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9.3 Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes
PRC Chemist Licensure Examination Exam
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9 Chapter 9: Physical Chemistry: Equilibrium, Phase Equilibria, Solutions, and Electrochemistry
9.1 Chemical Equilibrium: Equilibrium Constants, Le Chatelier's Principle, and the van 't Hoff Equation 9.2 Phase Rule, Phase Diagrams of One- and Two-Component Systems, and Clausius-Clapeyron Equation 9.3 Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes 9.4 Electrochemistry: Galvanic Cells, Standard Potentials, and the Nernst Equation 9.5 Electrolyte Conductance, Ionic Mobility, Transport Numbers, and Concentration Cells
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12.1 Nucleophilic Aliphatic Substitution ($S_N1$, $S_N2$) and Elimination ($E_1$, $E_2$) Mechanisms 12.2 Alcohols, Phenols, Ethers, and Epoxides: Structure, Synthesis, and Reactivity 12.3 Carbonyl Compounds I: Aldehydes and Ketones, Nucleophilic Addition, and Oxidation/Reduction 12.4 Carbonyl Compounds II: Carboxylic Acids and Derivatives, Enolate Chemistry, and Condensations 12.5 Amines: Basicity, Synthesis, Reactions, and Arenediazonium Chemistry 12.6 Structure Elucidation: NMR Spectroscopy, Combined Spectral Problems, and Qualitative Organic Tests 12.7 Organic Synthesis Strategy, Protecting Groups, Rearrangements, and Pericyclic Reactions
13 Chapter 13: Biochemistry: Cells, Biomolecules, Enzymes, and Membranes
13.1 Cells, Organelles, Water, and Biological Acid-Base Buffers 13.2 Amino Acids, Peptides, and Hierarchical Protein Structure 13.3 Protein Purification and Biochemical Analysis of Proteins, Carbohydrates, and Lipids 13.4 Enzyme Nomenclature and Classification, Cofactors, Coenzymes, and Vitamin Precursors 13.5 Enzyme Kinetics: Michaelis-Menten Model, Lineweaver-Burk Analysis, and Enzyme Inhibition 13.6 Carbohydrates, Lipids, Biological Membranes, and Membrane Transport
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9.3 Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes
Key Takeaways Explain This Topic AI
Ideal solutions obey Raoult's Law ( P i = x i P i ∗ P_i = x_i P_i^* P i = x i P i ∗ ) across all compositions, characterized by zero enthalpy of mixing ( Δ mix H = 0 \Delta_{\text{mix}} H = 0 Δ mix H = 0 ), zero volume change ( Δ mix V = 0 \Delta_{\text{mix}} V = 0 Δ mix V = 0 ), and an entirely entropy-driven spontaneity ( Δ mix G = n R T ∑ x i ln x i < 0 \Delta_{\text{mix}} G = nRT \sum x_i \ln x_i < 0 Δ mix G = n RT ∑ x i ln x i < 0 ).
In dilute real solutions, the solvent approaches Raoult's Law ( P 1 = x 1 P 1 ∗ P_1 = x_1 P_1^* P 1 = x 1 P 1 ∗ ) while the volatile solute obeys Henry's Law ( P 2 = K H , 2 x 2 P_2 = K_{H, 2} x_2 P 2 = K H , 2 x 2 ), reflecting constant solvated microscopic environments.
Colligative properties—vapor pressure lowering ( Δ P = x 2 P 1 ∗ \Delta P = x_2 P_1^* Δ P = x 2 P 1 ∗ ), boiling point elevation ( Δ T b = i K b m \Delta T_b = i K_b m Δ T b = i K b m ), freezing point depression ( Δ T f = i K f m \Delta T_f = i K_f m Δ T f = i K f m ), and osmotic pressure ( Π = i c R T \Pi = i c R T Π = i c RT )—depend strictly on the number ratio of dissolved solute particles to solvent molecules, not on solute chemical identity.
