Class 03 · Ion Association & the Bjerrum Equation

00 Overview

Class 2 asked how many solvent molecules travel with an ion. Class 3 — notebook pages 27–33, lecture of 30 Aug 2026 — asks the next question: when do two oppositely charged ions stop being separate ions at all? The answer is Bjerrum's: below a characteristic distance the electrostatic attraction outweighs thermal motion, and the pair counts as associated.

One distance, one integral, one constant

The distance (Sections 3–6): $r_B$ , where pair energy equals thermal energy $k_BT$ . The integral (Sections 7–8): the association constant $K_A$ , built by weighting every shell $4\pi r^2 dr$ between closest approach $a$ and $r_B$ with the Boltzmann factor. The constant (Sections 9–10): $K_A = \alpha / [c(1-\alpha)^2]$ , tied directly to what a conductivity meter reads through $\Lambda_c/\Lambda_c^{\circ} = 1-\alpha$ .

Sections 1–2 set up the ion-pair equilibrium and Bjerrum's five assumptions.
Sections 3–6 derive $U(r)$ , the Boltzmann factor, the Bjerrum distance and the association region $a \le r \le r_B$ .
Sections 7–8 derive $K_A$ and its dimensionless form.
Sections 9–10 define the associated fraction $\alpha$ and connect it to $K_A$ and conductivity.
Sections 11–16 live calculator, equation sheet, symbol table, rapid revision, raw scans, PYQ audit with the lecture's four solved problems.

01 The ion-pair equilibrium

For an electrolyte that dissociates into a cation and an anion,

\mathrm{A}^{z+}\mathrm{B}^{z-} \rightleftharpoons \mathrm{A}^{z+} + \mathrm{B}^{z-} \qquad (\text{ion pair} \rightleftharpoons \text{free ions})

the reverse process — ion-pair formation — is always competing with dissociation. In solution some ions remain free and some form ion pairs: oppositely charged ions held together not by a chemical bond but by plain Coulomb attraction, close enough that thermal motion cannot separate them.

Ion pair, in one line

An ion pair is a cation–anion pair whose separation is small enough that electrostatic attraction dominates thermal agitation — so the pair moves and conducts as one unit instead of two.

02 Basic assumptions of Bjerrum's theory

Ions are treated as charged particles.
The solvent is treated as a continuous dielectric medium (the same continuum idea as Born, Class 1).
The interaction between oppositely charged ions is mainly electrostatic.
At sufficiently small separation the ions are considered associated.
At larger separation they are considered free ions.

Notice what is missing: no shells, no solvent molecules, no covalency — pure point charges in a dielectric. That is why the whole theory collapses into one distance and one integral.

03 Electrostatic potential energy between two ions

Consider two ions $\mathrm{A}^{z+}$ and $\mathrm{B}^{z-}$ with charges $q_1 = z_+ e$ and $q_2 = -z_- e$ , at separation $r$ . By Coulomb's law the electrostatic potential energy is

U(r) = \frac{q_1 q_2}{4\pi\varepsilon_0\,\varepsilon_r\, r}

Since the charges are opposite, $q_1 q_2 = -z_+ z_- e^2$ , and therefore

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U(r) = -\frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, r}

The negative sign means the interaction is attractive. Two features of this curve do all the work later: it grows without bound as $r \to 0$ , and it decays like $1/r$ , so at large $r$ it becomes negligible compared with thermal energy.

Fig 1. The pair potential |U(r)| against separation: it crosses the thermal energy k(B)T exactly at the Bjerrum distance; the green band is the association region (Section 5).

04 The Boltzmann factor

At temperature $T$ , the probability of finding the two ions at separation $r$ is related to the Boltzmann factor:

\exp\!\left(-\frac{U(r)}{k_B T}\right) = \exp\!\left(\frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T\, r}\right)

This factor becomes very large when oppositely charged ions come close together — the statistical weight of close approach is enormous, which is exactly why pairs form at all.

05 The Bjerrum distance

Bjerrum introduced a characteristic distance at which the electrostatic interaction energy becomes comparable to thermal energy. Putting $U(r) = k_B T$ :

\frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, r} = k_B T \quad\Longrightarrow\quad r = \frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T}

This distance is called the Bjerrum distance or Bjerrum length:

r_B = \frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T} \qquad \xrightarrow{\;1:1\; \text{electrolyte},\ z_+ = z_- = 1\;} \qquad r_B = \frac{e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T}

Physical meaning

$r < r_B$ : electrostatic attraction is strong compared with thermal agitation → the ions stay together (association, ion pair).
$r > r_B$ : thermal motion becomes relatively more important → the ions behave as free ions (dissociation).

