Jan 01, 2016
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CHAPTER 4MATERIAL EQUILIBRIUM
ANIS ATIKAH BINTI AHMAD
PHYSICAL CHEMISTRY
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SUBTOPIC
Introduction to Material Equilibrium Entropy and Equilibrium The Gibbs and Helmholtz Energies Thermodynamic Relations for a System
Equilibrium Calculation of Changes in State Function Phase Equilibrium Reaction Equilibrium
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WHAT IS MATERIAL EQUILIBRIUM?
In each phase of the closed system, the number of moles of each substances present remains constant in time
No net chemical reactions are occurring in the system
No net transfer of matter from one part of the system to another
Concentration of chemical species in the various part of the system are constant
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Material equilibrium
Reaction equilibrium
Phase equilibrium
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ENTROPY AND EQUILIBRIUM Entropy, is a measure of
the "disorder" of a system. What "disorder refers to is really the number of different microscopic states a system can be in, given that the system has a particular fixed composition, volume, energy, pressure, and temperature.
While energy strives to be minimal, entropy strives to be maximal
Entropy wants to grow. Energy wants to shrink. Together, they make a compromise.
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ENTROPY AND EQUILIBRIUM Example: In isolated system (not in material
equilibrium) The spontaneous chemical reaction or
transport of matter are irreversible process that increase the ENTROPY
The process was continued until the system’s entropy is maximized.
Once it is maximized, any further process can only decrease entropy –(violate the second law)
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• isolated systems: is one with rigid walls that has no communication (i.e., no heat, mass, or work transfer) with its surroundings. An example of an isolated system would be an insulated container, such as an insulated gas cylinder
isolated (Insulated) System:
U = constant
Q = 0
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The system is not in material equilibrium but is in mechanical and thermal equilibrium
The surroundings are in material, mechanical and thermal equilibrium
System and surroundings can exchange energy (as heat and work) but not matter
Since system and surroundings are isolated , we have
dqsurr= -dqsyst (1)
Since, the chemical reaction or matter transport within the non equilibrium system is irreversible, dSuniv must be positive:
dSuniv = dSsyst + dSsurr > 0 (2)
Consider a system at T;
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The surroundings are in thermodynamic equilibrium throughout the process.
Therefore, the heat transfer is reversible, anddSsurr= dqsurr/T (3)
The systems is not in thermodynamic equilibrium, and the process involves an irreversible change in the system, therefore
DSsyst ≠dqsyst/T (4)
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Equation (1) to (3) give dSsyst > -dSsurr = -dqsurr/T = dqsyst/T (5)
Therefore dSsyst > dqsyst/T
dS > dqirrev/T (6) closed syst. in them. and mech. equilib.
dqsurr= -dqsyst (1)
dSsurr= dqsurr/T (3)
dSuniv = dSsyst + dSsurr >0 (2)
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When the system has reached material equilibrium, any infinitesimal process is a change from a system at equilibrium to one infinitesimally close to equilibrium and hence is a reversible process.
Thus, at material equilibrium we have, ds = dqrev/T (7)
Combining (6) and (7): ds ≥dq/T (8) material change, closed
syst. in
them & mech. Equilib
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The first law for a closed system isdq = dU – dw (9)
Eq 8 gives dq≤ TdS Hence for a closed system in mechanical and
thermal equilibrium we have dU – dw ≤ TdS Or
dU ≤ TdS + dw (10)
ds ≥ dq/T (8)
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A spontaneous process at constant-T-and-V is accompanied by a decrease in the Helmholtz energy, A.
A spontaneous process at constant-T-and-P is accompanied by a decrease in the Gibbs energy, G.
dA = 0 at equilibrium, const. T, V
dG = 0 at equilibrium, const. T, P
THE GIBSS & HELMHOLTZ ENERGIES
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dU TdS + SdT – SdT + dw
dU d(TS) – SdT + dw
d(U – TS) – SdT + dw
d(U – TS) – SdT - PdVat constant T and V, dT=0, dV=0
d(U – TS) 0
dU TdS + dw
Equality sign holds at material equilibrium
HELMHOLTZ FREE ENERGY
A U - TS
Consider material equilibrium at constant T and
V
dw = -P dV for P-V work
only
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For a closed system (T & V constant), the state function U-TS, continually decrease during the spontaneous, irreversible process of chemical reaction and matter transport until material equilibrium is reached
d(U-TS)=0 at equilibrium
HELMHOLTZ FREE ENERGY
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dU d(TS) – SdT – d(PV) + VdP
d(H – TS) – SdT + VdP
d(U + PV – TS) – SdT + VdP
at constant T and P, dT=0, dP=0
d(H – TS) 0
Consider material equilibrium for constant T & P, into with dw = -P dV
dU T dS + dw
dU T dS + S dT – S dT + P dV + V dP – V
dP
GIBBS FREE ENERGY
G H – TS U + PV – TS
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the state function H-TS, continually decrease during material changes (constant T and P) , until material equilibrium is reached.
