Sequences & Series — Maths for Thermodynamics & Kinetics

Adding up patterns

Sequences, series and convergence

A sequence is an ordered list of numbers; a series is the result of adding them up. Two types dominate chemistry: arithmetic (constant difference) and geometric (constant ratio). The arithmetic sum of n terms is straightforward; the geometric series has the remarkable property that infinitely many shrinking terms can converge to a finite value.

S_{n}^{arith} = \frac{n}{2} (a_{1} + a_{n}), S_{\infty}^{geom} = \frac{a}{1 - r} (∣ r ∣ < 1)

The convergence condition $∣ r ∣ < 1$ is crucial: if $∣ r ∣ \geq 1$ the terms grow (or stay constant) and the sum diverges. When $∣ r ∣ < 1$ each new term is smaller than the last, and the series settles to a limit.

In statistical thermodynamics, energy levels are evenly spaced (for a harmonic oscillator) and the Boltzmann factors $e^{- n ϵ / k T}$ form a geometric series. Summing over all levels gives the vibrational partition function, one of the most important results in physical chemistry — and it depends entirely on the geometric series formula.

Arithmetic

Constant difference

d

. Sum of

n

terms:

\frac{n}{2} (a_{1} + a_{n})

Geometric

Constant ratio

r

. Sum to

\infty

\frac{a}{1 - r}

when

∣ r ∣ < 1

Convergence

A series converges if its partial sums approach a finite limit. For geometric: need

∣ r ∣ < 1

Partition function

q_{vib} = \sum_{n = 0}^{\infty} e^{- n ϵ / k T} = \frac{1}{1 - e ^{- ϵ / k T}}

Partition functions

The vibrational partition function is a geometric series in

e^{- h ν / k T}

that sums to

1/ (1 - e^{- h ν / k T})

. Energy-level ladders and many statistical-thermodynamics results are series. Open the geometric series lab.

To decide whether a series whose ratio is not obviously constant converges, use the ratio test: form $L = lim_{n \to \infty} ∣ a_{n + 1} / a_{n} ∣$ . If $L < 1$ the series converges absolutely; if $L > 1$ it diverges; if $L = 1$ the test is inconclusive. For the vibrational series the ratio is always $e^{- h ν / k T} < 1$ at any positive temperature, so convergence is guaranteed.

Once the partition function $q_{vib}$ is in hand, the average vibrational energy follows from a clean trick: differentiate $ln q$ with respect to $β = 1/ (k_{B} T)$ .

⟨ E_{vib} ⟩ = - \frac{\partial ln q _{vib}}{\partial β} = \frac{h ν}{e ^{h ν / k_{B} T} - 1}

Ratio test

L = lim_{n \to \infty} \frac{a _{n + 1}}{a _{n}}

. Converges if

L < 1

, diverges if

L > 1

Average vibrational energy

⟨ E ⟩ = \frac{h ν}{e ^{h ν / k_{B} T} - 1}

; tends to

k_{B} T

at high

T

Common pitfalls

Using $S_{\infty} = a / (1 - r)$ when $∣ r ∣ \geq 1$ . At high temperature $r = e^{- h ν / k_{B} T} \to 1$ and the denominator goes to zero — the formula diverges. The high- $T$ partition function must then be handled via the $1 - e^{- x} \approx x$ approximation.
Off-by-one on the first term. The vibrational ladder starts at $n = 0$ (not $n = 1$ ), so the first term is $a = e^{0} = 1$ , not $a = e^{- h ν / k_{B} T}$ . Getting $a$ wrong shifts the entire result.
Confusing the nth term $a_{n} = a r^{n - 1}$ with the partial sum $S_{n} = a (1 - r^{n}) / (1 - r)$ . In a convergence argument always ask: which one are you working with?
Applying an arithmetic-series formula to a geometric problem (or vice versa). Check: does the sequence have a constant difference (arithmetic) or a constant ratio (geometric)?

Going deeper

The vibrational partition function is for a single harmonic mode. A non-linear polyatomic molecule with $N$ atoms has $3 N - 6$ vibrational modes, each contributing its own geometric series. The total vibrational partition function is the product of all these individual $q_{i}$ , because the modes are (to a first approximation) independent and $ln q_{total} = \sum_{i} ln q_{i}$ . This additivity of logarithms is why chemists prefer to work with $ln q$ rather than $q$ itself.

Geometric series

Adjust the common ratio r and number of terms — watch the partial sum converge to a/(1−r) as more terms are added.