This comparison — attraction versus $k_BT$ — is the physical basis for defining an ion pair.

Fig 2. Charge ladder: |z+ z−| = 1, 2, 4 for 1:1, 2:1, 2:2 electrolytes — the relative electrostatic attraction, and why MgSO₄ is the classic ion-pair former (Prob. 4).

visual lab 7

How wide is the pairing window?

The green band is the association region $a \le r \le r_B$ drawn to scale. Cool the solution, drop the dielectric constant, or raise the charge product — and watch the window open.

Temperature — 298 K

Dielectric ε(r) — 78.5

Charge product |z+ z−|

Contact a — 3.0 Å

r(B) = 7.14 Å pairing window = 4.14 Å r(B) ∝ 1/T, 1/ε(r), |z+ z−|

Sanity value: water (ε = 78.5) at 298 K, 1:1 electrolyte → r(B) ≈ 7.1 Å, the textbook number. Wider window = more separations count as associated = larger K(A) (Sections 7–8).

06 The association region: from a to r_B

Let the closest possible distance between the centres of the two ions be $a$ — the contact distance,

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a \approx r_+ + r_-

Bjerrum considers ions within a suitable association region as associated:

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a \le r \le r_B \quad \Rightarrow \quad \text{associated ion pair}

while ions at larger separations are treated as free. Separations below $a$ are simply not possible — the ion cores cannot interpenetrate.

Separation $r$	Status	Why
$r < a$	not possible	closest approach is the contact distance $a \approx r_+ + r_-$
$a \le r \le r_B$	ion pair (associated)	electrostatic attraction > thermal energy
$r > r_B$	free ions	thermal motion dominates

Fig 3. Bjerrum’s rule on one number line: below a impossible, between a and r(B) an ion pair, beyond r(B) free ions (Section 6).

visual lab 8

Pair or free? Slide the separation

One cation, one anion, water at your chosen temperature. Inside $r_B$ the Coulomb grip wins and the pair jiggles together; outside it, thermal motion pulls them apart into free ions.

Separation r — 5.0 Å

Temperature — 298 K

Dielectric ε(r) — 78.5

status: associated (ion pair) U(r)/k(B)T = -1.43 Boltzmann weight e^(rB/r) = 4.2

|U| ≫ k(B)T (ratio far below −1) means the pair survives thermal kicks; near 0 the ions wander free. Separations below the contact distance a are clamped — the cores cannot interpenetrate (Section 6).

07 Derivation of the Bjerrum association constant

Count the configurations

The number of possible configurations of two ions at separation $r$ is proportional to the spherical volume element:

dV = 4\pi r^2\, dr

Weight them electrostatically

The electrostatic Boltzmann factor is $\exp(r_B/r)$ , because $\frac{z_+ z_- e^2}{4\pi\varepsilon_0 \varepsilon_r k_B T} = r_B$ . So the effective weighted shell is

dW_{\text{effective}} = 4\pi r^2 \exp\!\left(\frac{r_B}{r}\right) dr

Integrate over the association region

The association constant $K_A$ is proportional to the volume available to the associated ions; therefore

K_A = 4\pi \int_{a}^{r_B} r^2 \exp\!\left(\frac{r_B}{r}\right) dr = 4\pi \int_{a}^{r_B} r^2 \exp\!\left(\frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T\, r}\right) dr

This is the Bjerrum equation for ion association — the basic Bjerrum expression.

08 Dimensionless form of the Bjerrum equation

Put $x = r/r_B$ , so $r = r_B x$ and $dr = r_B\,dx$ ; then $r^2 dr = r_B^3 x^2 dx$ and $r_B/r = 1/x$ . The limits become $r = a \Rightarrow x = a/r_B$ and $r = r_B \Rightarrow x = 1$ :

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K_A = 4\pi r_B^3 \int_{a/r_B}^{1} x^2\, e^{1/x}\, dx

This is a useful dimensionless form of the Bjerrum equation: all the physics of temperature, dielectric constant and charge sits inside $r_B^3$ , while the integral is a pure number once $a/r_B$ is fixed.

Fig 4. The dimensionless integrand x² e^(1/x): the Boltzmann factor dominates near contact, the spherical volume near r(B); the shaded area times 4π r(B)³ is K(A) (Section 8).