This is the minimisation of Gibbs free energy.
GIBBS FREE ENERGY,G=H-TS
d(H – TS) 0
G = H – TS = U + PV -
TS
GIBBS FREE ENERGY
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G H – TS U + PV – TS
dGT,P 0
Equilibrium reached
Constant T, P
Time
G
G decreases during the approach to equilibrium, reaching minimum at equilibrium
GIBBS FREE ENERGY
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As G of the system decrease at constant T & P, Suniv increases.
WHY?Consider a system in mechanical and thermal equilibrium which undergoes an irreversible chemical reaction or phase change at constant T and P.
systsurruniv SSS systsyst STH /
TGTSTH systsystsyst //)(
TGS systuniv / closed syst., const. T, V, P-V work only
GIBBS FREE ENERGY
The decrease in Gsyst as the system proceeds to equilibrium at constant T and P corresponds to a proportional increase in S univ
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const. T
dwSdTTSUd )(
dwSdTdA dwdAwA
wwby
Awby const. T, closed syst.
It turns out that A carries a greater significance than being simply a signpost of spontaneous change:
The change in the Helmholtz energy is equal to the maximum work the system can do: Aw max
Closed system, in thermal &mechanic. equilibrium
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G H – TS U + PV – TS
VdPPdVdwSdTdG
G U– TS + PV A + PV VdPPdVdAdG
VdPPdVdwSdTdG
PdVdwdG
const. T and P, closed syst.
If the P-V work is done in a mechanically reversible manner, then
VPnondwPdVdw
VPnondwdG VPnonbyVPnon wwG ,
const. T and P, closed syst.
Gw VPnonby ,
or
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For a reversible change
The maximum non-expansion work from a process at constant P and T is given by the value of -G
GwVPnon
max,
(const. T, P)
Gw VPnonby ,
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Thermodynamic Reactions for a System in Equilibrium
6 Basic Equations:
dU = TdS - PdV
H U + PV
A U – TS
G H - TS
VV T
UC
PP T
HC
closed syst., rev. proc., P-V work only
closed syst., in equilib., P-V work only
closed syst., in equilib., P-V work only
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The rates of change of U, H, and S with respect to T can be
determined from the heat capacities CP and CV.
VV T
STC
PP T
STC
Key properties
closed syst., in equilib.
Basic Equations
Heat capacities
(CP CV )
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dG = -SdT + VdP
dA = -SdT - PdV
dH = TdS + VdP
dU = TdS - PdV
The Gibbs Equations
closed syst., rev. proc., P-V work only
How to derive dH, dA and dG?
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The Gibbs Equations
dH = d(U + PV)
dH = TdS + VdP
= dU + d(PV)= dU + PdV + VdP= (TdS - PdV) + PdV + VdP
H U + PV
dH = ?
dU = TdS - PdV
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dA = d(U - TS)
dG = d(H - TS)
dG = -SdT + VdP
dA = -SdT - PdV
= dU - d(TS)= dU - TdS - SdT= (TdS - PdV) - TdS - SdT
= dH - d(TS)= dH - TdS - SdT= (TdS + VdP) - TdS - SdT
dA = ?
dU = TdS - PdV
dH = TdS+VdP
dG = ?
A U - TS
G H - TS
dVV
UdS
S
UdU
SV
TS
U
V
PV
U
S
ST
G
P
VP
G
T
PT
V
VPT
1
),(TP
V
VPT
1
),(
The Power of thermodynamics:
)( PdVTdSdU
Difficultly measured properties to be expressed in terms of easily measured properties.
The Gibbs equation dU= T dS – P dV implies that U is being considered a function of the variables S and V. From U= U (S,V) we have
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(dG = -SdT + VdP)
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The Euler Reciprocity
RelationsIf Z = f(x , y) , and Z has continuous second partial derivatives, then
y
z
xx
z
y
That is
NdyMdxdz xy y
zN
x
zM
yxx
N
y
M
The Gibbs equation (4.33) for dU is
dU = TdS - PdV
V
U
SS
U
V
dS = 0
PV
U
S
TS
U
V
dV = 0
Applying Euler Reciprocity,
SV V
T
S
P
dU = TdS - PdV
The Maxwell Relations (Application of Euler relation to Gibss equations)
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VS PS
TV
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These are the Maxwell
Relations
VT T
P
V
S
PT T
V
P
S
PS S
V
P
T
SV V
T
S
P
The first two are little used.
The last two are extremely valuable.
The equations relate the isothermal pressure and volume variations of entropy to measurable properties.
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DEPENDENCE OF STATE FUNCTIONS ON T, P, AND V
We now find the dependence of U, H, S and G on the variables of the system.
The most common independent variables are T and P.
