Solutions

1. Properties and Definition of Solutions

• Solution properties reflect properties of each component mixed.
• Particle size: < 1 nanometer (10⁻⁹ m).
• Composition: One solvent and one or more solutes (e.g., water + sugar + lemon + salt).
• Solute: The component that dissolves.
• Solvent: The component that does the dissolving.
• The physical state (phase) of a solution is determined by the solvent.

2. Identifying Solute and Solvent

• Criteria 1 (Different Phases): The component which has the same phase as the final solution is the solvent.
• Criteria 2 (Same Phase): The component present in the greater amount is the solvent.

3. Components of a Solution

• A solution consists of two primary components: the solute and the solvent.
• Solvent: The component present in a larger quantity (greater number of moles).
• Solute: The component dissolved in the solvent, present in a smaller quantity.

Examples of Solute and Solvent Identification :

Solution	Solute	Solvent
Tea	Sugar, Tea powder	Water
Shikanji	Lemon juice, Sugar	Water
Air (O2, N2, CO2)	O2, CO2	N2
Vinegar	Acetic acid	Water

4. Types of Solution by Physical State

• Gaseous solutions
• Liquid solutions
• Solid solutions

Gaseous Solutions :

Solute	Solvent	Example
Gas	Gas	O2 + N2
Liquid	Gas	Chloroform in N2
Solid	Gas	Camphor in N2

Liquid Solutions :

Solute	Solvent	Example
Gas	Liquid	O2 in water
Liquid	Liquid	Ethanol in water
Solid	Liquid	Glucose in water

Solid Solutions :

Solute	Solvent	Example
Gas	Solid	H2 in Pd
Liquid	Solid	Amalgam of Hg with Na
Solid	Solid	Cu in Au

5. Solubility of Gas in Liquid

• Solubility: The maximum amount of gas dissolved in a solvent per litre at a particular temperature.

6. Henry's Law Concepts

• The partial pressure of a gas in the vapour phase (p) is proportional to the mole fraction of the gas in the solution (χ).
• p = K_Hχ.
• Key characteristics:
  • Different gases have different K_H values at the same temperature.
  • The value of K_H increases with an increase in temperature.
  • K_H is a function of the nature of the gas.
  • For a given pressure, a lower value of K_H implies higher solubility of the gas in the liquid.

7. Henry's Law & Solubility Trends

• For gases, more solubility = less K_H (Henry's Law constant).
• K_H depends on the nature of the gas. Polar gases (e.g., SO₂) have higher solubility and lower K_H values compared to non-polar gases (e.g., H₂).
• In a P_gas vs X_gas graph, a steeper slope represents a higher K_H value and thus lower solubility.

8. Vapour Pressure of Pure Liquid

• Molecules in the bulk:
  • Are stable.
  • Have lower energy.
  • Are in equilibrium.
• Molecules at the surface:
  • Are not in equilibrium.
  • Are less stable.
  • Have high energy.
• Equilibrium process: Liq ⇌ Vap, where (rate)_evap = (rate)_cond.
• Vapour Pressure (V.P.): The pressure exerted by vapours on the surface of the liquid and the walls of the container at a given temperature (T).

9. Effect of Temperature on Vapour Pressure

• Vapour pressure of a pure liquid depends only on temperature.
• In general, as Temperature (T) increases, Vapour Pressure (V.P.) increases.
• The process Liq ⇌ Vap is endothermic (ΔH = positive).
• For an endothermic reaction, the equilibrium constant (K_p) increases with an increase in temperature. Since K_p = V.P. for this equilibrium, V.P. increases.

• The relationship between V.P. and T is exponential.

10. Molecular Basis (Maxwell-Boltzmann Distribution)

• E₀ is the minimum energy required for molecules to leave the liquid surface.
• The Maxwell-Boltzmann distribution shows that increasing the temperature increases the fraction of molecules having energy greater than or equal to E₀.
• Consequently, the rate of evaporation increases, leading to an increase in vapour pressure.
• Note: While PV = nRT, for vapour pressure P ∝ T is incorrect. The dependence is exponential, not linear.

11. Intermolecular Forces and Vapour Pressure

• Vapour pressure is inversely proportional to the strength of intermolecular forces.
• Intermolecular forces ↑ ⇒ V.P. ↓
• For two liquids A and B at the same temperature, if their intermolecular forces are f₁ and f₂ respectively:
• If f₁ > f₂, then (V.P.)_B > (V.P.)_A.

