1. SIMPLE MIXTURES
Chapter 5

2. Simple Mixtures
Often in chemistry, we encounter mixtures of substances that can react together.
Chapter 7 deals with reactions, but let’s first deal with properties of mixtures that don’t react.
We shall mainly consider binary mixtures – mixtures of two components.

3. Partial Molar Volumes
Imagine a huge volume of pure water at 25 °C. If we add 1 mol H2O, the volume increases 18 cm3 (or 18 mL).
So, 18 cm3 mol-1 is the molar volume of pure water.

4. Partial Molar Volumes
Now imagine a huge volume of pure ethanol and add 1 mol of pure H2O it. How much does the total volume increase by?

5. Partial Molar Volumes
When 1 mol H2O is added to a large volume of pure ethanol, the total volume only increases by ~ 14 cm3.
The packing of water in pure water ethanol (i.e. the result of H-bonding interactions), results in only an increase of 14 cm3.

6. Partial Molar Volumes
The quantity 14 cm3 mol-1 is the partial molar volume of water in pure ethanol.
The partial molar volumes of the components of a mixture varies with composition as the molecular interactions varies as the composition changes from pure A to pure B.

9. Partial Molar Volumes
When a mixture is changed by dnA of A and dnB of B, then the total volume changes by:

10. Partial Molar Volumes

11. Partial Molar Volumes

12. Partial Molar Volumes

13. Partial Molar Volumes How to measure partial molar volumes?
Measure dependence of the volume on composition.
Fit a function to data and determine the slope by differentiation.

14. Partial Molar Volumes Ethanol is added to 1.000 kg of water.
The total volume, as measured by experiment, fits the following equation:

15. Partial Molar Volumes

17. Partial Molar Volumes
Molar volumes are always positive, but partial molar quantities need not be. The limiting partial molar volume of MgSO4 in water is -1.4 cm3mol-1, which means that the addition of 1 mol of MgSO4 to a large volume of water results in a decrease in volume of 1.4 cm3.

18. Partial Molar Gibbs energies
The concept of partial molar quantities can be extended to any extensive state function.
For a substance in a mixture, the chemical potential is defined as the partial molar Gibbs energy.

20. Partial Molar Gibbs energies
For a pure substance:

21. Partial Molar Gibbs energies
Using the same arguments for the derivation of partial molar volumes,
Assumption: Constant pressure and temperature

22. Partial Molar Gibbs energies

23. Chemical Potential

24. Chemical Potential

25. Gibbs-Duhem equation

26. Gibbs-Duhem equation

27. Molarity and Molality
Molarity, c, is the amount of solute divided by the volume of solution. Units of mol dm-3 or mol L-1.
Molality, b, is the amount of solute divided by the mass of solvent. Units of mol kg-1.

28. Using Gibbs-Duhem
The experimental values of partial molar volume of K2SO4(aq) at 298 K are found to fit the expression:

29. Using Gibbs-Duhem

30. Using Gibbs-Duhem

31. Using Gibbs-Duhem

32. Using Gibbs-Duhem

33. Using Gibbs-Duhem

35. Thermodynamics of mixing
So we’ve seen how Gibbs energy of a mixture depends on composition.
We know at constant temperature and pressure systems tend towards lower Gibbs energy.
When we combine two ideal gases they mix spontaneously, so it must correspond to a decrease in G.

36. Thermodynamics of mixing

37. Dalton’s Law
The total pressure is the sum of all the partial pressure.

38. Thermodynamics of mixing

39. Thermodynamics of mixing

40. Thermodynamics of mixing

41. Thermodynamics of mixing

42. Gibbs energy of mixing
A container is divided into two equal compartments. One contains 3.0 mol H2(g) at 25 °C; the other contains 1.0 mol N2(g) at 25 °C. Calculate the Gibbs energy of mixing when the partition is removed.

43. Gibbs energy of mixing Two processes: 1) Mixing
2) Changing pressures of the gases.

44. Gibbs energy of mixing p p

45. Other mixing functions

47. Other mixing functions

48. Ideal Solutions
To discuss the equilibrium properties of liquid mixtures we need to know how the Gibbs energy of a liquid varies with composition.
We use the fact that, at equilibrium, the chemical potential of a substance present as a vapor must be equal to its chemical potential in the liquid.

49. Ideal Solutions
Chemical potential of vapor equals the chemical potential of the liquid at equilibrium.
If another substance is added to the pure liquid, the chemical potential of A will change.

50. Ideal Solutions

51. Raoult’s Law

52. Ideal Solutions

53. Non-Ideal Solutions

54. Ideal-dilute solutions
Even if there are strong deviations from ideal behaviour, Raoult’s law is obeyed increasingly closely for the component in excess as it approaches purity.

55. Henry’s law
For real solutions at low concentrations, although the vapor pressure of the solute is proportional to its mole fraction, the constant of proportionality is not the vapor pressure of the pure substance.

56. Henry’s Law
Even if there are strong deviations from ideal behaviour, Raoult’s law is obeyed increasingly closely for the component in excess as it approaches purity.

57. Ideal-dilute solutions
Mixtures for which the solute obeys Henry’s Law and the solvent obeys Raoult’s Law are called ideal-dilute solutions.

58. Properties of Solutions
We’ve looked at the thermodynamics of mixing ideal gases, and properties of ideal and ideal-dilute solutions, now we shall consider mixing ideal solutions, and more importantly the deviations from ideal behavior.

