Chapter 1 : Introduction and Definition of Terms

System
 Open system : exchange of energy and mass
 Closed system : exchange of energy but not mass
 Isolated system : no exchange of energy or mass

 Microscopic properties
→ e.g. molecule, position, velocity, mode of motion
 Macroscopic properties
→ e.g. temperature, pressure, volume, density
 Homogeneous or heterogeneous (in terms of phase)
→ phase : finite volume in which the properties are uniformly constant
 Number of components
→ component : chemical species of fixed composition
 Equilibrium or non-equilibrium
→ equilibrium : no net flow of matter or energy, no phase change, no
change in time and space

Macroscopic properties
 Extensive : dependent on size of system (e.g. V, n, m)
 Intensive : independent of size of system (e.g. T, P, ρ)

Ideal gas
 Ideal gas : a theoretical gas composed of a set of randomly moving,
non-interacting point particles
 PV =nRT or P V =RT (V =V / n : molar volume)

State function
 Adiabatic : no heat transfer with surroundings
 Isobaric : constant pressure
 Isothermal : constant temperature
 Isochoric : constant volume
 Isentropic : constant entropy

 Path-independent (∮ dV =0)
 Exact differential (dV )
Chapter 2 : The First Law of Thermodynamics
Energy and work
 Energy : the ability to do work
 w=−wmech =−∫ ⃗ F ext ∙ d l⃗ : work w is path-dependent, not a
F ext ∙ d l⃗ (δ wmech =⃗
state function)
→ work done by system on surroundings > 0
→ work done by surroundings on system < 0

The first law of thermodynamics

 1st law : the change in the internal energy of a closed system is equal to
the amount of heat supplied to the system minus the amount of work
done by the system on its surroundings
 ∆ U =U 2−U 1 =q−w (dU =δq−δw : internal energy U is a state function,
heat q and work w are path-dependent)

Total energy E
 ∆ E=∆ U + ∆(KE)+ ∆( PE)=Q−W

Application of the 1st law

 Isochoric : w=∫ PdV =0 , q ¿ ¿ V =∆U
 Isobaric : w=P ∆ V , q ¿ ¿ P=∆ H (enthalpy : H ≡U + PV )
 Isothermal : w=∫ PdV , q=w
 Adiabatic : δq =0, δw=PdV =−dU =−CV dT

Heat capacity
∂U
 Constant V : δq ¿ ¿V =( ∂ T ) dT =C V dT  dU =CV dT
V
∂H
 Constant P : δq ¿ ¿ P=( ∂ T ) dT =C P dT  dH =C P dT
P

 For 1 mol of monoatomic ideal gas : C P =

5
2 ()
R, C V =
3
2 ()
R  C P −CV =R

Isothermal vs. adiabatic

 Isothermal reversible process : PV =P1 V 1=RT
γ γ
 Adiabatic reversible process : PV =P1 V 1
→ Adiabat curve steeper than isotherm curve
→ Isotherm does more work (larger area)
→ After adiabatic expansion, T of the system drops because it
consumes U
Chapter 3 The Second Law of Thermodynamics
Reversible process
 Reversible process : one that is performed in such a way that, at the
conclusion of the process, both the system and the local surroundings
may be restored to their initial states without producing any changes in
the rest of the universe

Carnot engine
 Carnot cycle (between two thermal reservoirs)
→ 12 : isothermal reversible expansion
→ 23 : adiabatic reversible expansion
→ 34 : isothermal reversible compression
→ 41 : adiabatic reversible compression

The Clausius inequality

|q 2 ,rev| |q 1 ,rev|
 Reversible Carnot engine : + =0
T2 T1
|q 2 ,rev| |q 1 ,rev|
 Irreversible engine : + <0
T2 T1
 General case : an arbitrary closed path can be broken into a series of
Carnot cycles ABCD + EFGH + …
n
q δq
→ lim ∑ i =∮ ≤ 0
n → ∞ i=1 T i T

The second law of thermodynamics

δ q rev
 Entropy S (defined by dS= ) is a state function
T
 2nd law : the entropy of a system in an adiabatic enclosure can never
decrease; it increases during an irreversible process and remains
constant during a reversible process
 For an isolated system of constant U and V, equilibrium is at maximum S
(dS isolated =0 )
 S of the system increases for irreversible, natural, spontaneous heat
transfer