12. Strength of Intermolecular Forces

Order of force strength:

• Examples:
  • Ionic: Na⁺:::Cl⁻
  • Ion-Dipole: Na⁺---H₂O
  • Hydrogen Bond: H₂O---H₂O---H₂O
  • Dipole-Dipole: Acetone (CH₃-C(=O)CH₃ interaction)
  • Dipole-Induced Dipole: H₂O---Br-Br (δ+ δ-)
• General order: Hydrogen Bonding > Dipole-Dipole > Vander Waal attraction.
• (Force of attraction)_Polar > (Force of attraction)_Non-polar.
• For non-polar molecules: Larger size/surface area (higher molecular weight) leads to stronger forces of attraction.

13. Vapour Pressure of a Solution (Non-volatile Solute)

• When a non-volatile solute is added to a volatile solvent, the vapour pressure of the solution (P_s) is lower than that of the pure solvent (P°).
• Reasons for decrease in V.P.:
  • Number of solvent particles per unit surface area is less in the solution.
  • Due to solute-solvent interactions.

14. Raoult's Law

• The partial vapour pressure of each component in a solution is proportional to its mole fraction in the solution.

15. Vapour Pressure of a Solution of Two Volatile Liquids

• For a solution of two miscible, volatile liquids A and B:

• P_T = total vapour pressure, P°_A/P°_B = vapour pressure of pure components, X_A/X_B = mole fractions in the solution.

16. P_T in Terms of Single Mole Fraction

In terms of X_A :

In terms of X_B :

17. Composition of Vapour Phase

• For a liquid mixture, the vapour phase composition is determined by the partial pressures of components:
• Partial pressure of A (P_A) = P°_A X_A
• Partial pressure of B (P_B) = P°_B X_B
• Total Vapour Pressure (P_T) = P_A + P_B

• Ratio of vapour mole fractions:

X_A = mole fraction of A in liquid phase; Y_A = mole fraction of A in vapour phase.

18. Vapour Pressure of Binary Solutions

• For a solution of two volatile components 1 and 2, where x₁ and y₁ are mole fractions in liquid and vapour phase respectively, the relationship follows a linear plot for 1/x₁ vs 1/y₁:

19. Distillation of Ideal Solutions

• Distillation is a process to separate components of a mixture based on differences in volatility.
• If A is more volatile than B (P°_A > P°_B), the vapour will be richer in A than the liquid phase.
• By repeatedly condensing the vapour (which is richer in the more volatile component) and re-boiling, the liquid becomes progressively purer in the more volatile component.

Pressure-Composition Curve :

• The P-x_A graph displays three regions: liquid (L) at the top, vapour (V) at the bottom, and a liquid-vapour equilibrium region (L+V) in between.
• A zig-zag path on the graph illustrates the distillation process: moving from the liquid composition (x_A) to the vapour composition (y_A) in the L+V region, which then becomes the liquid composition (x'_A) for the next stage.

20. Ideal Solution: Fractional Distillation

• For the nth container in a fractional distillation process, the mole fraction of component A in the vapour phase (Y_A,n) is related to the initial mole fraction in the liquid phase (X_A,1) by:

Where A is the more volatile component (P°_A > P°_B). As n → ∞, the vapour becomes enriched with the more volatile component (Y_A,n → 1).

21. Composition of Vapour Phase

• When a liquid mixture evaporates, the composition of the first drop of vapour formed (Y_A, Y_B) relates to the liquid composition (X_A, X_B) as:

22. Composition of Last Drop of Liquid

• As evaporation proceeds, the composition of the remaining liquid changes. For the last drop of liquid remaining (X'_A, X'_B), the vapour pressure is determined by its composition:

• The relation between the final vapour composition and the final liquid composition (X'_A) follows the same principle:

23. Immiscible Volatile Liquids

• When two liquids are immiscible, they do not mix.
• The total vapor pressure is the sum of the pure vapor pressures of the individual components:

• The mole fraction of components in the vapor phase is given by:

24. Vapor Phase Composition (Immiscible)

• The ratio of moles of A (n_A) to moles of B (n_B) in the vapor phase is:

• In terms of weights (W_A, W_B) and molar masses (M_A, M_B):

W_A, W_B = weight of A and B in the vapor phase.

25. Ideal Solutions

• An ideal solution is formed when similar interactions are replaced by the same type of interaction.
• Condition: A–B interaction = A–A and B–B interaction.
• Example: Hexane + Heptane.

• ΔH_mix = 0 because energy required for bond breaking is equal to energy released during bond formation.
• ΔV_mix = 0 for A–A + B–B → A–B (e.g., 10 + 10 = 20).
• ΔG_mix < 0 since mixing occurs spontaneously.
• ΔS_mix > 0 because A–A + B–B → 2A–B increases randomness (symmetrical to unsymmetrical).