For electrolyte solutions, the van 't Hoff factor i i i accounts for ion dissociation and ion pairing, correlating with the degree of dissociation via α = i − 1 ν − 1 \alpha = \frac{i - 1}{\nu - 1} α = ν − 1 i − 1 .
Long-range electrostatic interactions in electrolyte solutions are quantified by the ionic strength I = 1 2 ∑ c i z i 2 I = \frac{1}{2}\sum c_i z_i^2 I = 2 1 ∑ c i z i 2 and mean activity coefficients via the Debye-Hückel Limiting Law: log 10 γ ± = − A ∣ z + z − ∣ I \log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I} lo g 10 γ ± = − A ∣ z + z − ∣ I (where A = 0.509 A = 0.509 A = 0.509 in water at 25°C).
Last updated: October 2026
Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes
A solution is a homogeneous mixture of two or more substances down to the molecular scale. In physical chemistry and on the PRC Chemist Licensure Examination, solution theory spans ideal solutions, volatile and non-volatile solute behaviors, colligative phenomena, macromolecular characterization by osmometry, and the non-ideality of electrolyte solutions treated by Debye-Hückel theory.
1. Ideal Solutions, Raoult's Law, and Thermodynamics of Mixing
An ideal solution is defined at the molecular level as a mixture in which the intermolecular forces between unlike molecules ( A − B A-B A − B ) are identical in magnitude to those between like molecules ( A − A A-A A − A and B − B B-B B − B ).
Raoult's Law
For an ideal solution in equilibrium with its vapor, the partial vapor pressure of each volatile component i i i is proportional to its mole fraction in the liquid phase ( x i x_i x i ):
P i = x i P i ∗ P_i = x_i P_i^* P i = x i P i ∗
where P i ∗ P_i^* P i ∗ is the saturation vapor pressure of pure component i i i at the specified temperature.
For a binary mixture of volatile components A A A and B B B , the total vapor pressure is:
P t o t = P A + P B = x A P A ∗ + x B P B ∗ = x A P A ∗ + ( 1 − x A ) P B ∗ = P B ∗ + x A ( P A ∗ − P B ∗ ) P_{tot} = P_A + P_B = x_A P_A^* + x_B P_B^* = x_A P_A^* + (1 - x_A) P_B^* = P_B^* + x_A (P_A^* - P_B^*) P t o t = P A + P B = x A P A ∗ + x B P B ∗ = x A P A ∗ + ( 1 − x A ) P B ∗ = P B ∗ + x A ( P A ∗ − P B ∗ )
The mole fraction of component A A A in the coexisting vapor phase ( y A y_A y A ) is given by Dalton's law:
y A = P A P t o t = x A P A ∗ x A P A ∗ + x B P B ∗ y_A = \frac{P_A}{P_{tot}} = \frac{x_A P_A^*}{x_A P_A^* + x_B P_B^*} y A = P t o t P A = x A P A ∗ + x B P B ∗ x A P A ∗
If component A A A is more volatile than B B B ( P A ∗ > P B ∗ P_A^* > P_B^* P A ∗ > P B ∗ ), then y A > x A y_A > x_A y A > x A , meaning the vapor is enriched in the more volatile component.
Thermodynamic Functions of Mixing for Ideal Solutions
When n A n_A n A moles of pure A A A and n B n_B n B moles of pure B B B mix to form an ideal solution at constant T T T and P P P :
Enthalpy of Mixing: No net bond breaking or making energetics occurs: Δ m i x H = 0 \Delta_{mix} H = 0 Δ mi x H = 0
Volume of Mixing: Molecules pack identically to their pure states: Δ m i x V = 0 \Delta_{mix} V = 0 Δ mi x V = 0
Entropy of Mixing: Driven entirely by configurational disorder ( x i < 1 ⟹ ln x i < 0 x_i < 1 \implies \ln x_i < 0 x i < 1 ⟹ ln x i < 0 ): Δ m i x S = − n R ∑ i x i ln x i > 0 \Delta_{mix} S = -nR \sum_{i} x_i \ln x_i > 0 Δ mi x S = − n R ∑ i x i ln x i > 0
Gibbs Free Energy of Mixing: Δ m i x G = Δ m i x H − T Δ m i x S = n R T ∑ i x i ln x i < 0 \Delta_{mix} G = \Delta_{mix} H - T\Delta_{mix} S = nRT \sum_{i} x_i \ln x_i < 0 Δ mi x G = Δ mi x H − T Δ mi x S = n RT ∑ i x i ln x i < 0 Because Δ m i x G < 0 \Delta_{mix} G < 0 Δ mi x G < 0 for all compositions, ideal mixing is spontaneous at all temperatures.