Combination of 1st and 2nd laws

 General : dU =δq−δw
 For a closed (no mass exchange) and single-component (no chemical
reactions; no chemical work) system :
→ δw=P ext dV (no other work than PV) ¿ Psys dV (no other work than PV
+ reversible)
→ δq =TdS (only when reversible)
  dU =T sys dS−P sys dV (energetic fundamental equation)
→ only true for no mass exchange + only PV work
1 P
 dS= dU + dV (entropic fundamental equation)
T T
→ only true for no mass exchange + only PV work

Laws of thermodynamics
 1st law
→ conservation of energy
→ shows the equivalence of work and heat
→ for cyclic process, ∮ dU =0  q=w
→ suggests one can convert heat into useful work
nd
 2 law
→ puts restrictions on the conversion of heat to work
→ suggests a directionality to natural processes
rd
 3 law : there exists a lower limit to the temperature that can be attained
by matter, called the absolute zero of temperature, and the entropy of all
substances is the same (zero) at that temperature
→ at 0 K, Carnot efficiency = 100%
→ but we cannot reach 0 K
Chapter 4 : The Statistical Interpretation of Entropy
Microscopic view of entropy
 Gibbs : entropy = a measure of the degree of mixed-up-ness (disorder)
→ more disorder = greater entropy (Ssolid < Sliquid < Sgas)

Atomistic meaning of entropy

 Classical thermodynamics : dS ¿¿ U ,V ≥ 0
 Spontaneous (natural, irreversible) direction of process
= increasing entropy (classical TD)
= going to the most probable distribution, or increasing Ω (total
number of microstates) (statistical TD)
 Equilibrium
= maximum entropy (classical TD)
= maximum Ω (statistical TD)
 Entropy must be related to number of microstates  S=k ln Ω
→ natural direction of heat flow = increasing S = increasing total Ω

Configurational entropy
 Up to now, we have discussed thermal (vibrational) entropy (: the change
in S accompanying redistribution of thermal energy among identical
particles (momentum space))
 Configurational entropy : number of ways in which the particles
themselves can be redistributed (geometric space)
 Entropy change of system
Reversible isothermal Irreversible adiabatic Reversible adiabatic
expansion free expansion expansion
Total ∆ S rev ,isotherm >0 ∆ Sirr , adiabatic >0 ∆ S rev ,adiabatic =0
Micro Sconf > 0 , Sth =0 Sconf > 0 , Sth =0 Sconf > 0 , Sth <0
Macro S xfer >0 , S gen=0 S xfer=0 , S gen >0 S xfer=0 , S gen=0
* ∆ S rev ,isotherm =∆ S irr , adiabatic
T2 V2
 Reversible adiabatic expansion : ∆ S=0, ∆ Sth =C V ln ≠ 0, ∆ S conf =R ln ≠0
T1 V1
1 P
→ classical : dS= dU + dV =0
T T
→ statistical : dS=d Sth + d S conf =0
V2
 Free (irreversible) expansion : ∆ S> 0, ∆ Sth =0, ∆ S conf =R ln
V1
Chapter 5 : Fundamental Equations and Their Relationships
Criterion for equilibrium : U, S
 Energy minimum principle : dU ¿ ¿ S ,V ,w =0 ≤ 0 (equilibrium : U = minimum,
ex

dU = 0)
 Entropy maximum principle : dS ¿¿ U ,V , w =0 ≥ 0 (equilibrium : S = maximum,
ex

dS = 0)

Auxiliary functions
 Enthalpy : U [ P ] =U + PV ≡ H ( P , S)
→ dH =TdS+VdP
 Helmholtz free energy : U [ T ]=U −TS ≡ A (T , V )
→ dA=−SdT −PdV
 Gibbs free energy : U [ T , P ] =U + PV −TS ≡ G (P ,T )
→ dG =−SdT +VdP

Criterion for equilibrium : H, A, G

 Enthalpy minimum principle : dH ¿ ¿ S , P ,w =0 ≤ 0 ex

 Helmholtz free energy minimum principle : dA ¿ ¿T , V ,w =0 ≤ 0 ex

 Gibbs free energy minimum principle : dG ¿ ¿T ,P ,w =0 ≤ 0 ex

Chemical potential
∂U
 Chemical potential : μi ≡( ∂ n )
i S ,V ,n j≠ i

→ dU =TdS−PdV + ∑ μ i d ni
∂U ∂H ∂A ∂G
 μi=( ) =( ) =( ) =( )
∂ ni S , V ,nj ≠i
∂ ni S , P , n
j ≠i
∂n i T ,V , n ∂n i T , P , n
j ≠i j≠ i