We can relate the temperature and pressure variations of H, S, and G to the measurable Cp,α, and κ
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Volume dependence of UThe Gibbs equation gives dU = TdS - PdV
PV
ST
V
U
T
T
T
T d
d
d
d
From Maxwell Relations
PV
ST
V
U
TT
VT T
p
V
S
Divided above equation by dVT, the infinitesimal volume change at constant T, to give
PT
PT
PT
V
U
VT
For an isothermal process dUT = TdST - PdVT
T subscripts indicate that the infinitesimal changes dU, dS, and dV are for a constant-T process
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from Gibbs equations, dH = TdS + VdP
Pressure dependence of H
Temperature dependence of U
Temperature dependence of H
PT T
V
P
S
VP
ST
P
H
TT
VTVVT
VT
P
H
PT
PP
CT
H
VV
CT
U
From Basic Equations
From Maxwell Relations
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Temperature and Pressure dependence of G
Temperature dependence of S
Pressure dependence of S
T
C
T
S P
P
From Basic Equations
The Gibbs equation (4.36) for dG is
dG = -SdT + VdP
dT = 0
VP
G
T
ST
G
P
dP = 0
VT
V
P
S
PT
From Maxwell Relations
The equations of this section apply to a closed system of fixed composition and also to a closed system where the composition changes reversibly
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Joule-Thomson Coefficient(easily measured quantities)
HJT P
T
PT
JT CP
H/
from (2.65)
)1(])[1(
T
C
VVTVC
PPJT
VTVVT
VT
P
H
PT
From pressure
dependence of H
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Heat-Capacity Difference(easily measured quantities)
PTVP T
VP
V
UCC
PT
PT
PT
V
U
VT
PVP T
VTCC
2TV
CC VP V
T
V
T
VV
P
P
1
From volume
dependence of U
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2TV
CC VP
1. As T 0, CP CV
Heat-Capacity Difference
2. CP CV (since > 0)
3. CP = CV (if = 0)
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TV
U
Ideal gases 0
TV
U
Solids 300 J/cm3 (25 oC, 1 atm)
Internal Pressure
Liquids 300 J/cm3 (25 oC, 1 atm)
PT
PT
PT
V
U
VT
Strong intermolecular forces in solids and liquids.
Solids, Liquids, & Non-ideal Gases
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CALCULATION OF CHANGES IN STATE FUNCTION
1. Calculation of ΔS Suppose a closed system of constant composition
goes from state (P1,T1) to state (P2,T2), the system’s entropy is a function of T and P
dPP
SdT
T
SdS
TP
VdPdTT
CdS P
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Integration gives:
Since S is a state function, ΔS is independent of the path used to connect states 1 and 2. A convenient path (Figure 4.3) is first to hold P constant at P1 and change T from T1 to T2. Then T is held constant at T2, and P is changed from P1 to P2.
For step (a), dP=0 and gives
For step (b), dT=0 and gives
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1
PPconstdTT
CS
T
T
Pa
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1
TTconstdPVSP
Pb
dPVdTT
CSSS P
2
1
2
112
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ΔU can be easily found from ΔH using :
ΔU = ΔH – Δ (PV) Alternatively we can write down the equation for ΔU similar to:
dPTVVdTCH P 2
1
2
1
2. Calculation of ΔH
dPP
HdT
T
HdH
TP
dPVTVdTCP )(
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3. Calculation of ΔG
For isothermal process:
Alternatively, ΔG for an isothermal process that does not involve an irreversible composition change can be found as:
A special case:
TconstSTHG
TconstVdPG
P
P
2
1
onlyworkVPPandTconstatprocessrevG ;0[Since ]TqSqH /,
VP
G
T
from slide 35
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Phase Equilibrium
A phase equilibrium involves the same chemical species present in different phase. [ eg:C6H12O6(s) C6H12O6(g) ]
-
-
Phase equilib, in closed syst, P-V work only
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For the spontaneous flow of moles of j from phase to phase
- Closed syst that has not yet reached phase equilibrium
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One EXCEPTION to the phase equilibrium,
Then, j cannot flow out of (since it is absent from ). The system will therefore unchanged with time and hence in equilibrium. So the equilibrium condition becomes:
Phase equilib, j absent from
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Reaction Equilibrium
A reaction equilibrium involves different chemical species present in the same phase.
Let the reaction be:
reactants products
a, b,…..e, f….. Are the coefficients
...... 121 mm fAeAbAaA
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Adopt the convention of of transporting the reactant to the right side of equation:
are negative for reactant and positive for products
During a chemical reaction, the change Δn in the no. of moles of each substance is proportional to its stoichometric coefficient v. This proportionality constant is called the extent of reaction (xi)
For general chemical reaction undergoing a definite amount of reaction, the change in moles of species i, , equals multiplied by the proportionality constant :
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The condition for chemical-reaction equilibrium in a closed system is
Reaction equilib, in closed system., P-V work only
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