Taylor & Maclaurin Series — Maths for Thermodynamics & Kinetics

Any function as a polynomial

The engine behind every approximation in physical chemistry

A Taylor (Maclaurin) series rebuilds a function near a point from its derivatives there, as an infinite polynomial. For a function centred on $x = 0$ (a Maclaurin series):

f (x) = f (0) + f^{'} (0) x + \frac{f ^{''} ( 0 )}{2 !} x^{2} + \frac{f ^{'''} ( 0 )}{3 !} x^{3} + \dots = k = 0 \sum \infty \frac{f ^{(k)} ( 0 )}{k !} x^{k}

Keep just the first few terms and you have a simple, accurate approximation near that point. The radius of convergence tells you how far from $x = 0$ the polynomial stays reliable. For $e^{x}$ and $sin x$ this is infinite; for $ln (1 + x)$ it is only $∣ x ∣ \leq 1$ .

The most useful truncations — the ones you will use almost every week — are $e^{x} \approx 1 + x$ (linear), $ln (1 + x) \approx x$ (linear), and $sin x \approx x$ (valid when $x$ is in radians and small). Each one turns a transcendental function into a straight line near zero.

e^{x}

1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots

; for small

x

e^{x} \approx 1 + x

ln (1 + x)

x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots

; for small

x

ln (1 + x) \approx x

sin x

x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots

; for small

x

sin x \approx x

Harmonic oscillator

Expand

V (r)

near

r_{0}

: linear term vanishes (minimum), quadratic term gives

V \approx \frac{1}{2} k_{e} (r - r_{0})^{2}

(1 + x)^{n}

1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots

; for small

x

\approx 1 + n x

(binomial series).

Taylor about a general point

f (x) \approx f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \dots

. Maclaurin is the special case

a = 0

Remainder / error

The Lagrange remainder after

N

terms is

R_{N} = \frac{f ^{(N + 1)} ( c )}{( N + 1 )!} x^{N + 1}

for some

c \in (0, x)

. It bounds the truncation error.

Virial equation

Z = 1 + B / V_{m} + C / V_{m}^{2} + \dots

— a Taylor series in

1/ V_{m}

that converges for large

V_{m}

(low pressure).

‘For small x…’

e^{x} \approx 1 + x

ln (1 + x) \approx x

, and the harmonic approximation of a bond's potential well are all truncated Taylor series. They turn intractable expressions into easy ones. Open the series approximator.

Common pitfalls

Expanding about the wrong point. Always confirm the expansion point before substituting. For the harmonic approximation of a bond potential expand about $r = r_{e}$ , not about $r = 0$ — the latter gives a divergent or meaningless result.
Forgetting the factorials. The $n$ th term is $f^{(n)} (0) x^{n} / n!$ , not $f^{(n)} (0) x^{n}$ . Dropping $n!$ inflates higher terms by large factors.
Using too few terms for large x. Near $x = 0$ one or two terms suffice; for $x = 1$ or larger the polynomial diverges from the true function rapidly. Always check the truncation error before trusting an approximation.
Mixing the linear approximation with the wrong sign. $ln (1 - x) \approx - x$ (negative), whereas $ln (1 + x) \approx + x$ (positive). Confusing these flips the sign of entropy or free-energy corrections.

Going deeper

The Lennard-Jones 12-6 potential $V (r) = 4 ε [(σ / r)^{12} - (σ / r)^{6}]$ has its minimum at $r_{0} = 2^{1/6} σ$ . Expanding to second order about that minimum gives the same parabolic form $V \approx \frac{1}{2} k (r - r_{0})^{2}$ as the Morse potential — confirming that the local curvature of any smooth well looks like a harmonic oscillator, regardless of the long-range form. The force constant $k = V^{''} (r_{0})$ is read directly from the second derivative — the worked Challenge example below derives it explicitly.

Series approximator

Switch between eˣ, sin x and cos x — add terms one by one and watch the polynomial converge to the true function. The dashed line is the approximation; solid is the truth.