Thermodynamic relations
Thermodynamic 1st derivative 2nd derivative
potential (equation of state) (Maxwell’s relations)
function

( )
dU =TdS−PdV ∂U ∂U 2
∂U 2 2
T =(
∂S V
) , −P=( )
∂V S ( 2 ) ; ∂ U =( ∂ U ) ;
∂S V ∂ S ∂V ∂V ∂S
2
∂U
( 2
)
∂V S
↓ ↓ ↓
∂T ∂T ∂P ∂P
( ) ;( ) =−( ) ; −( )
∂S V ∂V S ∂S V ∂V S
dH =TdS+VdP ∂H ∂H ∂T ∂T ∂V ∂V
T =( ) , V =( ) ( ) ;( ) =( ) ;( )
∂S P dP S ∂S P ∂P S ∂ S P ∂P S
dA=−SdT −PdV ∂A ∂A ∂S ∂S ∂P
−S=( ) , −P=( ) −( ) ; −( ) =−( )
∂T V ∂V T ∂T V ∂V T ∂T V ;
 ∂P
−( )
∂V T
dG=−SdT +VdP ∂G ∂G ∂S ∂S ∂V ∂V
−S=( ) , V =( ) −( ) ; −( ) =( ) ;( )
∂T P ∂P T ∂T P ∂ P T ∂T P ∂P T

( )
∂S CP
 =
∂T P T
1 ∂V
 Thermal expansion coefficient : α ≡ V ( ∂T )
P
1 ∂V
 Isothermal compressibility : κ T ≡− V ( ∂ P )
T

Gibbs-Helmholtz equation
 If any one of U(S,V), H(S,P), A(T,V), G(T,P) is known explicitly in terms of
its proper variables then we have complete information about the system
 G=H −TS


→ S=−(

A=U−TS
∂G
) , H=
∂T P
[( )]
∂( GT )
∂( )
1
T P

→ S=−(
∂A
) , U=
∂T V
[( )]
∂( TA )
∂( )
T
1
V
Chapter 6 : Heat Capacity, Enthalpy, Entropy, and the Third
Law of Thermodynamics
Enthalpy (T, composition)
T tr T2
∂H
 Single component : ( ∂T ) =c P  H ( T 2) −H ( T 1 ) =∫ c P , α dT + ∆ H tr +∫ c P , β dT
P T T
1 tr
T2

 Multi-components : ∆ H ( T 2 )−∆ H ( T 1 )=∫ ∆ c P dT (Kirchoff’s law)

T1

Enthalpy at standard state

 H doesn’t have an absolute value – only changes in H can be measured
 Convention : assign 0 to enthalpy of element i in its stable state at T =
298 K, P = 1 atm (standard state)
→ H 0i ( 298 K )=0

Entropy (T, composition)

 Single component :
∂S CP
→ at constant P : ( ) = 
∂T P T
T tr T2
∆ H tr
S ( T 2 , P )−S ( T 1 , P ) =∫ c P ,α d lnT + +∫ c d lnT
T1
T tr T P , β
tr

∂S CV
→ at constant V : ( ) = 
∂T V T
T tr T2
∆U tr
S ( T 2 , V ) −S ( T 1 ,V )=∫ c V , α d ln T + +∫ c d ln T
T 1
T tr T V ,β
tr

 Multi-components :
T2

→ at constant P : ∆ S ( T 2 , P ) −∆ S ( T 1 , P )=∫ ∆ c P d ln T (Kirchoff’s law)

T1
T2

→ at constant V : ∆ S ( T 2 ,V ) −∆ S ( T 1 , V ) =∫ ∆ c V d ln T (Kirchoff’s law)

T1

The 3rd law of thermodynamics

 Planck : the entropy of any homogeneous substance, which is in
complete internal equilibrium, may be taken to be zero at 0 K
 If not in complete internal equilibrium  S(0 K) may not be zero due to
residual (frozen-in) entropy where atomic mobility frozen not to achieve
homogeneous internal equilibrium state

Pressure dependence of H and S

 Negligible in condensed phase  low pressure dependence of H and S,
only need to consider temperature dependence
Chapter 7 : Phase Equilibrium in a One-Component System
Equilibrium
 The condition of a system must be time-independent; the system must
have achieved a stationary state
 The internal state of a system must not change if it is isolated from its
surroundings
→ the total entropy of the system can only increase while the system
approaches equilibrium
→ for an isolated system, the equilibrium state corresponds to the
state with maximum entropy