2. Dilute Real Solutions and Henry's Law
In real solutions, A − A A-A A − A , B − B B-B B − B , and A − B A-B A − B intermolecular potentials differ. However, in the limit of high dilution (where solute mole fraction x 2 → 0 x_2 \rightarrow 0 x 2 → 0 and solvent mole fraction x 1 → 1 x_1 \rightarrow 1 x 1 → 1 ):
Solvent Behavior (Raoult's Law): Every solvent molecule is surrounded virtually exclusively by other solvent molecules. Thus, the solvent behaves ideally and obeys Raoult's Law: P 1 = x 1 P 1 ∗ ( as x 1 → 1 ) P_1 = x_1 P_1^* \quad (\text{as } x_1 \rightarrow 1) P 1 = x 1 P 1 ∗ ( as x 1 → 1 )
Solute Behavior (Henry's Law): Every solute molecule is completely surrounded by solvent molecules, experiencing a constant solvated environment. The partial pressure of the solute is directly proportional to its mole fraction, but governed by Henry's Law constant ( K H , 2 K_{H, 2} K H , 2 ): P 2 = K H , 2 x 2 ( as x 2 → 0 ) P_2 = K_{H, 2} x_2 \quad (\text{as } x_2 \rightarrow 0) P 2 = K H , 2 x 2 ( as x 2 → 0 )
Alternatively, expressing solute concentration in molarity ( c c c ): c = k H P c = k_H P c = k H P
Because the dissolution of gases in liquids is generally an exothermic process ( Δ H s o l n < 0 \Delta H_{soln} < 0 Δ H so l n < 0 ), the solubility of gases in water decreases with increasing temperature, as predicted by Le Chatelier's principle.
3. Colligative Properties of Non-Electrolyte Solutions
Colligative properties are properties of solutions that depend solely on the number of solute particles dissolved per unit volume or mass of solvent, and are independent of the chemical identity, size, or charge of the solute particles.
Thermodynamically, all colligative properties originate from a single phenomenon: the lowering of the solvent's chemical potential in solution compared to pure liquid solvent:
μ 1 ( s o l n ) = μ 1 ∗ ( l ) + R T ln x 1 < μ 1 ∗ ( l ) ( since x 1 < 1 ) \mu_1(soln) = \mu_1^*(l) + RT \ln x_1 < \mu_1^*(l) \quad (\text{since } x_1 < 1) μ 1 ( so l n ) = μ 1 ∗ ( l ) + RT ln x 1 < μ 1 ∗ ( l ) ( since x 1 < 1 )
This stabilization of the liquid phase expands the liquid temperature range, lowering both the vapor pressure and the freezing point, while elevating the boiling point.
