two squares, such as

$K_c = \frac{x^2}{(c - x)^2}$

take the square root of both sides first. That turns it into a linear equation.

Pattern 2 — a single unknown on one side. If the expression is linear in $x$ after multiplying out, no quadratic appears at all.

If you find yourself facing a genuine quadratic, stop and re-read the question: something has been transcribed wrongly.

Worked example: Worked example: a perfect square

Hydrogen iodide decomposes in a $1.00\ \text{dm}^3$ vessel:

$2\text{HI}(g) \rightleftharpoons \text{H}_2(g) + \text{I}_2(g) \qquad K_c = 0.0200$

Initially the vessel contains $1.00\ \text{mol}$ of $\text{HI}$ . Calculate the equilibrium concentration of $\text{H}_2$ .

Have a go first?

Worked example: Worked example: a $K_p$ composition

Hydrogen and iodine are mixed in a 1:1 mole ratio and reach equilibrium at a total pressure of $2.0\ \text{atm}$ :

$\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g) \qquad K_p = 49$

Calculate the mole fraction of $\text{HI}$ at equilibrium.

Have a go first?

Check, then check again

Substitute your answer back into the expression — it takes ten seconds and catches nearly every slip.

Then sanity-check the chemistry: a large $K$ should give mostly products, and a small $K$ should leave most of the reactant unconverted. If the numbers say otherwise, the algebra is wrong.

Common mistakes

The change in amount is always $-x$ , whatever the coefficient.

The change follows the coefficients: for $2\text{HI}$ reacting, the change is $-2x$ .

The total number of moles always stays the same.

It stays the same only when the gas-molecule counts match. In the Haber equilibrium, 4 mol becomes 2 mol.

In the exam

Define $x$ in words before using it: "let $x$ mol of $\text{H}_2$ react".
Show the three-row table; it is quick and earns method marks.
Take the square root when both sides are perfect squares, and say that you are doing so.
Reject a negative root explicitly.
Finish with a substitution check and give the amount or concentration the question asked for — not $x$ itself.

Checking the idea

Multiple choiceQuestion 1 · Apply

Answer the 3 questions above to continue.