 For a system divided into two compartments by an isolating wall :

→ d U (1) + d U (2)=0 (total system = isolated)
→ d V (1) =0=dV (2) (rigid)
→ d ni(1 )=0=d ni(2) (impermeable)
 Thermal contact via diathermal wall
(1) (2 ) 1 (1) 1 (2) 1 1 (1)
→ d S ¿ ¿U ,V , N =d S + d S = (1 ) d U + ( 2) d U =( ( 1) − (2 ) )d U ≥ 0 (2nd
T T T T
law)
→ T : escaping tendency of U
→ T (1) >T (2)  d U (1) <0 (heat transferred 1  2)
→ T (1) <T (2)  d U (1) >0 (heat transferred 2  1)
→ T (1)=T (2 )  dS=0 (max. entropy)  thermal equilibrium
 Mechanical contact via non-rigid wall
(1) (2)
→ d S ¿ ¿U ,V , N =d S (1) + d S(2 )= P d V (1) + P d V (2)= 1 (P(1 )−P(2) ) d V (1) ≥ 0 (2nd
T T T
law)
→ P : escaping tendency of -V
→ P(1) > P(2)  d V (1) >0
→ P(1) < P(2)  d V (1) <0
→ P(1) =P(2)  dS=0 (max. entropy)  mechanical equilibrium
 Contact via permeable wall
−μ (1) μ ( 2)
→ d S ¿ ¿U ,V , N =d S (1) + d S(2 )= i d ni( 1)− i d ni( 2)= −1 ( μ i(1)−μi(2)) d ni(1) ≥ 0
T T T
nd
(2 law)
→ μi : escaping tendency of n
→ μi(1) > μ i(2)  d ni(1 )< 0
→ μi(1) < μ i(2)  d ni(1 )> 0
→ μi(1) =μi(2)  dS=0 (max. entropy)  diffusive equilibrium
 If the wall is flexible, permeable, and diathermal, at equilibrium, T, P, μi
 are all equal (T1 = T2, P1 = P2, μi,1 = μi,2)

Equilibrium criteria
 At constant T (thermal reservoir TR) : at equilibrium, d A ¿ ¿T =0 (minimum
Helmholtz free energy)
→ P(1 )=P(2 ) ; μi(1 )=μi(2 )
 At constant T and P (thermal reservoir TR, pressure reservoir PR) : at
equilibrium, d G¿ ¿ T , P=0 (minimum Gibbs free energy)
→ μi(1 )=μi(2 )

Component and phase

 Component : substance with fixed proportion of elements
→ e.g. water (H2O) : 2 elements (H, O) and 1 component (H2O 
composition fixed)
 Phase : any portion that is physically homogeneous within itself, bounded
by a surface so that it is mechanically separable
→ T < 273.15 K : ice (single phase)
→ T = 273.15 K : ice + water (two phases)
→ 273.15 K < T < 373.15 K : water (single phase)
→ T = 373.15 K : water + steam (two phases)
→ T > 373.15 K : steam (single phase)

Phase diagram of H2O

Equilibrium between ice and water

 Ice and water in equilibrium at 0*C, 1 atm  G’ (total G of entire system)
= minimum
→ heat added to melt 1 mol of ice : ∆ G=G H 2 O (l )−G H 2 O ( s)=0 
G H 2 O ( l)=G H 2O (s ) (molar G)
 For system of ice + water containing n moles of H2O (n=n H 2O (s) +n H 2 O (l) )
→ G ' =n H 2 O(s ) G H 2O (s) +n H 2 O (l) G H2 O (l)=[nH 2 O ( s) +n H 2O (l) ]GH 2 O (s ,l) at 0*C, 1 atm
→ the ratio of sol/liq changes but no change in G’
  d G' ¿¿ T , P=μ i d ni  G '=∑ ni μ i
→ the chemical potential of a species in a particular state (e.g. fixed
T, P) equals the molar Gibbs free energy of the species in the
particular state
→ at 1 atm, T > 0*C, ice spontaneously melts (↓G’) : μ H 2O (s) > μ H 2 O(l) (
G H 2 O (s) >G H 2O (l))
→ at 1 atm, T < 0*C, water spontaneously freezes : μ H 2O (s) < μ H 2 O(l) (
G H 2 O (s) <G H 2O (l))