The Four Colligative Properties
Property Governing Equation Physical Constants & Definitions
1. Vapor Pressure Lowering Δ P = P 1 ∗ − P 1 = x 2 P 1 ∗ \Delta P = P_1^* - P_1 = x_2 P_1^* Δ P = P 1 ∗ − P 1 = x 2 P 1 ∗ Δ P P 1 ∗ = x 2 \frac{\Delta P}{P_1^*} = x_2 P 1 ∗ Δ P = x 2 (Relative vapor pressure lowering equals solute mole fraction)
2. Boiling Point Elevation Δ T b = T b − T b ∗ = K b m \Delta T_b = T_b - T_b^* = K_b m Δ T b = T b − T b ∗ = K b m K b = R ( T b ∗ ) 2 M 1 Δ H v a p K_b = \frac{R (T_b^*)^2 M_1}{\Delta H_{vap}} K b = Δ H v a p R ( T b ∗ ) 2 M 1 ( Ebullioscopic constant ; for H 2 O H_2O H 2 O , K b = 0.512 ∘ C ⋅ kg/mol K_b = 0.512^\circ\text{C}\cdot\text{kg/mol} K b = 0.51 2 ∘ C ⋅ kg/mol )
3. Freezing Point Depression Δ T f = T f ∗ − T f = K f m \Delta T_f = T_f^* - T_f = K_f m Δ T f = T f ∗ − T f = K f m K f = R ( T f ∗ ) 2 M 1 Δ H f u s K_f = \frac{R (T_f^*)^2 M_1}{\Delta H_{fus}} K f = Δ H f u s R ( T f ∗ ) 2 M 1 ( Cryoscopic constant ; for H 2 O H_2O H 2 O , K f = 1.86 ∘ C ⋅ kg/mol K_f = 1.86^\circ\text{C}\cdot\text{kg/mol} K f = 1.8 6 ∘ C ⋅ kg/mol )
4. Osmotic Pressure Π = c R T = ( n V ) R T \Pi = c R T = \left(\frac{n}{V}\right) R T Π = c RT = ( V n ) RT Π \Pi Π is hydrostatic pressure preventing solvent influx across a semipermeable membrane
Exam Application: Macromolecular Molar Mass by Osmometry For polymers and proteins ( M > 10000 g/mol M > 10000\text{ g/mol} M > 10000 g/mol ), freezing point depression produces temperature changes of less than 0.001 ∘ C 0.001^\circ\text{C} 0.00 1 ∘ C , which cannot be measured accurately. In contrast, osmotic pressure produces hydrostatic column heights of several centimeters, making osmometry the primary colligative method for determining macromolecular weights: M = w R T Π V M = \frac{w R T}{\Pi V} M = Π V wRT where w w w is the mass of dissolved solute in grams.
4. Electrolyte Solutions and the van 't Hoff Factor ( i i i )
When ionic salts or ionizable acids dissolve, each formula unit dissociates into ν \nu ν ions, producing a greater number of colligative particles than expected from stoichiometry.
The van 't Hoff factor ( i i i ) is defined as:
i = Actual number of particles in solution Number of formula units dissolved = Measured colligative property Calculated property for non-electrolyte i = \frac{\text{Actual number of particles in solution}}{\text{Number of formula units dissolved}} = \frac{\text{Measured colligative property}}{\text{Calculated property for non-electrolyte}} i = Number of formula units dissolved Actual number of particles in solution = Calculated property for non-electrolyte Measured colligative property
Incorporating i i i into the colligative formulas: Δ T b = i K b m , Δ T f = i K f m , Π = i c R T \Delta T_b = i K_b m, \quad \Delta T_f = i K_f m, \quad \Pi = i c R T Δ T b = i K b m , Δ T f = i K f m , Π = i c RT
Degree of Dissociation ( α \alpha α )
For a weak electrolyte where each formula unit dissociates into ν \nu ν ions with a fractional degree of dissociation α \alpha α :
A ν ⇌ ν ions Initial: 1 0 Equilibrium: 1 − α ν α \begin{matrix} & A_\nu & \rightleftharpoons & \nu \text{ ions} \\ \text{Initial:} & 1 & & 0 \\ \text{Equilibrium:} & 1 - \alpha & & \nu\alpha \end{matrix} Initial: Equilibrium: A ν 1 1 − α ⇌ ν ions 0 να
Total particles at equilibrium = ( 1 − α ) + ν α = 1 + α ( ν − 1 ) = (1 - \alpha) + \nu\alpha = 1 + \alpha(\nu - 1) = ( 1 − α ) + να = 1 + α ( ν − 1 ) . Therefore:
i = 1 + α ( ν − 1 ) ⟹ α = i − 1 ν − 1 i = 1 + \alpha(\nu - 1) \implies \alpha = \frac{i - 1}{\nu - 1} i = 1 + α ( ν − 1 ) ⟹ α = ν − 1 i − 1
Ideal vs. Real Electrolytes: For strong electrolytes like N a C l NaCl N a Cl ( ν = 2 \nu = 2 ν = 2 ), i i i approaches 2.00 2.00 2.00 at infinite dilution. However, in real solutions ( 0.05 M 0.05\text{ M} 0.05 M ), i ≈ 1.90 i \approx 1.90 i ≈ 1.90 due to electrostatic attraction and the formation of transient ion pairs ( [ N a + C l − ] 0 [Na^+ Cl^-]^0 [ N a + C l − ] 0 ).