26. Raoult's Law Graph (Ideal)

• Assuming A is more volatile: P°_A > P°_B.

_AP_T P°_BP°_A X_A=00.2X_A=1 X_B=10.8X_B=0

27. Examples of Ideal Solutions

• Hexane + Heptane
• Benzene + Toluene
• Ethyl bromide + Ethyl Iodide
• Chlorobenzene + Bromobenzene

28. Non-Ideal Solutions (Positive Deviation)

• Characteristics:

• Graph behavior: The total vapour pressure curve lies above the straight line ideal Raoult's law behavior.
• P_T > P°_A X_A + P°_B X_B

29. Examples of Positive Deviation

• Ethanol + Cyclohexane
• Ethanol + water or Methanol + water
• Ethanol + Acetone
• CCl₄ + Benzene or CCl₄ + Toluene
• CCl₄ + CHCl₃
• CCl₄ + Methanol
• Benzene + Acetone
• Acetone + CS₂
• Acetaldehyde + CS₂

30. Negative Deviation from Raoult's Law

• Occurs when weaker solvent-solvent (A-A, B-B) interactions are replaced by stronger solute-solvent (A-B) interactions.
• Condition: Interaction A-B > A-A and B-B interactions.
• P_T < P_A°x_A + P_B°x_B.

31. Graph: Negative Deviation

• The total vapour pressure (P_T) curve lies below the ideal line (dashed).

_AP_T P_B°P_A° 01 0.2

32. Examples of Negative Deviation

• CHCl₃ + CH₃COCH₃ (Acetone)
• CHCl₃ + C₆H₆ (Benzene)
• CHCl₃ + C₂H₅OC₂H₅ (Diethylether)
• H₂O + HCl
• H₂O + HNO₃
• CH₃COCH₃ + Aniline
• Phenol + Aniline

33. Chloroform and Acetone Mixture

• Components: CHCl₃ (Dipole-Dipole) and CH₃COCH₃ (Dipole-Dipole).
• The mixture forms Hydrogen Bonds (H.B).
• Interaction strength: H.B > Dipole-Dipole.

34. Non-Ideal Solutions

• The main reason for deviation from ideal behavior in non-ideal solutions is the difference in intermolecular forces between the solute and solvent (A-B interaction) compared to pure components (A-A and B-B interactions).
• Positive Deviation : Occurs when the A-B attraction is weaker than the A-A and B-B attractions (A-B attraction < A-A, B-B).
• Negative Deviation : Occurs when the A-B attraction is stronger than the A-A and B-B attractions (A-B attraction > A-A, B-B).

35. Pressure-Composition Curves for Non-Ideal Solutions

• Ideal Solution: Shows a linear or standard convex curve connecting P°_B and P°_A.
• Negative Deviation:
  • Weak (-)ve deviation: Curve lies slightly below the ideal dashed line.
  • Strong (-)ve deviation: Curve lies significantly below the ideal dashed line.
• Positive Deviation:
  • Weak (+)ve deviation: Curve lies above the ideal dashed line.
  • Strong (+)ve deviation: Curve exhibits a peak where X_A = Y_A.

36. Azeotropic Mixture

• A binary solution containing two volatile liquids where the liquid and vapour phases have the same composition.
• A boiling liquid mixture at the azeotropic composition produces vapour with the exact same composition as the liquid.
• Distillation cannot separate the two liquids in an azeotropic mixture because the condensate retains the same composition as the original mixture.

37. Minimum Boiling Azeotrope

• These azeotropes are formed by solutions showing large positive deviation from Raoult's Law.
• The boiling point of the solution is lower than the boiling point of either pure component.
• Example shown: Water and Ethanol. The curve shows a maximum vapour pressure point at approximately 0.95 ethanol and 0.05 water.

38. Maximum Boiling Azeotrope

• These azeotropes are formed by solutions which show negative deviation from Raoult's Law.
• The boiling point of the azeotropic mixture is more than that of either individual component.

39. Colligative Properties

• Properties of a solution that depend only on the number of solute particles and not on their nature.
• Conditions:
  • Solute must be non-volatile.
  • Solution must be dilute.

40. Relative Lowering of Vapour Pressure (RLVP)

• Vapour pressure of solution (P_s) = P° × X_solvent
• Relative Lowering of Vapour Pressure (RLVP) formula:

Where: P° = v.p. of pure solvent, P_s = v.p. of solution, W = weight, M = molar mass.

