DAT General Chemistry: Vapor-Pressure Lowering and Raoult's Law

DAT High-Yield Concept

Vapor-Pressure Lowering: Adding a nonvolatile solute decreases the solvent's vapor pressure because solute particles block surface evaporation sites and increase solution entropy.
Raoult's Law (Nonvolatile Solute): Solution vapor pressure depends directly on solvent mole fraction: $P_{\text{soln}} = \chi_{\text{solvent}} P^\circ_{\text{solvent}}$ . Vapor pressure reduction is $\Delta P = \chi_{\text{solute}} P^\circ_{\text{solvent}}$ .
van 't Hoff Factor ( $i$ ): Dissociating electrolyte solutes yield multiple dissolved ions. The effective solute moles equal $i \times n_{\text{solute}}$ .
Volatile Liquid Mixtures: Total vapor pressure above two volatile liquids is $P_{\text{tot}} = \chi_A P_A^\circ + \chi_B P_B^\circ$ . Vapor composition is enriched in the more volatile component.
Non-Ideal Deviations: Positive deviations occur when solvent-solute interactions are weaker than pure-component interactions ( $\Delta H_{\text{mix}} > 0$ ); negative deviations occur when solvent-solute interactions are stronger ( $\Delta H_{\text{mix}} < 0$ ).

Vapor-Pressure Lowering by a Nonvolatile Solute

When a nonvolatile solute (a substance with negligible vapor pressure of its own) dissolves in a liquid solvent, the vapor pressure above the solution decreases relative to pure solvent at the same temperature. Solute particles occupy surface area at the liquid-gas interface, decreasing the rate of solvent vaporization while increasing the entropy of the liquid phase. This vapor-pressure lowering directly causes boiling-point elevation (covered in depth in its own lesson), because higher temperatures are required for solution vapor pressure to reach external atmospheric pressure.

Raoult's Law for Nonvolatile Solutes:

P_{\text{soln}} = \chi_{\text{solvent}} P^\circ_{\text{solvent}}

\Delta P = P^\circ_{\text{solvent}} - P_{\text{soln}} = \chi_{\text{solute}} P^\circ_{\text{solvent}}

Where

\chi_{\text{solvent}}

and

\chi_{\text{solute}}

are mole fractions in the liquid phase, and

P^\circ_{\text{solvent}}

is the vapor pressure of pure solvent.

Worked Example: Nonvolatile Electrolyte Calculation

Problem: Calculate the vapor pressure at $\pu{25.0 degC}$ of a solution prepared by dissolving $\pu{0.100 mol}$ of $\ce{CaCl2(s)}$ in $\pu{0.900 mol}$ of pure $\ce{H2O(l)}$ . Pure water vapor pressure at $\pu{25.0 degC}$ is $\pu{23.8 mmHg}$ .

Solution: Soluble ionic salts dissociate in solution according to: $\ce{CaCl2(s) -> Ca^{2+}(aq) + 2Cl^-(aq)}$ , producing $3$ moles of ions per mole of formula unit ( $i = 3$ ).

Effective solute particle moles: $n_{\text{particles}} = 3 \times \pu{0.100 mol} = \pu{0.300 mol}$ .
Total liquid moles: $n_{\text{tot}} = \pu{0.900 mol H2O} + \pu{0.300 mol particles} = \pu{1.200 mol}$ .
Solvent mole fraction: $\chi_{\text{solvent}} = \frac{\pu{0.900 mol}}{\pu{1.200 mol}} = 0.750$ .
Solution vapor pressure: $P_{\text{soln}} = (0.750)(\pu{23.8 mmHg}) = \pu{17.9 mmHg}$ .

Common DAT Exam Trap: Forgetting the van 't Hoff factor ( $i$ ) for electrolyte solutes, or mistakenly substituting $\chi_{\text{solute}}$ for $\chi_{\text{solvent}}$ when solving directly for $P_{\text{soln}}$ .

Binary Mixtures of Two Volatile Liquids

When two volatile liquids (Component A and Component B) form an ideal solution, both components exert partial vapor pressure above the mixture according to Raoult's law and Dalton's law of partial pressures:

P_A = \chi_A P_A^\circ, \quad P_B = \chi_B P_B^\circ

P_{\text{tot}} = P_A + P_B = \chi_A P_A^\circ + \chi_B P_B^\circ

Because the components vaporize independently in ideal solutions, the mole fraction of component A in the gas phase ( $y_A$ ) is given by $y_A = \frac{P_A}{P_{\text{tot}}}$ . The vapor phase is always enriched in the more volatile component (the component with higher pure vapor pressure).