09 Fraction of ions associated, α

When an electrolyte AB dissociates in a solvent, some ions remain free and some form ion pairs:

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\mathrm{A}^{+} + \mathrm{B}^{-} \rightleftharpoons \mathrm{AB}\ (\text{ion pair})

At equilibrium, with $c$ = molar concentration and $\alpha$ = fraction of ions associated:

	$\mathrm{A}^{+}$	$\mathrm{B}^{-}$	$\mathrm{AB}$ (ion pair)
Initially (mol)	$c$	$c$	$0$
After equilibrium (mol)	$c(1-\alpha)$	$c(1-\alpha)$	$c\alpha$

The fraction associated is the moles associated over the total moles initially taken:

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\alpha = \frac{\text{moles associated}}{\text{total moles initially taken}} = \frac{c\alpha}{c} = \alpha \qquad 0 \le \alpha \le 1

$\alpha = 0$ : no association — every ion free.
$\alpha \to 1$ : complete association — essentially all ions paired.

The complementary quantity is the degree of dissociation:

\alpha_d = 1 - \alpha

$\alpha$ counts ion pairing; $\alpha_d$ counts free ions. Keep the two labels straight — exams love swapping them.

Fig 5. Solution snapshot: free ions wander, associated pairs move as single units; α is the paired share of the pool (Section 9).

10 α ↔ K(A), and the connection with conductivity

For $\mathrm{A}^{+} + \mathrm{B}^{-} \rightleftharpoons \mathrm{AB}$ the association constant is

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K_A = \frac{[\mathrm{AB}]}{[\mathrm{A}^{+}][\mathrm{B}^{-}]} = \frac{c\alpha}{c(1-\alpha) \times c(1-\alpha)}

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K_A = \frac{\alpha}{c\,(1-\alpha)^2} \qquad K_A = \text{ion association constant (mol}^{-1}\,\text{L)}

Connection with conductivity

For a weakly associated 1:1 electrolyte, only free ions carry current, so

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\frac{\Lambda_c}{\Lambda_c^{\circ}} = 1 - \alpha \quad\Longrightarrow\quad \alpha = 1 - \frac{\Lambda_c}{\Lambda_c^{\circ}}

where $\Lambda_c$ = molar conductivity at concentration $c$ and $\Lambda_c^{\circ}$ = limiting molar conductivity. As ion association increases, free ions decrease, and $\Lambda_c$ decreases — the conductivity meter is literally counting the ions that escaped pairing.

Fig 6. Molar conductivity of an associating electrolyte: Λ falls below Λ°(c) exactly by the paired fraction, Λ(c)/Λ°(c) = 1 − α (Section 10).

visual lab 9

Association fraction ↔ what the meter reads

Set the association constant and the concentration: the exact quadratic gives α, and the ion pool below re-partitions into free ions and contact pairs. The conductivity ratio is simply Λ(c)/Λ°(c) = 1 − α.

K(A) — 60 L/mol

Concentration c — 0.050 mol/L

α (associated) = 0.566 α(d) = 1 − α = 0.434 Λ(c)/Λ°(c) = 0.434

Dilution check: send c → 0 and α → 0 (pairs dissolve, Λ → Λ°). Concentrate or raise K(A) and the pool pairs up while the conductivity ratio falls — exactly the trend of Section 10.

11 Play with the Bjerrum equation

The whole lecture in one panel: Bjerrum distance, the dimensionless integral (numerically integrated live), the association constant, and the paired fraction at your concentration.

Charge product |z+ z−|

Solvent ε(r) — 78.5

Temperature — 298 K

Closest approach a (Å) — 4.0

Concentration c (mol/L) — 0.050

Bjerrum distance r(B)

7.14 Å

a / r(B) (integral lower limit)

0.560

K(A) = 4π r(B)³ ∫ x² e^(1/x) dx

2.70 L/mol

α at c = 0.050 M

0.108

Try water (ε = 78.5), 298 K, 1:1, a = 4.0 Å: r(B) ≈ 7.1 Å and K(A) lands in the single-digit L/mol — weak pairing. Switch to 2:2 at the same conditions and K(A) explodes by ~r(B)³ × integral (hundreds–thousands): that is MgSO₄ territory. Reduced ε (dioxane-like) does the same. The integral is Simpson-rule over x ∈ [a/r(B), 1]; α solves K(A) = α / [c(1−α)²] exactly.

12 The equation sheet, at a glance

Every boxed result of the lecture on one screen; labels match the derivation above.