Alternative Form :

• Dividing by X_solvent gives an exact form (useful when P_s is given): P° − P_sP_s = n_soluten_solvent = W_solute / M_soluteW_solvent / M_solvent

41. Comparison of RLVP

• RLVP ∝ X_solute.
• Since X_solute = n_soluten_solute + n_solvent, for a fixed mass of solvent and equal masses of solutes, a smaller molar mass (M_solute) results in more moles (n_solute) and thus a higher RLVP.

42. Ostwald–Walker Experiment

• This experiment is used to determine the Relative Lowering of Vapour Pressure (RLVP).
• Dry air is passed sequentially through a solution, then pure solvent, and finally a U-tube containing an anhydrous dehydrating agent (CaCl₂).

W₁ = weight loss in solution container ; W₂ = weight loss in solvent container.
P_s = Vapour pressure of solution ; P° = Vapour pressure of pure solvent.

43. Elevation in Boiling Point

• Vapour pressure of a solution containing a non-volatile solute is lower than that of the pure solvent.
• To reach atmospheric pressure (1 atm), the solution requires more heat, thus increasing its boiling point.
• This elevation is a colligative property.

_bT_b'T_b''

T_b = B.P of solvent ; T_b' = B.P of solution 1 ; T_b'' = B.P of solution 2.

44. Ebullioscopic Constant (K_b)

• The boiling point elevation (ΔT_b) is proportional to molality (m):

• K_b (Ebullioscopic Constant / Molal boiling point elevation constant) is a property of the solvent only.
• Relationship for K_b:

M = molar mass of solvent ; T_b = boiling point of solvent ; ΔH_vap = heat of vaporisation of solvent.

• Since K_b ∝ T_b²ΔH_vap:
• (ΔH_vap)_H2O ≫ (ΔH_vap)_Benzene (due to H-bonding vs Vander Waal forces).
• Consequently, (K_b)_H2O < (K_b)_Benzene.

45. Depression in Freezing Point

• Freezing point: The temperature at which the first crystal of ice separates and the solid phase is in dynamic equilibrium with the liquid phase.
• Condition: Freezing point is the temperature where the vapour pressure of the liquid phase equals the vapour pressure of the solid phase.
• Only the solvent freezes.

46. Depression in Freezing Point

• Freezing point itself is not a colligative property; however, the depression in freezing point (ΔT_F) is a colligative property.
• Graphically, the freezing point corresponds to the temperature where the vapour pressure of the liquid solvent/solution intersects with the vapour pressure of the solid (frozen) phase.

• As solute concentration increases, the vapour pressure of the solution decreases, leading to a shift in the freezing point intersection to lower temperatures (T_F° > T_F' > T_F'').
• Entropy relationships at equilibrium (solvent ⇌ ice): ΔG = ΔH − TΔS. At equilibrium ΔG = 0, so T_F = ΔH_fusΔS.
• Since T_F ∝ 1ΔS and ΔS₁ < ΔS₂ < ΔS₃ (entropy differences between solution phases and ice), it follows that (T_F)_solvent > (T_F)_soln1 > (T_F)_soln2.

47. Cryoscopic Constant (K_F)

• The depression in freezing point is directly proportional to the molality (m) of the solution:

• K_F is the cryoscopic constant or molal depression in freezing point constant. It is a characteristic property of the solvent only.
• Formula for K_F:

Where M = molar mass of solvent, R = gas constant, T_F = freezing point of solvent, ΔH_fus = heat of fusion of solvent.

48. Concept of Ice Separation

• For the freezing process: ΔT_F = K_F ⋅ m.
• As molality (m) increases, the depression in freezing point (ΔT_F) increases, causing the freezing point (F.P.) of the solution to decrease.
• Key Concept: During the cooling of a solution, the temperature until which the solution is being cooled is considered the freezing point of the remaining liquid solution.

49. Molal Depression and Elevation Constants

where M = molar mass of solvent (kg), R = gas constant, T_b = boiling point, T_f = freezing point, ΔH = enthalpy of vaporization/fusion, ΔS = entropy of vaporization/fusion.

• Derived from ΔG = ΔH - TΔS. At freezing point (F.P.), ΔG = 0, so ΔH_fus = T_fΔS_fus ⇒ ΔS_fus = ΔH_fusT_f.

50. Osmotic Pressure

• Semipermeable membrane: Artificial (Cu₂Fe(CN)₆) or Natural (animal bladder).
• Osmosis: Movement of solvent molecules through a semipermeable membrane.
  • In terms of solvent concentration: High solvent conc. → Low solvent conc.
  • In terms of solution concentration: Low solute conc. → High solute conc.

[DIAGRAM: Container divided by a membrane. Left: 'Solvent'. Right: 'Solution' containing solute particles.]

