Physical Chemistry 2 A PORTFOLIO OF MAJOR CONCEPTS
LEANDRO E. NERO BS Chemical Engineering 3 ENGR. MICHELLE CANARIA Professor
Contents Physical Chemistry PORTFOLIO
1
2 3
SIMPLE MIXTURES
PHASE DIAGRAMS
1.1 1.2 1.3 1.4 1.5 1.6
Partial Molar Quantities Gibbs Energy of Solutions Thermodynamics of Mixing Chemical Potential of Liquids Colligative Properties Activities of Solutions
2.1 2.2 2.3 2.4
Terminologies One-Component System Phase Rule General Two-Component Systems Liquid-Liquid Diagrams Liquid-Solid Diagrams Ternary Systems
2.5 2.6 2.7
CHEMICAL EQUILIBRIA
3.1 3.2 3.3
Chemical Equilibrium Equilibria and Gibbs Energy Le Chatelier’s Principle
CHAPTER
1
SIMPLE MIXTURES
Chapter Overview 1.1
Partial Molar Quantities
1.2
Gibbs Energy of Solutions
1.3
Thermodynamics of Mixing
1.4
Chemical Potential of Liquids
1.5
Colligative Properties
1.6
Activities of Solutions
The majority of chemical processes are reactions that occur in solution. Important industrial processes often utilize solution chemistry. “Life” is the sum of a series of complex processes occurring in solution. Air, tap water, tincture of iodine, beverages, and household ammonia are common examples of solutions.
Figure 1.1 Mixture of CO2 and Argon
These two gases comprises an example of a simple mixture.
Simple Mixtures Chapter 1
1
Partial Molar Quantities
1.1
PARTIAL MOLAR VOLUME Volume (V) •
space occupied by a substance
Composition (n) •
amount of substance usually in moles
Partial molar volume of a any substance in a mixture is the
rate of change in volume per mole of that substance when added to a larger mixture. ∂V VJ = ∂nJ p, T, n’
(
(
V = VAnA + VBnB In this equation, A and B signies two signicant components
of a certain mixture.
Gibbs Energy of Solutions
1.2
The Gibbs free energy of a system at any moment in time is dened as the enthalpy of the system minus the product of the temperature times the entropy of the system. G = H - TS PARTIAL MOLAR GIBBS ENERGY In chemical solutions, partial molar gibbs energy is dened as chemical potential (μ). Chemical potential is the slope of Gibbs
energy against the amount of component of the substance at constant pressure and temperature. For binary mixtures, the formula for the total Gibbs energy of a mixture is: G = nAμA + nBμB
Figure 1.1 Sample Plot of μ The tangent line formed by the slope of a point in the curve of gibbs energy against composition is called chemical potential. http://bit.ly/2DnRdR7
When temperature and pressure are not held constant, the equation becomes: d G = μAd nA + μBd nB + ... fundamental equation of chemical thermodynamics
THE GIBBS-DUHEM EQUATION The signicance of the Gibbs–Duhem equation is that the
chemical potential of one component of a mixture cannot change independently of the chemical potentials of the other components. n
d μB = - nA d μ A B
Entropy (S) •
measure of disorder in a system
FURTHER SIGNIFICANCE OF CHEMICAL POTENTIAL Deriving from the formula of Gibbs free energy, chemical potential can also be used to calculate the change in internal energy (U) of the system. dU = -pdV - Vdp + SdT + TdS + dG dU = -pdV - Vdp + μAd nA + μBd nB + ... dU = μAd nA + μBd nB + ... at constant T and P
2
Simple Mixtures Chapter 1
1.3 Thermodynamics of Mixing In perfect gases, molar gibbs energy (G m) is just equal to the chemical potential (μ) of the gas. Փ
μ = μ + RT ln
p p
Փ Փ
The standard chemical potential (μ ) is the chemical potential of a pure gas at 1 bar while p is the standard pressure. Փ
Thus, the change in Gibbs energy of the mixture before and after mixing can be rewritten as: ∆mixG = nART ln
pA + nBRT ln pB p p
Using the concept of mole fraction where n A = x An: ∆mixG = nRT ( xA lnxA + xB lnxB )
nA
where xA = ( n + n ) A B
ENTROPY OF MIXING Entropy can also be correlated to chemical potential. known as the entropy of mixing dened by the equation: ∆mixS = -nR ln( x A lnxA + xB lnxB ) ENTHALPY OF MIXING In the mixture of two perfect gases, it is assumed that the soultion does not use up or produce heat making ∆mixH = 0
Enthalpy (H) • •
is a measurement of energy in a thermodynamic system. H = U + PV