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U(r) = -\frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, r}

r_B = \frac{z_+ z_-\, e^2}{4\pi\varepsilon_0\,\varepsilon_r\, k_B T}

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a \le r \le r_B \ \Rightarrow\ \text{ion pair}

K_A = 4\pi \int_{a}^{r_B} r^2 \exp\!\left(\frac{r_B}{r}\right) dr

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K_A = 4\pi r_B^3 \int_{a/r_B}^{1} x^2\, e^{1/x}\, dx

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K_A = \frac{\alpha}{c\,(1-\alpha)^2}

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\alpha = 1 - \frac{\Lambda_c}{\Lambda_c^{\circ}}

13 Symbol table

Symbol	Meaning	Typical unit
$z_+, z_-$	Charge numbers of cation and anion	dimensionless
$e$	Elementary charge	C
$q_1, q_2$	Ion charges, $+z_+e$ and $-z_-e$	C
$r$	Separation between ion centres	m, Å
$a$	Closest-approach (contact) distance, $a \approx r_+ + r_-$	m, Å
$r_B$	Bjerrum distance / length: $\|U\| = k_BT$	m, Å (≈ 7.1 Å in water, 298 K, 1:1)
$U(r)$	Electrostatic potential energy of the pair (negative = attractive)	J
$\varepsilon_0, \varepsilon_r$	Vacuum permittivity; relative permittivity of the solvent	F m⁻¹; dimensionless
$k_B$	Boltzmann constant	J K⁻¹
$T$	Absolute temperature	K
$K_A$	Bjerrum association constant	mol⁻¹ L
$x$	Dimensionless separation, $r/r_B$	dimensionless
$c$	Molar concentration of the electrolyte	mol L⁻¹
$\alpha$	Fraction of ions associated (ion pairing), $0 \le \alpha \le 1$	dimensionless
$\alpha_d$	Degree of dissociation (free-ion fraction), $1-\alpha$	dimensionless
$\Lambda_c, \Lambda_c^{\circ}$	Molar conductivity at $c$ ; limiting molar conductivity	S cm² mol⁻¹

14 Rapid revision — 12 lines before the exam

Ion pair = cation + anion held by Coulomb attraction alone; $\mathrm{A}^{z+}\mathrm{B}^{z-} \rightleftharpoons \mathrm{A}^{z+} + \mathrm{B}^{z-}$ .
Bjerrum assumptions: charged points, continuum dielectric, electrostatics only, small $r$ = associated, large $r$ = free.
$U(r) = -z_+z_-e^2/(4\pi\varepsilon_0\varepsilon_r r)$ ; negative sign = attractive.
Boltzmann weight $\exp(-U/k_BT) = \exp(r_B/r)$ → huge at close approach.
$r_B$ : where $|U| = k_BT$ ; $r_B = z_+z_-e^2/(4\pi\varepsilon_0\varepsilon_r k_BT)$ ; 1:1 → $e^2/(4\pi\varepsilon_0\varepsilon_r k_BT)$ ≈ 7.1 Å in water at 298 K.
$r < r_B$ : attraction wins (pair). $r > r_B$ : thermal motion wins (free).
Association region $a \le r \le r_B$ with $a \approx r_+ + r_-$ ; $r < a$ impossible.
$K_A = 4\pi \int_a^{r_B} r^2 e^{r_B/r} dr = 4\pi r_B^3 \int_{a/r_B}^{1} x^2 e^{1/x} dx$ (dimensionless form).
$\alpha$ = fraction associated ( $0 \le \alpha \le 1$ ); $\alpha_d = 1-\alpha$ = fraction dissociated.
$K_A = \alpha/[c(1-\alpha)^2]$ , units mol⁻¹ L; conductivity link $\Lambda_c/\Lambda_c^{\circ} = 1-\alpha$ .
Association ↑ when: $T$ ↓, $\varepsilon_r$ ↓, $|z_+z_-|$ ↑. Same three make $K_A$ ↑; dilution makes $K_A$ look ↓ in effect via $\alpha$ .
Charge ladder: 1:1 → $|z_+z_-| = 1$ ; 2:1 → 2; 2:2 → 4, so $|U_{2:2}| \approx 4|U_{1:1}|$ (MgSO₄ pairs hard).

Next up: Class 4 notes land here after the next lecture — the hub grows one page per class.

← All classes

15 Original notebook scans

Digitised from the class notebook of 30 Aug 2026 — notebook pages 27–33 (CamScanner spreads S1–S7, the fair copy) plus the in-class phone photos N1–N6; every equation above was cross-checked against these pages. Tap any thumbnail to open the full scan.