Variation of G with T at constant P

| | | −C P
2
∂G ∂G ∂S
 =−S< 0 ; =− = <0 (non-linear)
∂T P 2
∂T P ∂T P T
 S H 2O (l) > S H 2 O (s)
→ liquid phase more disordered  steeper slope in G vs. T
 ∆ G(s →l )=∆ H (s →l )−T ∆ S(s → l) where ∆ H (s →l )=H l−H s, etc.
 At melting temperature Tm, Gs =Gl

Variation of G with P at constant T


∂G
∂P T|=V >0 ;
| |
∂2 G
∂ P T ∂P T
2
=
∂V
=−V κT <0


∂P |
∂ ∆ G(s → l)
T
=∆ V (s → l)

 For most solids, ∆ V (s →l )> 0

 exceptions : H2O, Sb, Bi, I2 (∆ V (s →l )< 0)

G as a function of T and P
 For any infinitesimal change in T and P : dG s=dG l
 S( l)−S (s ) ∆ H ( s →l )
( )dP ∆ S (s → l)
→ Clapeyron equation : = = = where
dT eq V ( l)−V (s ) ∆ V (s →l ) T eq ∆ V (s →l )
∆ H =T eq ∆ S at equilibrium

 For H2O, ∆ V (s →l )< 0 and ∆ H (s →l )> 0  ( dPdT ) < 0

eq

Equilibrium between vapor and condensed phase

∆ H (cond → vap )
 ∆ V (cond → vap)=V vap−V cond ≅ V vap  ( )
dP
dT eq
=
T V vap

 For ideal vapor (e.g. PV = RT) : ( )

dP
=
P∆ H
dT eq RT 2
dP ∆H
→ Clausius-Clapeyron equation : =d ln P= dT
P R T2
−∆ H
→ if ΔH is independent of T (Cp,v = Cp,l) : ln P= +constant
RT
 Here, P is equilibrium pressure for solid (liquid) ↔ vapor
→ P = saturated vapor pressure exerted by the condensed phase at T
→ P relates to the tendency of particles to escape from the
solid/liquid
→ a substance with a high vapor pressure at normal temperatures is
often referred to as volatile
→ the vapor pressure of a single component in a mixture in the
system is called partial pressure
A
 If ∆ C P ≠ f ( T ) : ln P= + B lnT +C
T

Phase diagram of H2O

 Gibbs’ phase rule : F = C – P + 2
→ C = # of components, P = # of
phases, F = # of degrees of
freedom
→ e.g. at triple point (point O) : F = 1 –
3+2=0
→ e.g. for solid phase : can freely
change both T and P  F = 2

Gibbs-Duhem equation
c
 Gibbs-Duhem equation : 0=SdT −VdP+ ∑ ni d ni
i=1

→ intensive variables are not all independently variable within a phase

: if you change T and P and all μi except one, the last one is set
automatically
 → can be applied to each phase separately or to the whole system
consisting of multiple phases

Phase diagrams (H2O)

Chapter 8 : The Behavior of Gases
Real gas

 At T < Tcr, two phases coexist  At T = Tcr (critical point) : 1st and
→ isothermal compression of 2nd derivatives = 0
vapor at A  vapor starts to  At T > Tcr
condense at B → supercritical fluid : no
→ as V ↓, more liquid and no physical distinction between
change in P and T the liquid and gaseous
 PBC = equilibrium states
vapor pressure at T8 → isothermal compression
 VC = molar volume of  molar volume of
liquid at PBC, T8 system decreases
 VB = molar volume of monotonously
vapor at PBC, T8  density of system
 liquid with molar increases
volume VC increases continuously
in amount  no phase separation
→ eventually at C, 100% liquid to occur
→ further decrease in V  → liquefaction only possible by
increase in P of liquid : cooling (compression X)
steeper slope (
− ( )
∂P
∂V T
=1/V κ ) of liquid than
T

vapor because water is hard

to compress

Deviation from ideality

PV
 Compressibility factor : Z= (= 1 for ideal gas)
RT
→ ↑T, ↓P : behavior of gas approaches ideal because distance
 between molecules increases
→ Z > 1 : for same V, larger P than P of ideal gas

Van der Waals gas

 Ideal gas : assemblage of volume-less and noninteracting particles
 Real gas : particles occupy a finite volume and are surrounded by force
fields which cause them to interact with one another
a
V ( ) V
a
→ for 1 mol of gas : P+ 2 (V −b )=RT where 2 = interaction term

and b = finite volume term

( )
2
an
→ general : P+ 2 ( V −bn )=nRT
V
8a P = a
 At the critical point (1st and 2nd derivatives = 0) : T cr = , cr 2
27 bR 27 b