5. Non-Ideal Electrolytes: Ionic Strength and Debye-Hückel Theory
In electrolyte solutions, long-range Coulombic forces ( V ( r ) ∝ 1 / r V(r) \propto 1/r V ( r ) ∝ 1/ r ) prevent ions from behaving independently even at concentrations below 0.01 M 0.01\text{ M} 0.01 M . Peter Debye and Erich Hückel developed the ionic atmosphere model, where every central cation is statistically surrounded by a net excess of anions, and vice versa.
Ionic Strength ( I I I )
The electrostatic field strength of an electrolyte solution depends on both ion concentrations and their charges, quantified by the ionic strength :
I = 1 2 ∑ i c i z i 2 [ or I = 1 2 ∑ i m i z i 2 ] I = \frac{1}{2} \sum_{i} c_i z_i^2 \quad \left[ \text{or } I = \frac{1}{2} \sum_{i} m_i z_i^2 \right] I = 2 1 ∑ i c i z i 2 [ or I = 2 1 ∑ i m i z i 2 ]
where c i c_i c i is molar concentration and z i z_i z i is ionic valence.
1 : 1 1:1 1 : 1 Salt ( N a C l NaCl N a Cl , c c c ): I = 1 2 [ c ( 1 ) 2 + c ( − 1 ) 2 ] = c I = \frac{1}{2} [c(1)^2 + c(-1)^2] = c I = 2 1 [ c ( 1 ) 2 + c ( − 1 ) 2 ] = c
2 : 1 2:1 2 : 1 Salt ( C a C l 2 CaCl_2 C a C l 2 , c c c ): I = 1 2 [ c ( 2 ) 2 + 2 c ( − 1 ) 2 ] = 1 2 [ 4 c + 2 c ] = 3 c I = \frac{1}{2} [c(2)^2 + 2c(-1)^2] = \frac{1}{2} [4c + 2c] = 3c I = 2 1 [ c ( 2 ) 2 + 2 c ( − 1 ) 2 ] = 2 1 [ 4 c + 2 c ] = 3 c
2 : 2 2:2 2 : 2 Salt ( M g S O 4 MgSO_4 M g S O 4 , c c c ): I = 1 2 [ c ( 2 ) 2 + c ( − 2 ) 2 ] = 1 2 [ 4 c + 4 c ] = 4 c I = \frac{1}{2} [c(2)^2 + c(-2)^2] = \frac{1}{2} [4c + 4c] = 4c I = 2 1 [ c ( 2 ) 2 + c ( − 2 ) 2 ] = 2 1 [ 4 c + 4 c ] = 4 c
3 : 1 3:1 3 : 1 Salt ( F e C l 3 FeCl_3 F e C l 3 , c c c ): I = 1 2 [ c ( 3 ) 2 + 3 c ( − 1 ) 2 ] = 1 2 [ 9 c + 3 c ] = 6 c I = \frac{1}{2} [c(3)^2 + 3c(-1)^2] = \frac{1}{2} [9c + 3c] = 6c I = 2 1 [ c ( 3 ) 2 + 3 c ( − 1 ) 2 ] = 2 1 [ 9 c + 3 c ] = 6 c