Worked Example: Volatile Liquid Mixture Calculation

Problem: A liquid mixture contains $\pu{2.00 mol}$ of liquid A ( $P_A^\circ = \pu{100.0 mmHg}$ ) and $\pu{3.00 mol}$ of liquid B ( $P_B^\circ = \pu{40.0 mmHg}$ ) at $\pu{25.0 degC}$ . Calculate the total vapor pressure and the mole fraction of component A in the vapor phase.

Solution:
Total liquid moles = $\pu{2.00 mol} + \pu{3.00 mol} = \pu{5.00 mol}$ .
Liquid mole fractions: $\chi_A = \frac{\pu{2.00 mol}}{\pu{5.00 mol}} = 0.400$ ; $\chi_B = \frac{\pu{3.00 mol}}{\pu{5.00 mol}} = 0.600$ .
Partial pressures: $P_A = (0.400)(\pu{100.0 mmHg}) = \pu{40.0 mmHg}$ ; $P_B = (0.600)(\pu{40.0 mmHg}) = \pu{24.0 mmHg}$ .
Total pressure: $P_{\text{tot}} = \pu{40.0 mmHg} + \pu{24.0 mmHg} = \pu{64.0 mmHg}$ .
Vapor mole fraction: $y_A = \frac{\pu{40.0 mmHg}}{\pu{64.0 mmHg}} = 0.625$ .

Deviations from Raoult's Law (Ideal vs. Non-Ideal Solutions)

An ideal solution occurs when intermolecular attractive forces between unlike molecules ( $A\text{--}B$ ) equal those between like molecules ( $A\text{--}A$ and $B\text{--}B$ ), yielding zero enthalpy change upon mixing ( $\Delta H_{\text{mix}} = 0$ ). Non-ideal solutions exhibit distinct deviations:

Positive Deviation ( $P_{\text{exp}} > P_{\text{ideal}}$ ): $A\text{--}B$ attractions are weaker than $A\text{--}A$ and $B\text{--}B$ attractions. Molecules escape into vapor easily, making mixing endothermic ( $\Delta H_{\text{mix}} > 0$ ).
Negative Deviation ( $P_{\text{exp}} < P_{\text{ideal}}$ ): $A\text{--}B$ attractions are stronger than $A\text{--}A$ and $B\text{--}B$ attractions (e.g., hydrogen bonding between acetone and chloroform). Molecules are retained tightly in liquid, making mixing exothermic ( $\Delta H_{\text{mix}} < 0$ ).

Total vapor pressure diagram of a mixture of two solvents; three examples: a) following Raoult's law, b) positive deviation from Raoult's law (with azeotrope), and c) negative deviation from Raoult's law (with azeotrope) (Source: Д.Ильин : vectorization · CC0 · Wikimedia Commons)

Mnemonic for Deviations:
Positive deviation = Push away (weaker attractions, higher vapor pressure).
Negative deviation = Nearer together (stronger attractions, lower vapor pressure).

Property	Ideal Solution	Positive Deviation	Negative Deviation
Intermolecular Forces	$A\text{--}B = A\text{--}A = B\text{--}B$	$A\text{--}B < A\text{--}A, B\text{--}B$	$A\text{--}B > A\text{--}A, B\text{--}B$
Total Vapor Pressure	Follows Raoult's Law	Higher than predicted	Lower than predicted
Enthalpy of Mixing ( $\Delta H_{\text{mix}}$ )	$0$	$> 0$ (Endothermic)	$< 0$ (Exothermic)
Volume of Mixing ( $\Delta V_{\text{mix}}$ )	$0$	$> 0$ (Expansion)	$< 0$ (Contraction)

Critical Thinking Questions

Why does dissolving $\pu{1.0 mol}$ of $\ce{Al(NO3)3}$ lower the vapor pressure of water significantly more than dissolving $\pu{1.0 mol}$ of glucose in the same mass of water?
If mixing liquid X and liquid Y produces a solution with a total vapor pressure lower than calculated by Raoult's law, will the temperature of the container increase or decrease during mixing? Explain.
How does fractional distillation utilize the difference between liquid composition ( $\chi_A$ ) and vapor composition ( $y_A$ ) to separate two volatile liquids?

References

Sources returned by Google Search grounding during lesson verification.

Raoult's Law - Chemistry LibreTexts — chem.libretexts.org
Ideal and Non-ideal Solutions - BYJU'S — byjus.com

Vapor-Pressure Lowering and Raoult's Law

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