51. Osmotic Pressure (π)

• Osmotic pressure is the extra pressure applied on the more concentrated solution side to stop the process of osmosis.
• It can also be measured by the hydrostatic pressure generated by the height (h) of the liquid column.

where h = height of liquid column, ρ = density, g = acceleration due to gravity.

• At equilibrium: (rate)_{solvent → solution} = (rate)_{solution → solvent}.

52. Reverse Osmosis

• When the pressure applied on the solution side (π') is greater than the osmotic pressure (π), the solvent flows from the solution side to the solvent side.

53. Calculation of Osmotic Pressure

• C = concentration (molarity), R = gas constant, T = temperature.
• Depending on units:
  • If π is in atm: R = 0.082 atm L / K-mol
  • If π is in Pa or N/m²: R = 8.314 J / K-mol
• Standard conversion: 1 atm = 1.013 × 10⁵ N/m².

54. Hypertonic, Hypotonic and Isotonic Solutions

• For two solutions (1 and 2) with concentrations C₁, C₂ and temperatures T₁, T₂:

• If π_{soln 1} > π_{soln 2}:
  • Soln 1 is Hypertonic.
  • Soln 2 is Hypotonic.
  • At the same temperature: C₁ > C₂.

55. Resultant Osmotic Pressure of Mixed Solutions

• When two solutions of the same substance with different osmotic pressures (π₁, π₂) and volumes (V₁, V₂) are mixed:

56. Van't Hoff Factor (i)

• Due to association or dissociation of solute particles, the observed colligative property differs from the theoretical (calculated) colligative property.

• For any solute:
  i = M_calM_obs

57. Colligative Properties with Van't Hoff Factor

• i) ΔT_b = i K_b m
• ii) ΔT_f = i K_f m
• iii) π = i CRT
• iv) RLVP = i n_solutei n_solute + n_solvent ≈ i n_soluten_solvent

58. Dissociation

• Reaction: A_n ⇌ nA
• If α is the degree of dissociation:

n = number of particles the solute dissociates into.

59. Association

• Reaction: nA ⇌ A_n
• If β is the degree of association:

n = number of particles associating together.

• Example of association: CH₃COOH (non-polar solvent).
• Reaction: 2CH₃COOH ⇌ (CH₃COOH)₂
• [DIAGRAM: Dimerization of acetic acid via two hydrogen bonds between carboxylic acid groups.]

60. Non-Electrolytes

• For solutes like glucose, sucrose, or urea:

• Here, (C.P.)_calculated = (C.P.)_observed.

61. Vapour Pressure of Electrolyte Solutions

• For an aqueous solution of an electrolyte, the lowering of vapour pressure is calculated using the effective number of particles.

Where P° is vapour pressure of pure solvent, P_s is vapour pressure of solution, and χ_solute is mole fraction of solute.

62. Relative Lowering of Vapour Pressure & Solubility

• For a salt with solubility S, dissolving as: XY₂₍ₛ₎ ⇌ X₍ₐq₎ + 2Y₍ₐq₎
• Number of solute particles formed = S + 2S = 3S

where S = solubility of the salt

63. Quick Formula Revision

Quantity	Formula
Clausius–Clapeyron	log(P2/P1) = (ΔHvap/2.303R)(1/T1 − 1/T2)
Raoult's law (non-volatile solute)	Ps = P° Xsolvent
RLVP	(P° − Ps)/P° = Xsolute ; (P° − Ps)/Ps = nsolute/nsolvent
Two volatile liquids	PT = P°AXA + P°BXB = P°B + XA(P°A − P°B)
Vapour composition	YA = P°AXA/PT ; YA/(1 − YA) = (P°A/P°B) XA/(1 − XA)
PT from vapour composition	1/PT = YA/P°A + YB/P°B
n-stage distillation	YA,n/(1 − YA,n) = (P°A/P°B)n XA,1/(1 − XA,1)
Immiscible liquids	PT = P°A + P°B ; WA/WB = P°AMA/(P°BMB)
Ostwald–Walker	(P° − Ps)/P° = W2/(W1 + W2)
Elevation in B.P.	ΔTb = iKbm ; Kb = MRTb2/(1000 ΔHvap) (M in g mol−1)
Depression in F.P.	ΔTf = iKfm ; Kf = MRTf2/(1000 ΔHfus)
Osmotic pressure	π = iCRT ; π = hρg ; mixing: π = (π1V1 + π2V2)/(V1 + V2)
van't Hoff factor	i = observed C.P./calculated C.P. = Mnormal/Mobserved
Dissociation / association	i = 1 + (n − 1)α ; i = 1 − (1 − 1/n)β