Binomial & Small-x Approximations — Maths for General Chemistry

The shortcut that powers approximations

(1 + x)ⁿ ≈ 1 + nx

The full binomial expansion is $(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \frac{n ( n - 1 ) ( n - 2 )}{3 !} x^{3} + \dots$ . When $∣ x ∣ ≪ 1$ , the $x^{2}$ and higher terms are negligible — they are smaller by a factor of at least $∣ x ∣$ each time — leaving the simple linear approximation:

(1 + x)^{n} \approx 1 + n x (∣ x ∣ ≪ 1)

It works for any power $n$ — positive, negative or fractional — making roots and reciprocals easy to estimate without a calculator. The key question is always: is $x$ truly much less than 1 for the system at hand?

In chemistry the approximation appears most often when the extent of reaction or dissociation is small compared with the initial concentration, turning a quadratic equilibrium expression into a trivial linear one. But you must always verify the approximation afterwards by checking that the neglected term is indeed small.

First-order binomial

(1 + x)^{n} \approx 1 + n x

— valid when

∣ x ∣ ≪ 1

Square root

1 + x = (1 + x)^{1/2} \approx 1 + x /2

Reciprocal

\frac{1}{1 + x} = (1 + x)^{- 1} \approx 1 - x

Validity rule

After using the approximation, verify

x < 5%

of the largest quantity in the expression.

Where chemists lean on it

The weak-acid “small-x” approximation, the Debye–Hückel limiting law, and quick estimates of roots all use it. It is a special case of the Taylor series. Open the approximation lab and watch the error grow with x.

A useful way to quantify when the approximation is safe: the next term in the full expansion is $\frac{n ( n - 1 )}{2} x^{2}$ . If this is less than 1 % of the leading term $n x$ , i.e. $\frac{∣ n - 1∣}{2} ∣ x ∣ < 0.01$ , then the one-term approximation is accurate to 1 %. In weak-acid problems this translates to the familiar 5 % rule: if $x / C_{0} < 0.05$ the linearised expression is adequate.

Debye–Hückel basis

ln γ_{\pm} = - A ∣ z_{+} z_{-} ∣ I

. Valid when

I ≲ 0.01 mol k g^{- 1}

— a first-order truncation of the extended law.

Negative exponent

(1 + x)^{- 2} \approx 1 - 2 x

, useful for pressure-dependent partition functions and molar volume reciprocals.

Common pitfalls

Applying it when |x| is not ≪ 1. For a 5 % solution ( $x \approx 0.05$ ) the error in $(1 + x)^{2}$ is about 0.25 % — fine. But at 50 % dissociation the error is 25 % — catastrophic. Always quote the error or percentage before accepting the approximation.
Dropping the nx term entirely. The approximation is $(1 + x)^{n} \approx 1 + n x$ , not simply 1. Forgetting the correction shifts a calculated equilibrium constant by a factor of $n x$ .
Sign of n for reciprocals. $(1 + x)^{- 1} \approx 1 - x$ (n = −1, so the sign flips). Mistaking n = +1 here reverses the direction of a pressure or concentration correction.
Not verifying after solving. Always substitute the answer back and check that the discarded higher terms really are small. If the 5 % threshold is exceeded, go back and solve the full quadratic (or cubic).

Approximation lab

Drag the exponent n and the test point x — the live error readout shows when the straight-line approximation breaks down.

The straight line 1 + nx hugs the curve only near x = 0

(1+x)ⁿ 1 + nx

(1 + x)^{n} \approx 1 + n x

power n2.0

evaluate at x0.30

exact (1+x)ⁿ1.690

approx 1+nx1.600

error5.3%

Inside the green band (small x) the straight line and the curve are indistinguishable. Push x out and the error grows — exactly why the equilibrium “small-x” shortcut must be checked.

Worked example 1Weak acid: small-x approximation

Ethanoic acid ( $K_{a} = 1.8 \times 1 0^{- 5}$ ) at concentration $C_{0} = 0.10 mol L^{- 1}$ . Using the small-x approximation, find $[H^{+}]$ and verify the approximation is valid.

Worked example 2Debye–Hückel limiting law

The mean activity coefficient of a 1:1 electrolyte at ionic strength $I = 0.010 mol k g^{- 1}$ is given by the DHLL: $lo g γ_{\pm} = - A ∣ z_{+} z_{-} ∣ I$ with $A = 0.509$ (water, 298 K). Calculate $γ_{\pm}$ .

Worked examples

Pull the ideas together — try each problem before revealing the full solution.

Worked example 3Pressure correction for a real gas

The pressure of a van der Waals gas is $p = \frac{n RT}{V - nb} - \frac{a n ^{2}}{V ^{2}}$ . For one mole of CO₂ at 300 K in a 10 L vessel ( $a = 3.64 L^{2} atm mo l^{- 2}$ , $b = 4.27 \times 1 0^{- 2} L mo l^{- 1}$ ), use $(1 - b / V)^{- 1} \approx 1 + b / V$ to estimate the pressure to first order and compare with the ideal-gas value.

Worked example 4Osmotic pressure: dilute-solution approximation

The osmotic pressure is $Π = - (RT / V_{m}^{*}) ln x_{1}$ . For a dilute solution, $x_{1} = 1 - x_{2}$ and $ln (1 - x_{2}) \approx - x_{2}$ . If $x_{2} = n_{2} / (n_{1} + n_{2}) \approx n_{2} / n_{1}$ for small solute mole fraction, show that $Π \approx n_{2} RT / V$ (van’t Hoff equation).

Worked example 5Validity check: concentrated weak acid

Repeat the acetic acid calculation from Example 1 but with $C_{0} = 5.0 \times 1 0^{- 4} mol L^{- 1}$ . Should the small-x approximation be trusted?

Worked example 6Relativistic mass correction: binomial to first order

The relativistic mass of an electron is $m = m_{e} / 1 - (v / c)^{2} = (1 - (v / c)^{2})^{- 1/2} m_{e}$ . For an electron in a 1s orbital of a heavy atom where $v / c = 0.58$ (gold, Z = 79), compare the first-order binomial estimate of $m / m_{e}$ with the exact relativistic value.

Worked example 7Debye–Hückel activity coefficient at moderate ionic strength

Calculate the mean activity coefficient $γ_{\pm}$ of CaCl₂ at $m = 0.050 mol k g^{- 1}$ using the DHLL: $lo g_{10} γ_{\pm} = - A ∣ z_{+} z_{-} ∣ I$ , with $A = 0.509$ , $z_{+} = + 2$ , $z_{-} = - 1$ , and ionic strength $I = \frac{1}{2} \sum c_{i} z_{i}^{2}$ . Then use the binomial approximation on the antilog to estimate $γ_{\pm}$ .

ChallengeChallenge — find the Ka and C threshold where the 5% rule breaks

For a weak acid HA with $K_{a}$ and initial concentration $C_{0}$ , the small-x approximation gives $[H^{+}] \approx K_{a} C_{0}$ . Show that the approximation is valid when $K_{a} / C_{0} < 0.05$ , and find the exact crossover concentration $C_{0}^{*}$ for chloroacetic acid ( $K_{a} = 1.36 \times 1 0^{- 3}$ ).

Check yourself

Four questions on the binomial approximation and its chemistry applications.

Question 1 of 4 · Score 0

For small x, the binomial approximation says (1 + x)ⁿ ≈

(1 + x)^{n} \approx 1 + n x

Choose an answer.