3 : 2 3:2 3 : 2 Salt ( A l 2 ( S O 4 ) 3 Al_2(SO_4)_3 A l 2 ( S O 4 ) 3 , c c c ): I = 1 2 [ 2 c ( 3 ) 2 + 3 c ( − 2 ) 2 ] = 1 2 [ 18 c + 12 c ] = 15 c I = \frac{1}{2} [2c(3)^2 + 3c(-2)^2] = \frac{1}{2} [18c + 12c] = 15c I = 2 1 [ 2 c ( 3 ) 2 + 3 c ( − 2 ) 2 ] = 2 1 [ 18 c + 12 c ] = 15 c
Mean Ionic Activity and the Debye-Hückel Limiting Law (DHLL)
Because individual ion activities cannot be measured independently due to electroneutrality, we use the mean ionic activity coefficient γ ± \gamma_\pm γ ± :
γ ± = ( γ + ν + γ − ν − ) 1 / ν where ν = ν + + ν − \gamma_\pm = \left( \gamma_+^{\nu_+} \gamma_-^{\nu_-} \right)^{1/\nu} \quad \text{where } \nu = \nu_+ + \nu_- γ ± = ( γ + ν + γ − ν − ) 1/ ν where ν = ν + + ν −
For very dilute solutions ( I < 0.01 M I < 0.01\text{ M} I < 0.01 M ), the Debye-Hückel Limiting Law states:
log 10 γ ± = − A ∣ z + z − ∣ I \log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I} lo g 10 γ ± = − A ∣ z + z − ∣ I
In aqueous solution at 25.0 ∘ C 25.0^\circ\text{C} 25. 0 ∘ C ( 298.15 K 298.15\text{ K} 298.15 K ), the solvent constant is A = 0.509 kg 1 / 2 mol − 1 / 2 A = 0.509\text{ kg}^{1/2}\text{mol}^{-1/2} A = 0.509 kg 1/2 mol − 1/2 (or L 1 / 2 mol − 1 / 2 \text{L}^{1/2}\text{mol}^{-1/2} L 1/2 mol − 1/2 ).
For higher concentrations ( 0.01 M < I < 0.1 M 0.01\text{ M} < I < 0.1\text{ M} 0.01 M < I < 0.1 M ), the Extended Debye-Hückel equation accounts for finite ion diameter ( a ∘ a^\circ a ∘ ):
log 10 γ ± = − A ∣ z + z − ∣ I 1 + B a ∘ I \log_{10} \gamma_\pm = -\frac{A |z_+ z_-| \sqrt{I}}{1 + B a^\circ \sqrt{I}} lo g 10 γ ± = − 1 + B a ∘ I A ∣ z + z − ∣ I
- 6. Worked Examination Calculations
- Problem A: Molar Mass of a Protein via Membrane Osmometry
Task: A 1.250 g 1.250\text{ g} 1.250 g sample of a purified enzyme is dissolved in sufficient isotonic buffer to yield 100.0 mL 100.0\text{ mL} 100.0 mL of solution. At 25.0 ∘ C 25.0^\circ\text{C} 25. 0 ∘ C , the osmotic pressure is measured as 4.350 mmHg 4.350\text{ mmHg} 4.350 mmHg . Determine the molar mass of the enzyme.