Mixture of ideal gases

 Variation of the molar Gibbs free energy of a closed system consisting of
1 mol of ideal gas of fixed composition with P at constant T
→ dG=VdP=RTd ln P
P
→ G ( P2 , T )=G ( P1 ,T ) + RT ln ( 2 ) where G ( P1 , T )=G ( 1 atm ,T of interest )=G 0
P1
0
→ G ( P , T )=G + RT ln P
nA
 Mole fraction : X A = where nA = number of moles of A and
n A + nB + nC
X A + X B+ X C =1
 Partial pressure : contribution to the total pressure by each individual gas
( Ptot = p A + p B + pC )
p
→ X i = i or pi=Ptot ∙ X i
Ptot
 Partial molar quantity : molar value of any extensive state function of a

component of a mixture (Qi= ∂ n

∂ Q'
i n ,T,P
|for an arbitrary quantity of the
j

mixture)
→ e.g. Gi=
∂ G'
∂ ni |
nj , T , P
=μi (note : U i=
∂U '
∂ ni |
n j, T , P
≠ μi ;
∂U '
∂ ni |
n j , S ,V
=μi)

Pure vs. mixture (in equilibrium)

Pure system (single component) Mixture system (multiple components)
T T =T A=T B =…
V' ' ' '
V =V A =V B=…
P P= p A + pB + …
n n=n A + nB + …
 X nA
X A= ,X ,…
n A + n B + nC + … B
G' G' =G'A +G'B +…=n A G A + nB GB + …
molar Gibbs free energy partial molar Gibbs free energy
G'
( )
'
G= ∂G
GA= ,G ,…
n ∂ n A T , P , n ,n , … B
B C

Partial molar Gibbs free energy of ideal gas

0 0 0 0
 G A ( p A )=G A + RT ln p A =G A + RT ln X A + RT ln P where G A =G A =G A ( p A =1 ) =
standard partial molar Gibbs free energy of gas A
→ standard state : 1 mole of pure gas at 1 atm and T of interest
0 0
 μ A ( p A )=μ A + RT ln p A=μ A + RT ln X A + RT ln P where μ0A = standard chemical
potential of gas A

ΔG of mixing for ideal gas

0
 Before mixing : G i=G i + RT ln Pi where Pi = pressure of pure gas i
0
 After mixing : G i=G i + RT ln pi where pi = partial pressure of species I in
the gas mixture

pi pi
∆ G ( mixed−unmixed ) =∑ ni Gi−∑ n i Gi=∑ ni RT ln =∑ ni RT ln =∑ ni RT ln X i <0
' mix
Pi Ptot
(since Xi < 1)  spontaneous mixing
→ only entropic contribution to ∆ G' mix because ideal gas = no
interaction = no change in enthalpy
ni
 Per mole : ∆ G =∑ RT ln X i =∑ X i RT ln X i
mix
n A + nB + …

Nonideal gases : fugacity

 Ideal gas : dG =VdP=RTd ln P
RT
 Nonideal gas : V ≠  dG is not a linear term of d(lnP)
P
f f
 Fugacity f : dG =RTd ln f =RTd ln P+ RTd ln where RTd ln = deviation
P P
from the ideal
f
→ →1 as P →0  gas becomes more ideal as P  0
P
→ G=G °+ RT ln f where G ° = standard state (f = 1 at T of interest)
 From both virial equations of state for real gas and the definition of
f C P 2 D P3
fugacity : ln =BP+ + +…
P 2 3

Summary
0 0
 Ideal gas : μ A =μ A + RT ln p A =μ A + RT ln X A + RT ln Ptot ( for mixture )
 → μ0A : pure, 1 atm
0
 Nonideal gas : μ A =μ A + RT ln f A
Chapter 9 : The Behavior of Solutions
Summary
0 0
 Ideal solution : μ A =μ A + RT ln a A =μ A + RT ln X A (for mixture)
→ μ0A : pure, any atm (atm doesn’t matter because condensed phase
is hardly compressible)
→ Ptot irrelevant (for mixture equation) because ΔP doesn’t matter for
condensed phase
→ ideal : A-B = A-A = B-B interaction 정도가 같을 때 (breaking A-B
bonding is equally as difficult as breaking A-A bonding)
0
 Nonideal solution : μ A + RT ln X A + RT ln γ A