59. Ideal Solutions

60. Ideal Solutions

61. Ideal Solutions

62. Real Solutions
Real solutions are composed of particles for which A-A, A-B and B-B interactions are all different.
There may be enthalpy and volume changes when liquids mix.
DG=DH-TDS
So if DH is large and positive or DS is negative, then DG may be positive and the liquids may be immiscible.

63. Excess Functions
Thermodynamic properties of real solutions are expressed in terms of excess functions, XE.
An excess function is the difference between the observed thermodynamic function of mixing and the function for an ideal solution.

64. Real Solutions

65. Real Solutions
Benzene/cyclohexane
Tetrachloroethene/cyclopentane

66. Colligative Properties
A colligative property is a property that depends only on the number of solute particles present, not their identity.
The properties we will look at are: lowering of vapor pressure; the elevation of boiling point, the depression of freezing point, and the osmotic pressure arising from the presence of a solute.
Only applicable to dilute solutions.

67. Colligative Properties
All the colligative properties stem from the reduction of the chemical potential of the liquid solvent as a result of the presence of solute.

68. Thermodynamics of mixing

70. Boiling Point Elevation
How do we figure out where the new boiling point is when a solute is present?
Look for the temperature at which at 1 atm, the vapor of pure solvent vapor has the same chemical potential as the solvent in the solution.

71. Boiling Point Elevation
Let’s denote solvent A and solute B.
Equilibrium is established when:
See Justification 5.1 (Atkins)

73. Boiling Point Elevation
DT makes no reference to the identity of the solute, only to its mole fraction.
So the elevation of boiling point is a colligative property.

74. Boiling Point Elevation

76. Freezing Point Depression
Let’s denote solvent A and solute B.
Equilibrium is established when:
Same calculation as before (Justification 5.1)

77. Freezing Point Depression

79. Cryoscopy

80. Solubility
Although solubility is not strictly a colligative property (because solubility varies with the identity of the solute), it may be estimated using the same techniques.
When a solid solute is left in contact with a solvent, it dissolves until the solution is saturated with the dissolved solute.

82. Solubility
See Justification 5.2

83. Osmosis
Osmosis refers to the spontaneous passage of a pure solvent into a solution separated from it by a semi-permeable membrane.
In this case, the membrane is permeable to the solvent but not to the solute.

85. Osmosis
The osmotic pressure, P, is the pressure that must be applied to the solution to stop the influx of solvent.
Examples of osmosis includes the transport of fluids across cell membranes and dialysis.
See Justification 5.3

86. van’t Hoff equation
For molecules of large molar mass, such as polymers or biological macromolecules, a viral-like expansion used to correct for non-ideality. B is the osmotic viral coefficient.

87. Solvent Activities
The general form of the chemical potential of a real or ideal solvent is given by:
For an ideal solution, when the solvent obeys Raoult’s law, then:

88. Solvent Activities
If a solution does not obey Raoult’s law, we can still use a form of the chemical potential equation:
The quantity aA is the activity of A. It can be considered an “effective” mole fraction.

89. Solvent Activities
Because this equation is true for real or ideal solvent, we can easily see that:

90. Activities

91. Calculating solvent activity
The vapor pressure of M KNO3(aq) at 100 °C is kPa so the activity of water in the solution at this temperature is

92. Solvent Activities
Because all solvents obey Raoult’s law increasingly close as the concentration of solute approaches zero, the activity of the solvent approaches the mole fraction as xA=1

93. Solvent Activities
A way of expressing this convergence is to introduce the activity coefficient, g

94. Solute Activities
Ideal-dilute solutions obeys Henry’s law has a vapor pressure given by pB=KBxB, where KB is Henry’s law constant.

95. Solute Activities
For real solutions we can replace xB with aB

96. Solute Activities
The selection of a standard state is entirely arbitrary, so we are free to choose one that best suits our purpose and description of the composition of the system

97. Activities of regular solutions
Ignore section 5.8

98. Ion Activities
If the chemical potential of a univalent M+ is denoted m+ and that of a univalent anion X- is denoted m-, the total molar Gibbs energy of the ions in the electrically neutral solution is the sum of these partial molar quantities.

99. Ion Activities
For a real solution of M+ and X- of the same molality

100. Ion Activities
Because experimentally a cation cannot exist in solution without an anion, it is impossible to separate the product g+g- into contributions from each ion, we introduce the mean activity coefficient

101. Ion Activities
The individual chemical potentials of the ions are:

102. Ion Activities
If a compound MpXq that dissolves to give a solution of p cations and q anions.

103. Ion Activities

104. Debye-Hückel theory
The departure from ideal behavior in ionic solutions can be mainly attributed to the Coulombic interaction between positively and negatively charged ions.
Oppositely charged ions attract one another. As a result anions are more likely found near cations in solution, and vice versa.

105. Debye-Hückel theory

106. Debye-Hückel theory
Although overall the solution is neutrally charged, but near any given ion there is an excess of counter ions.
Averaged over time this causes a spherical haze around the central ion, in which counter ions outnumbers ions of the same charge as the central ion, has a net charge the same but magnitude but opposite sign of the central ion, and is called the ionic atmosphere.

107. Debye-Hückel theory
The chemical potential of any given central ion is lowered as a result of its electrostatic interaction with its ionic atmosphere.
This lowering of chemical potential is due to the activity of the solute and can be identified with RTlng±.

108. Debye-Hückel theory
At very low concentrations the activity coefficient can be calculated from the Debye-Hückel limiting law

109. Debye-Hückel theory
When the ionic strength is too high for the limiting law to be valid, the activity coefficient can be estimated from the Debye-Hückel extended law

110. Debye-Hückel theory

111. Debye-Hückel theory