Watch the shrinking terms add up toward a finite total

1.00

0.50

0.25

0.13

0.06

0.03

k = 0 \sum 5 1 \cdot 0. 5^{k}

common ratio r0.50

number of terms n6

partial sum (n terms)1.9688

sum to infinity a/(1−r)2.0000

Because r < 1 the terms shrink geometrically, so even infinitely many of them add to a finite number — exactly what makes a partition-function sum tractable.

Worked example 1Vibrational partition function for HCl

The vibrational frequency of HCl is $ν = 8.97 \times 1 0^{13} Hz$ . At $T = 298 K$ , calculate the vibrational partition function $q_{vib} = 1/ (1 - e^{- h ν / k T})$ . (Given: $h = 6.626 \times 1 0^{- 34} J s$ , $k_{B} = 1.381 \times 1 0^{- 23} J K^{- 1}$ .)

Worked example 2Energy of a geometric energy-level ladder

For a harmonic oscillator with equally spaced levels $ϵ = k_{B} T \cdot 0.80$ (so $x = 0.80$ ), find the mean vibrational energy using $⟨ E ⟩ = \frac{x ϵ}{e ^{x} - 1}$ and confirm the series ratio $r = e^{- x}$ is less than 1.

Worked examples

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

Worked example 3Arithmetic series — energy levels in a particle-in-a-box

The energy levels of a particle in a 1-D box are $E_{n} = n^{2} E_{1}$ where $E_{1} = 6.02 \times 1 0^{- 21} J$ . These are not arithmetic in $n$ but the gaps $Δ E_{n} = E_{n + 1} - E_{n} = (2 n + 1) E_{1}$ form an arithmetic sequence. Find the sum of the first five energy gaps (i.e., from gap 1 to gap 5).

Worked example 4Convergence check — rotational partition function (high T limit)

The rotational partition function for a linear molecule at high temperature is approximated as a sum: $q_{rot} \approx \sum_{J = 0}^{\infty} (2 J + 1) e^{- J (J + 1) θ_{r} / T}$ . For CO at $T = 300 K$ the rotational temperature $θ_{r} = 2.77 K$ . Show that the ratio of the $J = 1$ term to the $J = 0$ term is less than 1, confirming the series will converge.

Worked example 5Nuclear spin partition function

For a nucleus with spin $I$ , the nuclear spin partition function is the sum of $(2 I + 1)$ equal Boltzmann factors (all with energy $\approx 0$ at chemical temperatures). Write the series and evaluate it for $^{1} H$ ( $I = \frac{1}{2}$ ) and $^{14} N$ ( $I = 1$ ).

Worked example 6Ratio test for convergence of a partition-function term

Apply the ratio test to the vibrational series $\sum_{n = 0}^{\infty} e^{- nh ν / k_{B} T}$ for HF at $T = 500 K$ . The fundamental vibrational wavenumber of HF is $\tilde{ν} = 4138 c m^{- 1}$ . Show that the series converges and find the limiting ratio $L$ .

Worked example 7Arithmetic series: total energy input in a stepped heating protocol

A calorimetry protocol applies pulses of heat in an arithmetic sequence: the first pulse delivers $q_{1} = 50 J$ , each subsequent pulse delivers $d = 15 J$ more than the last. Find the total energy input after $n = 8$ pulses and the energy of the 8th pulse.

Worked example 8Average vibrational energy from the partition function

Starting from $q_{vib} = (1 - e^{- x})^{- 1}$ with $x = h ν / k_{B} T$ , derive $⟨ E_{vib} ⟩$ by differentiating $ln q_{vib}$ with respect to $β = 1/ (k_{B} T)$ . Express the result in terms of $h ν$ and evaluate for CO at $T = 1000 K$ ( $\tilde{ν}_{CO} = 2143 c m^{- 1}$ ).

ChallengeChallenge — derive ⟨E_vib⟩ for a diatomic and show the Einstein solid limit

For a mole of a diatomic solid with vibrational frequency $ν$ , the molar vibrational energy is $U_{vib} = N_{A} ⟨ E ⟩ + \frac{1}{2} N_{A} h ν$ (the zero-point contribution is constant). Using $⟨ E ⟩ = h ν / (e^{h ν / k_{B} T} - 1)$ , show that the molar heat capacity $C_{V} = d U / d T$ reduces to the Dulong–Petit limit $C_{V} \to R$ per mode when $T ≫ θ_{E} = h ν / k_{B}$ .

Check yourself

Four questions linking sequences, series, convergence and the partition function.

Question 1 of 4 · Score 0

In a geometric series each term is the previous one multiplied by a constant ratio r. The sum to infinity (for |r| < 1) is:

S_{\infty} = \frac{a}{1 - r}

Choose an answer.