Mixed and Comprehensive Questions

CHEExam Practice

These problems combine concentration terms, all four colligative properties, the van't Hoff factor and osmotic-flow direction into multi-step numericals. The key skill is choosing the right formula chain, tracking the factor $i$ for electrolytes, and keeping units consistent throughout.

Relative lowering of vapour pressure

\dfrac{p^\circ - p_s}{p^\circ} = x_{solute} = \dfrac{n_2}{n_1 + n_2}

p^\circ

pure-solvent vapour pressure,

p_s

solution vapour pressure,

n_2

moles solute,

n_1

moles solvent

Boiling-point elevation and freezing-point depression

\Delta T_b = i K_b m \qquad \Delta T_f = i K_f m

i

van't Hoff factor,

K_b,K_f

molal constants,

m

molality

Osmotic pressure

\pi = i C R T

C

molarity,

R = 0.0821

L atm K

^{-1}

mol

^{-1}

T

in kelvin

Molar mass from depression / elevation

M_2 = \dfrac{i \, K \, w_2 \times 1000}{\Delta T \times w_1}

w_2

solute mass (g),

w_1

solvent mass (g),

K = K_b

K_f

van't Hoff factor (dissociation)

i = 1 + (n-1)\alpha

n

ions per formula unit,

\alpha

degree of dissociation

Master colligative set: relative lowering $\dfrac{p^\circ - p_s}{p^\circ} = x_{solute}$ ; $\Delta T_b = i K_b m$ ; $\Delta T_f = i K_f m$ ; $\pi = iCRT$ — all depend on number of particles, not their nature.
Molality is preferred over molarity in colligative work because it uses mass of solvent and is temperature-independent (volume changes with $T$ , mass does not).
For an electrolyte giving $n$ ions, $i = 1 + (n-1)\alpha$ (dissociation) and $i = 1 - (1 - \tfrac{1}{n})\alpha$ for association; e.g. $Na_2SO_4 \rightarrow 2Na^+ + SO_4^{2-}$ has ideal $i = 3$ .
Molar mass route from $\Delta T_f$ or $\Delta T_b$ : $M = \dfrac{i K \times w_2 \times 1000}{\Delta T \times w_1}$ , where $w_2$ is solute mass and $w_1$ solvent mass in grams.
Osmotic-flow direction: water moves from lower to higher effective particle concentration $iC$ , e.g. $0.1\,M$ $NaCl$ ( $iC = 0.2$ ) is more concentrated in particles than $0.1\,M$ glucose ( $iC = 0.1$ ), so water flows toward the $NaCl$ .
Henry's law link: dissolved gas amount $x_{gas} = \dfrac{p_{gas}}{K_H}$ ; convert mole fraction to moles using moles of water in $1\,\text{L}$ ( $\approx 55.5$ mol).
Plan multi-part numericals by identifying the measured property first ( $\Delta T_f$ , $\Delta T_b$ , $\pi$ or $p_s$ ), back-solving for $i$ or molality, and only then moving to molar mass or degree of dissociation.
Percentage dissociation from data: get $i$ from the observed property, then $\alpha = \dfrac{i-1}{n-1}$ and multiply by $100$ ; e.g. a $0.25\,m$ $Na_2SO_4$ giving $\Delta T_f = 1.08\,\text{K}$ (with $K_f = 1.86$ ) yields $i \approx 2.32$ .
A freezing point quoted as $-0.93^\circ\text{C}$ means $\Delta T_f = 0.93\,\text{K}$ ; never substitute the negative temperature directly into $\Delta T_f = iK_f m$ .
All four colligative properties scale with the same particle count, so for a given solution the ratios $\Delta T_b : \Delta T_f : \pi$ are fixed by $K_b$ , $K_f$ and $RT$ — useful for cross-checking an answer.
Keep a unit checklist: $K_b,K_f$ in $\text{K kg mol}^{-1}$ , $m$ in $\text{mol kg}^{-1}$ , $C$ in $\text{mol L}^{-1}$ , masses converted (solvent to kg for molality, kept in g for the molar-mass formula).
Common trap: forgetting $i$ for ionic solutes inflates the apparent molar mass and underestimates $\Delta T$ — always check whether the solute is an electrolyte before plugging numbers.

Where the marks go

Mixing molality and molarity within one problem — colligative formulae for $\Delta T$ need molality, but $\pi = iCRT$ needs molarity; converting requires the solution density.
Substituting a negative freezing-point temperature into $\Delta T_f = iK_f m$ instead of its magnitude relative to the pure solvent.
Dropping $i$ for the electrolyte in a multi-step chain, so the back-calculated molar mass comes out abnormally high.
Forgetting to convert solution volume from mL to L before using $C = n/V$ in the osmotic-pressure step.
Using $\alpha = i - 1$ (correct only for a 2-ion salt) instead of the general $\alpha = \dfrac{i-1}{n-1}$ for salts giving three or more ions.

How the board asks it

Numericalmolar mass from $\Delta T_f = i K_f m$
A solution of $1.25\,\text{g}$ of a non-electrolyte in $20\,\text{g}$ of water freezes at $-1.02\,^\circ\text{C}$ . Calculate the molar mass of the solute. ( $K_f = 1.86\,\text{K kg mol}^{-1}$ )
Numericalvan't Hoff factor and percentage dissociation from $\Delta T_f$
$0.25\,\text{m}$ aqueous $Na_2SO_4$ shows a freezing-point depression of $1.08\,\text{K}$ . Taking $K_f = 1.86\,\text{K kg mol}^{-1}$ , calculate the van't Hoff factor $i$ and the percentage dissociation of the salt.
Numericalosmotic pressure $\pi = iCRT$ with mL-to-L conversion
$2.5\,\text{g}$ of $K_2SO_4$ (molar mass $174$ ) is dissolved to make $250\,\text{mL}$ of solution. Assuming complete dissociation, calculate the osmotic pressure at $300\,\text{K}$ . ( $R = 0.0821\,\text{L atm K}^{-1}\,\text{mol}^{-1}$ )
Numericaladditive colligative effect of a mixture of two non-electrolytes
$0.6\,\text{g}$ of urea (molar mass $60$ ) and $1.8\,\text{g}$ of glucose (molar mass $180$ ) are dissolved in $100\,\text{g}$ of water. If $K_b = 0.52\,\text{K kg mol}^{-1}$ , calculate the elevation in boiling point of the resulting solution.
Give reasonseffective particle concentration $iC$ and osmotic flow
Account for the fact that a $0.1\,\text{M}$ $NaCl$ solution exerts nearly twice the osmotic pressure of a $0.1\,\text{M}$ glucose solution at the same temperature.
Conversionmolality-to-molarity link via density
An aqueous solution of glucose (molar mass $180$ ) is $0.5\,\text{m}$ and has a density of $1.04\,\text{g mL}^{-1}$ . Calculate the molarity of the solution.

PreviousVan't Hoff Factor and Abnormal Molar Mass

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.