Van't Hoff Factor and Abnormal Molar Mass

CHEColligative Properties

The van't Hoff factor $i$ corrects colligative formulae when a solute dissociates or associates, making the observed molar mass differ from the true value. ISC numericals use freezing-point data to back-calculate $i$ , then the degree of dissociation or association.

Definition of van't Hoff factor

i = \dfrac{\text{observed colligative property}}{\text{calculated (normal) property}} = \dfrac{M_{normal}}{M_{observed}}

M_{normal}

true molar mass,

M_{observed}

apparent molar mass

Degree of dissociation

i = 1 + (n-1)\alpha \quad\Rightarrow\quad \alpha = \dfrac{i-1}{n-1}

n

ions per formula unit,

\alpha

degree of dissociation

Degree of association

i = 1 - \left(1 - \dfrac{1}{n}\right)\alpha

n

monomers forming one associated unit,

\alpha

degree of association

Modified colligative laws

\Delta T_f = i K_f m \qquad \pi = i C R T

include

i

for every electrolyte or associating solute

Definition: $i = \dfrac{\text{observed colligative property}}{\text{calculated (normal) value}} = \dfrac{\text{normal molar mass}}{\text{observed molar mass}}$ .
Dissociation into $n$ ions: $i = 1 + (n-1)\alpha$ , so $i > 1$ ; e.g. ideal values $KCl = 2$ , $MgCl_2 = 3$ , $K_2SO_4 = 3$ .
Association of $n$ molecules into one: $i = 1 - \left(1 - \dfrac{1}{n}\right)\alpha$ , so $i < 1$ ; e.g. benzoic acid dimerises in benzene, ideal $i = 0.5$ .
Non-electrolytes (glucose, urea, sucrose) do not dissociate or associate, so $i = 1$ .
Find $i$ from data: $i = \dfrac{\Delta T_f \text{(observed)}}{K_f \, m}$ , then solve $\alpha$ from the appropriate $i$ relation.
Abnormal molar mass: dissociation gives an observed molar mass lower than true (more particles); association (e.g. acetic acid in benzene $\approx 120$ vs true $60$ ) gives roughly double the expected value.
Abnormal molar mass arises whenever the actual number of particles differs from that assumed; the observed (apparent) molar mass equals $M_{normal}/i$ .
For a weak electrolyte, $i$ lies between $1$ and its ideal maximum because dissociation is incomplete; $i$ also drifts toward $1$ at higher concentration due to inter-ionic attraction.
For dimerising solutes $n = 2$ , so $\alpha = 2(1-i)$ ; complete dimerisation ( $\alpha = 1$ ) gives $i = 0.5$ , doubling the apparent molar mass.
Strong electrolytes show $i$ slightly below the ideal integer (e.g. $NaCl$ near $1.9$ rather than $2$ ) because ions are not fully independent in solution.
Same data can be read two ways: a measured $i$ first, then $\alpha$ via the correct dissociation or association formula chosen by whether $i>1$ or $i<1$ .
Trap: carboxylic acids associate (H-bonded dimers) in non-polar solvents like benzene but dissociate in water — the same acid gives opposite $i$ behaviour in different solvents.

Where the marks go

Using $\alpha = i - 1$ for salts giving more than two ions — the correct expression is $\alpha = \dfrac{i-1}{n-1}$ , so $n=3$ salts need the full denominator.
Confusing the association formula with the dissociation formula: association gives $i < 1$ and uses $\left(1 - \tfrac{1}{n}\right)$ , not $(n-1)$ .
Forgetting that benzoic/acetic acid dimerise in benzene (so $i < 1$ ) but dissociate in water (so $i > 1$ ) — the solvent decides the behaviour.
Treating the apparent molar mass as the true one; dissociation lowers it and association raises it relative to the real value.
Assuming $i$ equals the exact ideal integer for strong electrolytes; real values fall slightly short because of inter-ionic interactions.

How the board asks it

Numericaldepression of freezing point for a mixture of non-electrolytes
$0.6\ g$ of urea ( $M = 60$ ) and $1.8\ g$ of glucose ( $M = 180$ ) are dissolved in $100\ g$ of water. Given $K_f = 1.86\ K\,kg\,mol^{-1}$ , calculate the depression in freezing point of the solution.
Numericaldegree of dissociation from $i = 1 + (n-1)\alpha$
$0.01\ mol$ of $K_2SO_4$ dissolved in $1\ kg$ of water lowers the freezing point by $0.0558\ ^\circ C$ . Calculate the van't Hoff factor and the degree of dissociation of $K_2SO_4$ , given $K_f = 1.86\ K\,kg\,mol^{-1}$ .
Numericalapparent molar mass and association, $\alpha = 2(1-i)$
$2\ g$ of benzoic acid dissolved in $25\ g$ of benzene shows a depression in freezing point of $1.62\ K$ . Calculate the molar mass of benzoic acid in benzene ( $K_f = 4.9\ K\,kg\,mol^{-1}$ ) and hence its degree of association.
Give reasonsassociation into H-bonded dimers in a non-polar solvent
Account for the fact that the experimentally determined molar mass of acetic acid in benzene is about $120\ g\,mol^{-1}$ , nearly double its expected value of $60\ g\,mol^{-1}$ .
Define / statedefinition of $i$ and its value for dissociation, association and non-electrolytes
Define the van't Hoff factor $i$ . State its value (greater than, less than, or equal to $1$ ) for $KCl$ in water, for benzoic acid in benzene, and for a $0.1\ M$ glucose solution.
Predict the productideal $i$ controls extent of freezing-point depression
Arrange equimolar aqueous solutions of $glucose$ , $KCl$ and $MgCl_2$ in increasing order of their freezing points, giving the van't Hoff factor assumed for each.

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.