Raoultian (ideal) solution

 At equilibrium, evaporation rate = condensation rate
→ r e ( A )=r c ( A )=k p 0A (pure A) where p0A = equilibrium vapor pressure
 In mixture of A and B where A-A = B-B = A-B interactions :
A −B
re ( A )=r e ( A ) X A=k p A
→ p A =X A p0A (Raoultian behavior)
→ evaporation rate of A in mixture (r e A −B ( A )) < evaporation rate of
pure A (r e ( A )) by XA due to less A per mole in mixture

Henrian solution
 If A-B bond energy more negative than A-A, B-B (stronger interaction
between A-B) and if dilute solution A in B (small [A] = A most likely
0 A −B
surrounded by B) : r e ( A )=r c ( A )=k p A (pure A) and r e ( A )=r e ' ( A ) X A =k p A
where r e ' ( A ) <r e ( A ) (harder to evaporate)
r ' ( A)
→ pA = e X p 0 =γ X p0 (Henrian behavior)
re ( A ) A A A A A
→ evaporation of A more difficult  equilibrium vapor pressure of A
smaller than ideal  γ A <1
 If A-B bonding weaker than A-A and B-B : γ A >1

Raoult’s vs. Henry’s law

  εA-B > εA-A or εB-B : A-A, B-B more stable (γA > 1)
→ A in liquid wants to leave  equilibrium vapor pressure of A > that
of ideal
 εA-B < εA-A or εB-B : A-B more stable (γA < 1)
→ harder to evaporate
fi pi
 Activity of i : a i= (non-ideal gas)= (ideal gas) where fi = fugacity of
fi° pi°
component i in solution at T, fio = fugacity of pure i (standard state) at T,
pi = vapor pressure of i in mixture, pio = equilibrium vapor pressure of
pure i
→ Raoultian a i=X i (liquid solution, ideal)
→ Henrian : a i=γ i X i (liquid solution, nonideal)

Partial molar quantity

( )
'
∂Q
 Partial molar quantity : Q i= =f ( X i )
∂n i T , P , Xj ≠i

 Graphical determination of Qi from Q

for a binary system at constant T, P :
dQ
Q A =Q ( X B )− X
d XB B ;
dQ
QB =Q ( X B ) + (1−X B )
d XB
→ intercept at XB = 0 or 1 of
tangential line
]

G of formation of a solution
 For a binary system (per mole) at
constant T, P : G= X A G A + X B GB ,
dG =G A d X A +G B d X B
dG
→ G A =G− X ;
d XB B
dG
GB =G+ (1− X B )
d XB

ΔG due to formation of mixture (gas)

p
 Ideal gas : G=G ° + RT ln ( )
p°
f
 Real gas : G=G ° + RT ln( )=G° + RT ln a (or μ=μ ° + RT ln a )
f°
  For mixture : G A =G A ° + RT ln a A (or μ A =μ A ° + RT ln a A )
 If nA moles of A and nB moles of B are mixed (ideal gas) :
→ after mixing : Gi=Gi ° + RT ln p i (pi : partial P of gas mixture / 1 atm)
→ before mixing : Gi=Gi °+ RT ln p i ° (pio : P of pure gas i / 1 atm)
p p
→ ∆ G ' M ,id =n A ( G A −G A ) + nB ( G B−GB )=n A RT ln ( A )+ nB RT ln ( B )
pA° pB °
o o M ,id
→ if P = pA = pB before/after mixing : ∆ G ' =n A RT ln X A + nB RT ln X B
→ molar ∆ G M ,id =X A RT ln X A + X B RT ln X B

ΔG due to formation of mixture (solution)

 After mixing : Gi=Gi °+ RT ln ai
 Before mixing : Gi=Gi °
M M M
 ∆ G ' =n A ( G A−G A ° )+ nB ( G B−G B ° ) =n A ∆G A +n B ∆ G B =n A RT ln a A +n B RT ln aB
M
where G A −G A ≡ ∆G A =RT ln a A
→ molar
∆ G M =X A RT ln a A + X B RT ln a B= X A RT ln X A + X B RT ln X B (ideal solution + ideal gas)=X A R
→ ∆ G M =X A ∆ G MA + X B ∆ G MB