Solution: Π = 4.350 mmHg 760.0 mmHg/atm = 5.7237 × 10 − 3 atm \Pi = \frac{4.350\text{ mmHg}}{760.0\text{ mmHg/atm}} = 5.7237 \times 10^{-3}\text{ atm} Π = 760.0 mmHg/atm 4.350 mmHg = 5.7237 × 1 0 − 3 atm Using the van 't Hoff osmotic pressure equation Π = n V R T = w M V R T \Pi = \frac{n}{V} R T = \frac{w}{M V} R T Π = V n RT = M V w RT : M = w R T Π V = ( 1.250 g ) ( 0.082057 L ⋅ atm / ( mol ⋅ K ) ) ( 298.15 K ) ( 5.7237 × 10 − 3 atm ) ( 0.1000 L ) M = \frac{w R T}{\Pi V} = \frac{(1.250\text{ g})(0.082057\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}))(298.15\text{ K})}{(5.7237 \times 10^{-3}\text{ atm})(0.1000\text{ L})} M = Π V wRT = ( 5.7237 × 1 0 − 3 atm ) ( 0.1000 L ) ( 1.250 g ) ( 0.082057 L ⋅ atm / ( mol ⋅ K )) ( 298.15 K ) M = 30.582 5.7237 × 10 − 4 = 53430 g/mol = 5.34 × 10 4 g/mol M = \frac{30.582}{5.7237 \times 10^{-4}} = 53430\text{ g/mol} = 5.34 \times 10^4\text{ g/mol} M = 5.7237 × 1 0 − 4 30.582 = 53430 g/mol = 5.34 × 1 0 4 g/mol
Comparison: What would the freezing point depression be for this same solution in water ( K f = 1.86 ∘ C ⋅ kg/mol K_f = 1.86^\circ\text{C}\cdot\text{kg/mol} K f = 1.8 6 ∘ C ⋅ kg/mol )? m ≈ n 0.100 kg = 1.250 / 53430 0.100 = 2.34 × 10 − 4 mol/kg m \approx \frac{n}{0.100\text{ kg}} = \frac{1.250 / 53430}{0.100} = 2.34 \times 10^{-4}\text{ mol/kg} m ≈ 0.100 kg n = 0.100 1.250/53430 = 2.34 × 1 0 − 4 mol/kg Δ T f = ( 1.86 ) ( 2.34 × 10 − 4 ) = 0.000435 ∘ C \Delta T_f = (1.86)(2.34 \times 10^{-4}) = 0.000435^\circ\text{C} Δ T f = ( 1.86 ) ( 2.34 × 1 0 − 4 ) = 0.00043 5 ∘ C A Δ T f \Delta T_f Δ T f of 0.0004 ∘ C 0.0004^\circ\text{C} 0.000 4 ∘ C is indistinguishable from thermal noise on a standard lab thermometer, whereas Π = 4.35 mmHg \Pi = 4.35\text{ mmHg} Π = 4.35 mmHg corresponds to a column height of 5.9 cm 5.9\text{ cm} 5.9 cm of water ( h = P ρ g h = \frac{P}{\rho g} h = ρ g P ), proving why osmometry is the definitive colligative tool for biochemistry.
Problem B: Degree of Dissociation of a Weak Acid
Task: A 0.0500 m 0.0500\text{ m} 0.0500 m aqueous solution of monochloroacetic acid ( C H 2 C l C O O H CH_2ClCOOH C H 2 ClCOO H , ν = 2 \nu = 2 ν = 2 ) freezes at − 0.1080 ∘ C -0.1080^\circ\text{C} − 0.108 0 ∘ C . For water, K f = 1.860 ∘ C ⋅ kg/mol K_f = 1.860^\circ\text{C}\cdot\text{kg/mol} K f = 1.86 0 ∘ C ⋅ kg/mol . Calculate the van 't Hoff factor i i i and the percent dissociation α \alpha α .