1.4 Chemical Potential of Liquids Experiments show that the ratio of vapor pressure can be related to the chemical composition of the liquid. French chemist Francois Raoult formulated this law: pA = xAp* Raoult’s Law p* is the vapor pressure of liquid in pure form xA is the mole fraction of substance A
Mixtures that obey Raoult’s law composition are called ideal solutions. μA = μA* + RT ln x A
althroughout its
For ideal dilute solutions, chemist William Henry found out that at very low pressures, the constant of proportionality is dierent, thus formulating the law: pB = xB KB
pB =bB KB
Henry’s Law xB is the mole fraction of the solute KB is the empirical constant called Henry’s coefficient bB is the molality of the solution
Figure 1.2 Graphical Comparison of Raoult’s and Henry’s Law Physical Chemistry, Atkins p. 146
Molality (b) •
a unit of concentration dened as moles of solute per kilogram of solvent
Simple Mixtures Chapter 1
3
Colligative Properties
1.5
In dilute solutions, colligative properties are proerties that depends on the amount of solute present, not its identity. The word colligative denotes ‘depending on the collection.’ Physical Chemistry, Atkins p.150
http://bit.ly/2n1ATzd
BOILING POINT ELEVATION The boiling points of solutions are all higher than that of the pure solvent. Dierence between the boiling points of the pure solvent and the solution is proportional to the concentration of the http://bit.ly/2BivUhU solute particles ∆T = Kbb KB is the boiling point constant of the solvent
FREEZING POINT DEPRESSION Figure 1.3 Eect of Solute on Vapor Pressure With the presence of solute, boiling temperature of the solution will increase.
The freezing points of solutions are all lower than that of the pure solvent and is directly proportional to the molality of the solute. ∆T = Kfb Kf is the freezing point constant of the solvent
SOLUBILITY Technically, solubility is not a colligative property but it can be calculated the same manner as colligative properties. This refers to the solute Substance B, not solvent Substance A. ln xB =
∆fusH 1
R
(T
f
1 T
(
OSMOSIS Osmosis is the spontaneous passage or diusion of water or
other solvents through a semipermeable membrane. Osmotic Pressure (Π) is the pressure required to stop the inux of solvent given by the equation: Π = [B] RT Van’t Ho Equation
Figure 1.4 Osmosis The solvent will spontaneously pass through the semipermeable membrane towards the solvent particles in the concentrated solution.
4
Simple Mixtures Chapter 1
[B] = nB / V
1.6 Activities of Solutions Activity (a) is a measure of the eective concentration of
a species under non-ideal (e.g., concentrated) conditions. This determines the real chemical potential for a real solution rather than an ideal one. THE SOLVENT ACTIVITY The thermodynamic activity of a solvent can be related with activity through the equation: μA = μA* + RT ln aA A here signifies that the equations refers to the solvent
Comparing this equation to the previous, it can be concluded that: aA =
pA pA*
As the value of xA -> 1, aA -> xA. This convergence denotes: aA = γAxA γA is called the activity coecient. An activity coecient
is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. THE SOLUTE ACTIVITY p aB = B KB
Figure 1.5 A Graph on Raoult’s Law The values of the thermodynamic activity of chloroform and acetone at varying composition using Raoult’s law.
aB = γBxB
Substance B, the solute, obeys Henry’s law. Then as its concentration goes to zero, a B -> xB and γB -> 1. ACTIVITIES OF NON-IDEAL SOLUTIONS It has been experimentally determined that if Raoukt’s Law is used in nding γ for regular solutions, deviations are imminent. For non-ideal solutions, this equation is used: ln γA = βxB2
ln γB = βxA2
Margules Equation
β is a constant that implies if the reaction is endothermic or exothermic
ION ACTIVITIES IN SOLUTION Peter Debye and Erich Hückel developed a theory that would allow us to calculate the mean ionic activity coecient of the solution, γ±γ± , and could explain how the behavior of ions in solution contribute to this constant. log γ± = -|z +z- |AI0.5
I=
Figure 1.5 A Graph on Henry’s Law The values of the thermodynamic activity of chloroform and acetone at varying composition using Henry’s Law.