Raoultian (ideal) solution

M ,id M ,id M ,id
 a i=X i  ∆ G ' =n A RT ln X A + nB RT ln X B ; ∆ G A =RT ln X A , ∆ G B =RT ln X B
 Change in volume due to mixing :
∆ V ' M ,id =( n A V A +nB V B )−( n A V A °+n B V B ° )=n A ∆ V MA , id +nB ∆V MB , id =0
M
 Change in enthalpy due to mixing : ∆ H A =H A −H A °=0 for ideal solution
(no change in bonding energy)
→ ∆ H M ,id =∆ H MA =∆ H MB =0
M ,id
 Change in entropy due to mixing : ∆ S =−R(X A ln X A + X B ln X B )
→ ∆ S MA =S A−S A ° =−R ln X A ≠ 0 for ideal solution (increase in
configurational entropy)
M ,id M ,id M ,id M ,id
 ∆G =RT ( X A ln X A + X B ln X B ) =∆ H −T ∆ S =−T ∆ S

Summary of symbols
 Q ' : sum of quantities of all species
 Q : sum of quantities of all species per mole
 ∆ Q ' M : after mixing – before mixing
 ∆ Q M : after mixing – before mixing per mole
 Qi ° : molar quantity of species i in pure form (before mixing)
 Qi : partial molar quantity of species i in solution (after mixing)
 ∆ QiM : after mixing – before mixing per mole of species i (Qi−Qi ° )

Non-ideal solution
 Non-ideal solution has change in molar enthalpy after mixing
 → ∆ H iM >0, endothermic : A-A, B-B stable (clustering)
→ ∆ H iM <0, exothermic : A-B stable  compound formation
(ordering)

Regular solution
 Assumptions :
→ no change in molar volume when mixed
→ only nearest-neighbor interactions
→ random mixing
→ no change in interatomic distance and bond energy when mixed
E +E
 Ω=z N 0 ( E AB− AA BB ) where N 0=N A + N B and ∆ H M =Ω X A X B
2
E +E
→ Ω>0 , ∆ H M >0 : E AB > AA BB : clustering favored (mixing unfavored)
2
E +E
→ Ω<0 , ∆ H M <0 : E AB < AA BB : ordering favored (mixing favored)
2
→ microscopic origin of Ω : difference between A-B bonding and
average of A-A and B-B bonding
M M
 ∆ S =−R (X A ln X A + X B ln X B ) ; ∆ G =Ω X A X B + RT (X A ln X A + X B ln X B )

 ∆ G M vs. X A : symmetric around 50% because 1) according to equation

and 2) physical reason that 50-50 mix maximizes entropy
 Temperature decrease  less negative red curve
→ if ∆ G M is positive at some point  no mixing
Chapter 10 : Gibbs Free Energy Composition and Phase
Diagrams of Binary Systems
Gibbs free energy and activity
 γ>1:Ω>0
(mixing
unfavored)
 γ<1:Ω<0
(mixing
favored)

ΔGM of regular
solution
 Ω < 0 : mixing
(A-B) favored
  Ω > 0 : non-
mixing (A-A, B-
B) favored
 At high enough
T, mixing still
occurs (close to
pure A or B,
where entropy
term dominates)
 At low enough
T, entropy term
is too small and
points of non-
mixing exist

Curvature of ΔGM and stability of phase

 Convex downward (Ω < 0)
 1) a system whose average composition is
X B=c ' consists of n moles of A1-aBa and m
moles of A1-bBb (not homogeneous)
→ a : ΔGM (per mole) of A1-aBa
→ b : ΔGM (per mole) of A1-bBb
→ c : ΔGM (per mole) of system
 2) a homogeneously mixed system whose
average composition is X B=c '
→ d : ΔGM (per mole) of system
→ lever rule : system goes from 1) to 2) because homogeneous
mixing of 2) more preferred (lower ΔGM on graph)
 cannot be separated once mixed
 curvature : convex downward  a + b always higher on
curve than d
 Convex upward (Ω > 0)
 γ →α + β : decomposes into two phases (to
reduce ΔG because ΔGM > 0)
 Common tangent line (blue) – intercepts :
(chemical potential of A in α = chemical
potential of A in β) and (chemical potential
of B in α = chemical potential of B in β)
→ Common tangent = most preferred
ΔGM (largest reduction)
Complete solubility phase diagram
Partial solubility phase diagram
 Conditions for complete solubility :
→ same structure
→ comparable size for substitution
→ similar electronegativity
→ same valence

Negligible solubility diagram

 Even if A and B structures are extremely different, they can still mix
somewhere due to entropy