Solution: Δ T f = 0.1080 ∘ C \Delta T_f = 0.1080^\circ\text{C} Δ T f = 0.108 0 ∘ C i = Δ T f K f m = 0.1080 ∘ C ( 1.860 ∘ C ⋅ kg/mol ) ( 0.0500 mol/kg ) = 0.1080 0.09300 = 1.1613 i = \frac{\Delta T_f}{K_f m} = \frac{0.1080^\circ\text{C}}{(1.860^\circ\text{C}\cdot\text{kg/mol})(0.0500\text{ mol/kg})} = \frac{0.1080}{0.09300} = 1.1613 i = K f m Δ T f = ( 1.86 0 ∘ C ⋅ kg/mol ) ( 0.0500 mol/kg ) 0.108 0 ∘ C = 0.09300 0.1080 = 1.1613
For a monoprotic acid ( C H 2 C l C O O H ⇌ H + + C H 2 C l C O O − CH_2ClCOOH \rightleftharpoons H^+ + CH_2ClCOO^- C H 2 ClCOO H ⇌ H + + C H 2 ClCO O − ), ν = 2 \nu = 2 ν = 2 : α = i − 1 ν − 1 = 1.1613 − 1 2 − 1 = 0.1613 ⟹ 16.13 % \alpha = \frac{i - 1}{\nu - 1} = \frac{1.1613 - 1}{2 - 1} = 0.1613 \implies 16.13\% α = ν − 1 i − 1 = 2 − 1 1.1613 − 1 = 0.1613 ⟹ 16.13%
Problem C: Ionic Strength and Mean Activity Coefficient
Task: Calculate the ionic strength and mean ionic activity coefficient γ ± \gamma_\pm γ ± of a 0.0050 M 0.0050\text{ M} 0.0050 M aqueous solution of C a ( N O 3 ) 2 Ca(NO_3)_2 C a ( N O 3 ) 2 at 25 ∘ C 25^\circ\text{C} 2 5 ∘ C .
Solution: C a ( N O 3 ) 2 ( a q ) → C a 2 + ( a q ) + 2 N O 3 − ( a q ) Ca(NO_3)_2(aq) \rightarrow Ca^{2+}(aq) + 2NO_3^-(aq) C a ( N O 3 ) 2 ( a q ) → C a 2 + ( a q ) + 2 N O 3 − ( a q ) .
[ C a 2 + ] = 0.0050 M [Ca^{2+}] = 0.0050\text{ M} [ C a 2 + ] = 0.0050 M , z + = + 2 z_+ = +2 z + = + 2
[ N O 3 − ] = 2 × 0.0050 = 0.0100 M [NO_3^-] = 2 \times 0.0050 = 0.0100\text{ M} [ N O 3 − ] = 2 × 0.0050 = 0.0100 M , z − = − 1 z_- = -1 z − = − 1
I = 1 2 [ ( 0.0050 ) ( + 2 ) 2 + ( 0.0100 ) ( − 1 ) 2 ] = 1 2 [ 0.0200 + 0.0100 ] = 0.0150 M I = \frac{1}{2} \left[ (0.0050)(+2)^2 + (0.0100)(-1)^2 \right] = \frac{1}{2} [ 0.0200 + 0.0100 ] = 0.0150\text{ M} I = 2 1 [ ( 0.0050 ) ( + 2 ) 2 + ( 0.0100 ) ( − 1 ) 2 ] = 2 1 [ 0.0200 + 0.0100 ] = 0.0150 M
Using the Debye-Hückel Limiting Law ( A = 0.509 A = 0.509 A = 0.509 ): log 10 γ ± = − A ∣ z + z − ∣ I = − ( 0.509 ) ∣ ( + 2 ) ( − 1 ) ∣ 0.0150 = − ( 0.509 ) ( 2 ) ( 0.12247 ) \log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I} = -(0.509) |(+2)(-1)| \sqrt{0.0150} = -(0.509)(2)(0.12247) lo g 10 γ ± = − A ∣ z + z − ∣ I = − ( 0.509 ) ∣ ( + 2 ) ( − 1 ) ∣ 0.0150 = − ( 0.509 ) ( 2 ) ( 0.12247 ) log 10 γ ± = − 0.12468 \log_{10} \gamma_\pm = -0.12468 lo g 10 γ ± = − 0.12468 γ ± = 10 − 0.12468 = 0.7504 ≈ 0.750 \gamma_\pm = 10^{-0.12468} = 0.7504 \approx 0.750 γ ± = 1 0 − 0.12468 = 0.7504 ≈ 0.750