1 b ( pz+2 + q z-2) 2 b
Փ
p and q are the number of moles of the ion in the compound z is the charge of the ion Փ
b is molality at standard state
Simple Mixtures Chapter 1
5
CHAPTER
2
PHASE DIAGRAMS
Chapter Overview Phase diagram is a graphical representation of the physical states of a substance under dierent conditions of temperature and pressure. A typical phase diagram has pressure on the y-axis and temperature on the x-axis. As we cross the lines or curves on the phase diagram, a phase change occurs. In addition, two states of the substance coexist in equilibrium on the lines or curves.
6
Phase Diagrams Chapter 2
2.1
Terminologies
2.2
One-Component System
2.3
Phase Rule
2.4
General Two-Component Systems
2.5
Liquid-Liquid Diagrams
2.6
Liquid-Solid Diagrams
2.7
Ternary Systems
2.1 Terminologies Phase diagram is a graphical representation of the physical states of a substance under dierent conditions of temperature and pressure. A typical phase diagram has pressure on the y-axis and temperature on the x-axis. As we cross the lines or curves on the phase diagram, a phase change occurs. In addition, two states of the substance coexist in equilibrium on the lines or curves. Phase boundaries are specic conditions of pressure and temperature wherein two phases of the same substance exists at equilibrium. Triple point is the specic temperature and pressure wherein three phases exist at equilibrium. It is the point where the phase boundaries meet. Critical point determines the maximum pressure and temperature where there are two distinguishable phases. Above the critical point, the substance is now called a supercritical uid hp://bit.ly/2D74IHM which is neither liquid nor vapor.
Figure 2.1 General Phase Diagram This gure shows the major parts of a phase diagram. Every substance have dierent phase diagrams and all of them contain these major parts.
phase boundaries
2.2 One-Component System DIAGRAM INTERPRETATION FOR WATER h t t p : / / b i t . l y / 2 D 9 A 9 k L
This diagram tells us that for water: at 100°C and 1 atm, water has two phases, liquid and vapor, at equilibrium. It also tells us that 0.006atm and 0.01°C, water will exist as three phases at equlibrium. Moving along a constant temperature line reveals relative densities of the phases. Moving along a constant pressure line reveals relative energies of the phases. When moving from the left of the diagram to the right, the relative energies increases. Phase Diagrams Chapter 2
7
Phase Rule
f P 3 6 5 D 2 / y l . t i b / / : p h
Figure 2.2 Phase Diagram of CO2
This gure determines the pressure and temperature ranges wherein carbon dioxide will take to form its specic phase.
2.3
Phase (P) is a substance with uniform physical and chemical composition althroughout. Variance (F) is the number of intensive variables that can’t be changed without disturbing the number of phases in equilibrium. Gibbs’ phase rule was proposed by Josiah Willard Gibbs. The rule applies to non-reactive multi-component heterogeneous systems in thermodynamic equilibrium and is given by the equality where F is the number of degrees of freedom, C is the number of components and P is the number of phases in thermodynamic equilibrium with each other. F=C-P+2
If F=0, the system is called invariance system.
Two-Component Systems
2.4
Applying Dalton’s Law of Partial Pressures for two gases A and B, it is implied that: P = PA + PB
Liquid Compisition (x) •
mole fraction of the component in the liquid phase
Vapor Compisition (y) •
substituting Raoult’s Law equations,
PA = xAPA*
PB = xBPB*
P = xAPA* + xBPB*
mole fraction of the component in the vapor phase
The vapour composition (y) of each component is given by the formula below. Also, like mole fraction, y A + yB = 1. yA =
PA P
yB =
PB P
x PA* yA = x P * + A(1-x )PB* A A A PRESSURE-COMPOSITION (P-Z) DIAGRAM P-z phase diagrams are held at a constant temperature. The a-b line in Figure 2.3 is called a tieline. Inside the area of the two curves, both liquid and vapor compositions are present as implied by the letter ‘z’. P-x line
or Saturated Liquid Curve or Bubble Curve
Figure 2.3 P-z Two-Component System At high pressures, the components are in liquid form. On the other hand, low pressure indicate the presence vapor.