N1 · K(A) derivation + dimensionless form

N2 · meaning of rB, region diagram

N3 · K(A) = α/c(1−α)², conductivity link

N4 · U(r), Boltzmann factor, rB

N5 · boxed K(A), trends, α table

N6 · 30/08/26 header, assumptions, targets

S1 · p27 · assumptions, U(r)

S2 · p28 · Boltzmann factor, rB

S3 · p29 · meaning, region, derivation start

S4 · p30 · K(A) integral, dimensionless, region box

S5 · p31 · Prob 1–4, charge table

S6 · p32 · U vs r plot, α, ICE table

S7 · p33 · K(A)=α/c(1−α)², Λ link, summary

16 PYQ bank · University papers 2020–2024

Same audit as Classes 1–2: every page of the five M.Sc. Semester-I question papers (2020–2024 — all subjects MSCH-101…106, 60 scanned pages) OCR'd and verified by hand; the Physical Chemistry paper each year is MSCH-104 (Physical General-I). The rule is unchanged: only questions that actually appeared, tagged with year and repeat count.

Year	Physical paper (MSCH-104)	Pages verified	Class-3 questions (ion association / Bjerrum)
2020	Physical General I (new + old syllabus copies)	p. 12–15	0 — group theory, QM, stat-thermo, rotational/vibrational spectroscopy, fullerenes
2021	Physical General I (+ internal)	p. 7–8	0 — group theory, QM, partition functions, nanotubes, Raman
2022	Physical General I	p. 9–10	0 — symmetry, operators, spectroscopy, stat-thermo
2023	Physical General I	p. 9–10	0 — point groups, operators, rotors, Raman, fullerenes, stat-thermo
2024	Physical General I	p. 12–13	0 — group theory, matrices, rotors, polarizability, partition functions

Honest verdict

Across 2020–2024 the university never asked an ion-association / Bjerrum question in these papers — MSCH-104 in those years is entirely group theory, quantum mechanics, spectroscopy and statistical thermodynamics — so, per the rule, the year-tagged bank stays empty rather than being padded with look-alikes. The moment one appears it lands here with its year and repeat count. For practice, the lecture's own four problems are answered in full below.

Lecture's own problems — answered in full

Prob. 1reasoning · temperatureSection 5

Q. Why does $K_A$ increase at low temperature?

Ans. From $r_B = z_+z_-e^2/(4\pi\varepsilon_0\varepsilon_r k_B T)$ we get $r_B \propto 1/T$ . As temperature decreases, the Bjerrum distance increases — the attraction-dominated zone widens — so more ion pairs qualify as associated and $K_A$ increases.

Prob. 2reasoning · dielectricSection 5

Q. How does the dielectric constant affect ion association?

Ans. $r_B \propto 1/\varepsilon_r$ . With an increase in $\varepsilon_r$ the Bjerrum distance decreases (the solvent screens the attraction), so ion association decreases; with a decrease in $\varepsilon_r$ association increases. High-dielectric solvents like water keep ions apart; low-dielectric solvents let them pair.

Prob. 3reasoning · chargeSection 5

Q. How does ionic charge affect ion association?

Ans. $r_B \propto |z_+ z_-|$ . Higher ionic charge produces stronger electrostatic attraction, so as $|z_+z_-|$ increases the Bjerrum distance grows and ion association increases.

Prob. 4comparison · 2:2 vs 1:1Section 3

Q. Why do 2:2 or 2:1 electrolytes generally have much stronger electrostatic association than a simple 1:1 electrolyte?

Ans. The magnitude of attraction scales as $|U| \propto z_+z_-$ , so higher charge products bind far harder:

Electrolyte (example)	Charges	$z_+z_-$	Relative electrostatic attraction
1:1 (Na⁺Cl⁻)	+1, −1	1	1
2:1 (Mg²⁺Cl⁻)	+2, −1	2	2 times
2:2 (Mg²⁺SO₄²⁻)	+2, −2	4	4 times

Hence approximately $U_{2:1} \approx 2\,U_{1:1}$ and $U_{2:2} \approx 4\,U_{1:1}$ — which is why MgSO₄ is the textbook ion-pair former.

00 Overview

01 The ion-pair equilibrium

02 Basic assumptions of Bjerrum's theory

03 Electrostatic potential energy between two ions

04 The Boltzmann factor

05 The Bjerrum distance

Physical meaning

How wide is the pairing window?

06 The association region: from a to rB

Pair or free? Slide the separation

07 Derivation of the Bjerrum association constant

Count the configurations

Weight them electrostatically

Integrate over the association region

08 Dimensionless form of the Bjerrum equation

09 Fraction of ions associated, α

10 α ↔ K(A), and the connection with conductivity

Connection with conductivity

Association fraction ↔ what the meter reads

11 Play with the Bjerrum equation

12 The equation sheet, at a glance

13 Symbol table

14 Rapid revision — 12 lines before the exam

15 Original notebook scans

16 PYQ bank · University papers 2020–2024

Lecture's own problems — answered in full

06 The association region: from a to r_B