Abnormal molecular mass

Every colligative-property formula you have met — $\Delta T_f = K_f m$, $\Delta T_b = K_b m$, $\Pi = MRT$, and the relative lowering of vapour pressure equal to the mole fraction $x_2$ — was derived assuming that one formula unit of solute dissolved becomes exactly one solute particle. Molecular solutes like glucose or urea obey this assumption well, so their colligative-property data hand back the correct molar mass. Ionic and hydrogen-bonding solutes often do not, and the mismatch is measured by the van't Hoff factor:

where: the "normal" molar mass is the formula mass computed from the periodic table, and the "observed" molar mass is whatever the plain, unmodified colligative-property formula returns when you feed in the experimental $\Delta T_f$, $\Delta T_b$, $\Pi$, or vapour-pressure lowering.

Dissociation pushes $i$ above 1. An electrolyte such as NaCl or $K_2SO_4$ splits into ions on dissolving, so a single formula unit yields two or more particles. The particle count rises, the observed colligative property is larger than the normal (undissociated) prediction, and — because observed molar mass sits in the denominator of $i$ — the molar mass you calculate by (wrongly) assuming no dissociation comes out smaller than the true formula mass. For a solute giving $n$ particles per formula unit on complete dissociation, with a fraction $\alpha$ (the degree of dissociation) actually dissociating:

Association pulls $i$ below 1. Some solutes do the opposite: carboxylic acids such as acetic or benzoic acid, dissolved in a nonpolar solvent like benzene, hydrogen-bond in pairs to form dimers. Two formula units become one particle, so the particle count falls, the observed colligative property is smaller than expected, and the calculated molar mass comes out larger than the true formula mass — famously, benzoic acid in benzene appears to weigh almost double its real $122\ \text{g mol}^{-1}$. For association into aggregates of $n$ units with a fraction $\alpha$ (the degree of association) actually aggregating:

Every colligative-property formula is modified by inserting $i$ once dissociation or association is present:

where: $K_f$ and $K_b$ are the solvent's molal depression and elevation constants, $m$ is molality on a formula-unit basis, $M$ is the formula-unit molar concentration, $R$ is the gas constant, $T$ is the absolute temperature, and $p_1^0, p_1$ are the pure solvent's and solution's vapour pressures.

A 0.1 molal aqueous solution of $K_2SO_4$ shows a freezing-point depression of $\Delta T_f = 0.518\ \text{K}$. Given $K_f$ for water $= 1.86\ \text{K kg mol}^{-1}$, calculate the van't Hoff factor $i$, the apparent degree of dissociation, and the molar mass $K_2SO_4$ would appear to have from this experiment (true molar mass of $K_2SO_4 = 174\ \text{g mol}^{-1}$).

Step 1. Find the normal (undissociated) freezing-point depression, as if $i=1$: $\Delta T_{f,\text{normal}} = K_f \times m = 1.86 \times 0.1 = 0.186\ \text{K}$.

Step 2. Find $i$ from the ratio of observed to normal depression: $i = \dfrac{\Delta T_{f,\text{observed}}}{\Delta T_{f,\text{normal}}} = \dfrac{0.518}{0.186} = 2.785$.

Step 3. $K_2SO_4 \rightarrow 2K^+ + SO_4^{2-}$ gives $n=3$ particles per formula unit on complete dissociation. Using $i=1+(n-1)\alpha$: $2.785 = 1 + (3-1)\alpha \Rightarrow 1.785 = 2\alpha \Rightarrow \alpha = 0.8925$, so $K_2SO_4$ is about $89.3\%$ dissociated in this solution.

Step 4 (competitive stretch). Since $i$ also equals normal molar mass over observed molar mass, the experiment would appear to report an observed molar mass of $M_{\text{observed}} = \dfrac{M_{\text{normal}}}{i} = \dfrac{174}{2.785} = 62.5\ \text{g mol}^{-1}$ — far below the true $174\ \text{g mol}^{-1}$, exactly as expected for a salt that is mostly dissociated into ions. Final answer: $i = 2.785$, degree of dissociation $\alpha \approx 89.3\%$, apparent molar mass $\approx 62.5\ \text{g mol}^{-1}$.

• Mixing up the direction: $i>1$ signals dissociation (molar mass appears too low), and $i<1$ signals association (molar mass appears too high) — students often swap these.
• Using the wrong $n$ for a given solute — always write the dissociation or association equation first ($K_2SO_4 \to 2K^+ + SO_4^{2-}$ gives $n=3$; a dimer gives $n=2$) before plugging into the $i$ formula.
• Applying the plain, unmodified colligative-property formula to a known strong electrolyte and being "surprised" the molar mass comes out wrong — that mismatch is exactly what abnormal molecular mass explains, not an error to fix away.
• Treating the observed (experimental) molar mass as the "real" one — the true formula mass is fixed by the periodic table; it is the ratio of the two that yields $i$, never the other way round.

Ask the tutor: want to work through a strong-electrolyte problem with $Al_2(SO_4)_3$ or a weak-acid partial-dissociation case — ask the AI tutor to set one up for you.