8
Phase Diagrams Chapter 2
P-y line
or Saturated Vapor Curve or Dew Point Curve
The Lever Rule
x A
z A
y A
The lever rule is used to nd the relative amounts of
components A and B. length of tieline from the point lα to the saturated liquid curve lα = z - x A A lβ
Figure 2.4 T-z Two-Component System
length of tieline from the point to the saturated vapor curve
At high temperatures, the components are in vapor form. On the other hand, low temperature indicate the presence of liquid.
lβ = y - z A A
TEMPERATURE-COMPOSITION (T-Z) DIAGRAM T-z phase diagrams are held at a constant pressure. Distillation Figure 2.5 Fractional Distillation The number labels in both a and b represents the number of theoretical steps underwent by the mixture until the desired composition is achieved.
Distillation is a process involving the conversion of a liquid into vapour that will be condensed back to liquid form. Simple Distillation involves the separation of a volatile
liquid from a non-volatile solute. In fractional distillation, the two substances are separated by boiling and condensation in repeated successive cycles. Each vaporization and condensation process is called a theoretical plate. Phase Diagrams Chapter 2
9
Azeotropic Mixtures
A mixture is said to be an azeotrope when there is a point where the liquid and vapor phase have the same composition.
Liquid-Liquid Diagrams
2.5
Partially Miscible Liquids • liquids that do not mix in all proportions at any temperature Figure 2.8 T-X diagram of Hexane and Nitobenzene Hexane and nitrobenzene served as substance A and B, respectively.
Figure 2.6 Azeotrope No matter how many times an azeotrope mixture is distilled, the composition at the azeotropic point will never change.
A plait point point where the two liquids have the same composition
Upper Critical Point (T UC)
indicates the maximum tempreature that the two components are separated
At point A, the tieline was created to determine a’ and a’’. At a’ the hexane-rich composition of the mixture is determined. On the other hand, at a’’ the nitrobenzene-rich composition will be found. The lever rule can determine the ratio of the two. Figure 2.9 T-X diagram of Water and Trimethylamine Water, as the solvent, became substance A while trimethylamine is the substance B because it is the solute.
Figure 2.10 Other Kinds of Binary Systems This diagram shows two fully miscible liquids before boiling happens. Physical Chemistry 8e, Atkins, p. 188
10
Phase Diagrams Chapter 2
Lower Critical Point (T LC)
indicates the lowest tempreature that the two components are separated
Upper and lower critical points may also exist in solid-solid diagrams. These are called critical solution temperatures. Some systems have both T UC and TLC. Weak complexes can lead to partial miscibilty.
2.6 Liquid-Solid Diagrams Immiscible liquids are liquids that almost do not mix at liquid form such as in Figure 2.11.
The mechanism of the diagram is as follows: at point a1 from point a1 to a2
from point a2 to a3 from point a3 to a4
from point a4 to a5
The mixture is purely liquid. Solid B begins to deposit in the solution. Liquid A also starts to increase. More solid B is formed. Liquid A is richer than before. There is very little liquid now, the solution is almost solid and the liquid composition is given by point e. The solution is now a two-phase system of pure solids A and B.
The “e” in Figure 2.11 indicates the eutectic composition. It is the composition of the mixture at its lowest melting point.
2.7 Ternary Systems
Figure 2.11 T-X diagram of Two Immiscible Liquids This diagram shows liquids A and B that both completely freezes at the same temperature.
Figure 2.12 Ternary Diagram of Substances X, Y and Z A point in the diagram determines the composition of each component in the mixture.
A ternary system is one with three components. We can independently vary the temperature, the pressure, and two independent composition variables for the system as a whole. A two-dimensional phase diagram for a ternary system is usually drawn for conditions of constant T and p. The sum of zA, zB, and zC must be 1. The method of representing composition with a point in an equilateral triangle works because the sum of the lines drawn from the point to the three sides, perpendicular to the sides, equals the height of the triangle. The proof is shown in Fig 2.13.