Function:

true function 3-term polynomial

e^{x} = \sum \frac{x ^{k}}{k !}

number of terms3

With one term the polynomial only matches at x = 0; add more and it hugs the true curve over a wider range. Near x = 0 even a 1–2 term approximation is excellent — which is why “for small x” shortcuts work.

Worked example 1Error of the linear approximation for the Boltzmann factor

In a reaction at $T = 298 K$ a perturbation shifts the energy by $Δ E = 0.50 kJ mo l^{- 1}$ . Compute $x = Δ E / RT$ and compare the exact factor $e^{- x}$ with the linearised approximation $1 - x$ .

Worked example 2Truncating the virial equation of state

The virial equation $Z = p V_{m} / RT = 1 + B / V_{m} + C / V_{m}^{2} + \dots$ is a Taylor expansion in $1/ V_{m}$ . For nitrogen at 300 K and 10 atm, $V_{m} = 2.462 L mo l^{- 1}$ and $B = - 4.2 \times 1 0^{- 2} L mo l^{- 1}$ . Estimate $Z$ keeping only the first correction term, and compare with the ideal-gas value.

Worked examples

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

Worked example 3Harmonic approximation of a Morse potential

The Morse potential is $V (r) = D_{e} (1 - e^{- a (r - r_{e})})^{2}$ . Expand $e^{- a (r - r_{e})}$ to second order in $x = r - r_{e}$ and show that near $r_{e}$ the potential becomes parabolic with force constant $k_{e} = 2 D_{e} a^{2}$ .

Worked example 4Low-field expansion of the Langevin function

The Langevin function $L (x) = coth x - 1/ x$ appears in the theory of electric polarisability. Show that for small $x$ (weak field), $L (x) \approx x /3$ by expanding $coth x$ as a series.

Worked example 5pH of a buffer: linearising the Henderson–Hasselbalch equation

For an acetic acid / acetate buffer ( $p K_{a} = 4.76$ ) with $[A^{-}] / [HA] = 1.00$ at pH 4.76, a small amount of strong acid is added so the ratio changes to $0.90$ . Using $pH = p K_{a} + lo g (r)$ and $ln (1 + x) \approx x$ (with $r = 1 + δ$ ), estimate the pH drop.

Worked example 6Maclaurin expansion of sin x to three non-zero terms

Write out the Maclaurin series for $sin x$ to three non-zero terms, and estimate the error when $x = 0.50 rad$ (a typical bending coordinate in vibrational spectroscopy) by comparing the one-term and three-term approximations with the exact value.

Worked example 7Expanding e^x and ln(1+x) to estimate the change in Kₐ with temperature

For a weak acid, $ln K_{a} (T) = ln K_{a} (T_{0}) - \frac{Δ H ^{\circ}}{R} (\frac{1}{T} - \frac{1}{T _{0}})$ . If $T = T_{0} + δ T$ with $δ T ≪ T_{0}$ , expand $1/ T$ to first order in $δ T / T_{0}$ and show that $ln K_{a} (T) \approx ln K_{a} (T_{0}) + \frac{Δ H ^{\circ} δ T}{R T _{0}^{2}}$ . Evaluate for acetic acid at 298 K with a 10 K rise and $Δ H^{\circ} = - 0.41 kJ mo l^{- 1}$ .

ChallengeChallenge — expand the Lennard-Jones potential to get the force constant k

The Lennard-Jones potential is $V (r) = 4 ε [(\frac{σ}{r})^{12} - (\frac{σ}{r})^{6}]$ . It has a minimum at $r_{0} = 2^{1/6} σ$ where $V (r_{0}) = - ε$ . By expanding $V (r)$ to second order about $r_{0}$ , find the effective harmonic force constant $k = V^{''} (r_{0})$ in terms of $ε$ and $r_{0}$ .

Check yourself

Four questions on Maclaurin expansions and their chemistry applications.

Question 1 of 4 · Score 0

The Maclaurin series for eˣ is:

e^{x} = k = 0 \sum \infty \frac{x ^{k}}{k !}

Choose an answer.