Figure 2.13
Phase Diagrams Chapter 2
11
CHAPTER
3
CHEMICAL EQUILIBRIA
Chapter Overview Chemical equilibria are of importance in explaining a great many natural phenomena, and they play important roles in many industrial processes. In this chapter we will explore chemical equilibria in some detail. Here we will learn how to express the equilibrium position of a reaction in quantitative terms.
12
Chemical Equilibria Chapter 3
3.1
Chemical Equilibrium
3.2
Equilibria and Gibbs Energy
3.3
Le Chatelier’s Principle
3.1 Chemical Equilibrium Chemical equilibrium is a state in which the rate of the forward reaction equals the rate of the backward reaction. In other words, there is no net change in concentrations of reactants and products. This kind of equilibrium is also called dynamic equilibrium.
Equilibrium Constant (K) •
Dimensionless number that indicates the reaction rate at equilibrium
Reaction Quotient (Q) •
Dimensionless number that indicated reaction rate at any point of the reaction
GIven the reaction above, the equilibrium constant can be calculated through the concetration of reactants by:
GIven the reaction above, the equilibrium constant can be calculated through the activity of each component by: if in equilibrium
if not in equilibrium
RELATIONSHIP OF Qc and Kc If
Qc = Kc
the reaction is at equilibrium;
Qc < Kc
the reaction is not at equilibrium and there’s a net forward reaction;
Qc > Kc
the reaction is not at equilibrium and there’s a net reaction in the opposite direction.
Equilibrium Constant (Kc) in terms of concentration •
Constant calculated by using the molar concentration of the products and reactants
Equilibrium Constant (Kp) in terms of pressure •
Constant calculated by using the partial pressures of the products and reactants
RELATIONSHIP OF Kp and Kc Kp = Kc (RT) ∆n
where
∆n = (c+d)-(a+b)
3.1 Equilibria and Gibbs Energy The relationship of chemical equilibrium and gibbs energy is dened as:
At equilibrium, the reaction has an equal rate of creating products and reactants making dG r = 0. Thus, at equilibrium:
Gibbs Free Energy (dG r) •
The energy that drives the reaction towards a direction
Standard Gibbs Free Energy (dGrº) •
The Gibbs free energy of the reaction at standard condition
Standard Pressure (Pº) In terms of pressure, equlibria can be calculated by:
•
Pressure at standard conditions
P P º
Chemical Equlibria Chapter 3
13
How dGr Predicts the Direction of the Reaction
If
dGr < 0
reaction is spontaneous to the products side reaction is non spontaneous to the reactants side exergonic or work-producing
dGr > 0
reaction is non spontaneous to the products side reaction is spontaneous to the reactants side endergonic or work-consuming
dGr = 0
reaction is in equilibrium neither exergonic nor endergonic not producing nor consuming work
Le Chatelier’s Principle
3.3
The Le Châtelier's principle states that: when factors that inuence an equilibrium are altered, the equilibrium will shift to a new position that tends to minimize those changes
EFFECT OF CHANGES IN CONCENTRATION •
Figure 3,1 Change in Pressure If the system is compressed, the total pressure of the system will change and may aect the direction of the reaction.
•
If the concentration of reactants is increased, the reaction will tend to produce more products to minimize the eects of the stress. If the concentration of the products is decreased, or if products are removed, the reaction will create more products to minimize the stress.
EFFECT OF CHANGES IN PRESSURE • •
If the total gas pressure is increased, the reaction will proceed either way depending on the stoichiometry of the reaction. If an inert gas that is not part of the equilibrium reaction is added, there will be no eect to the reaction at equlibrium.
EFFECT OF CHANGES IN TEMPERATURE •
If the reaction is exothermic (∆Hr < 0), then: if heat is applied, reactants are formed if heat is removed, products are formed
•
If the reaction is endothermic (∆Hr > 0), then: if heat is applied, products are formed if heat is removed, reactants are formed
Figure 3,2 Change in Concentration If components are removed while the reaction is at equilibrium, the reaction will tend to minimize the stress.
14
Chemical Equilibria Chapter 3
