CHEMICAL POTENTIAL
• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students.
• I plan to go through these slides in one 90-minute lecture.
Zhigang Suo, Harvard University
The play of thermodynamics
ENTROPY
energy
temperature
heat capacity
Helmholtz function
space
pressure
compressibility
enthalpy
matter
chemical potential
charge
electrical potential
capacitance
Gibbs function
thermal expansion
Joule-Thomson coefficient
2
plan
•
•
•
•
•
Definition of chemical potential
Examples of chemical potential
Equilibrium of two systems
Equilibrium of a chemical reaction
Equilibrium of phases
3
Model an open system as a family of isolated systems
weights
valve
for H2O
gas
open system
valve
for N2
liquid
H2O tank
N2 tank
2O
gas
liquid
a family of isolated systems
of four independent variables:
U, V, NH2O, NN2
fire
• The wine contains many species of molecules (components) and two phases.
• The wine is an open system, exchanging energy, space, and two components
with the rest of the world.
• Make the wine an isolated system by insulating the bottle, jam the piston, and
shut the valves.
• A system isolated for a long time reaches a state of thermodynamic equilibrium.
• Define the entropy of the isolated system: S = log (number of quantum states).
• Isolating the wine at various values of (U,V, NH2O, NN2), we obtain a family of
isolated systems of four independent variables.
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• Model the family of isolated systems by function S(U,V, NH2O, NN2).
Derivative
2O
gas
1. an operation in calculus
2. a thing based on something else
liquid
a family of
isolated systems
S(U, V, NA, NB)
dS =
(
¶S U ,V , N A , N B
¶U
) dU + ¶S (U ,V , N A , N B ) dV + ¶S (U ,V , N A , N B ) dN
¶V
¶N A
A+
(
¶S U ,V , N A , N B
¶N B
) dN
B
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Name derivatives by Gibbs equations
Define temperature:
Define pressure:
Define chemical potential:
Define chemical potential
Calculus:
(
¶S U ,V , N A , N B
¶U
¶S U ,V , N A , N B
(
¶V
¶S U ,V , N A , N B
(
(
¶N A
¶S U ,V , N A , N B
¶N B
dS =
)= 1
T
)=P
T
) = - mA
T
) = - mB
T
m
m
1
P
dU + dV - A dN A - B dN B
T
T
T
T
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Notes on chemical potentials
(
¶S U ,V , N A , N B
(
¶N A
¶S U ,V , N A , N B
¶N B
•
•
•
•
•
•
) = - mA
T
) = - mB
T
These equations define the two chemical potentials.
Each chemical potential is a child of entropy and a component.
Chemical potential of a component is an intensive property of a system.
T appears in the definition by convention.
Negative sign appears in the definition by convention. Thus, an isolated system increases
entropy when a component goes from a place of high chemical potential to a place of low
chemical potential.
Grammar: The chemical potential of a component in a system (e.g., mA is the chemical
potential of water in the wine, and mB is the chemical potential of nitrogen in the wine).
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Why don’t we know chemical potential
as well as temperature?
• Blame our parapets. Our parents tell us a lot
about temperature, but never tell us about
chemical potential. (But they do tell us about
humidity, and smells of many kinds.)
• Blame our world. The world confuses us with
many species of molecules.
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Breed equations (Gibbs 1878)
Gibbs equation:
Solve for dU:
Calculus:
dS =
m
m
1
P
dU + dV - A dN A - B dN B
T
T
T
T
dU = TdS - PdV + m AdN A + m BdN B
T=
(
¶U S,V , N A , N B
-P =
mA =
mB =
)
¶S
¶U S,V , N A , N B
)
¶V
¶U S,V , N A , N B
)
(
(
(
¶N A
¶U S,V , N A , N B
)
¶N B
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Breed more equations
Gibbs equation:
A Legendre transform defines the
Gibbs function:
Combine the above two equations:
dU = TdS - PdV + m AdN A + m BdN B
G =U -TS + PV
dG = -SdT +VdP + m AdN A + m BdN B
-S =
Calculus:
V=
(
¶G T ,P, N A , N B
¶T
¶G T ,P, N A , N B
mA =
mB =
(
)
)
¶P
¶G T ,P, N A , N B
(
(
¶N A
¶G T ,P, N A , N B
¶N B
)
)
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On being extensive
Gibbs equation
All independent variables are extensive:
Increase all extensive properties proportionally:
Derivative with respect to l:
Solve for U:
Definition of the Gibbs function:
Combine the above two equations:
dS =
m
m
1
P
dU + dV - A dN A - B dN B
T
T
T
T
l S (U,V , N A , N B ) = S ( lU, lV , l N A , l N B )
S=
m
m
1
P
U + V - A N A - B NB
T
T
T
T
U = TS - PV + m A N A + m B N B
G =U -TS + PV
G = m A N A + mB N B
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plan
•
•
•
•
•
Definition of chemical potential
Examples of chemical potential
Equilibrium of two systems
Equilibrium of a chemical reaction
Equilibrium of phases
12
Pure substance
Define Gibbs function per molecule (or per mole):
Define chemical potential of a pure substance:
Compare the two definitions:
Recall the definition of the Gibbs function:
Recall the Gibbs equation:
(
)
(
¶G T ,P, N
)
G T,P, N = Ng T ,P
m=
(
)
¶N
m = g (T ,P )
m = g = u -Ts + Pv
dm = dg = -sdT + vdP
1. For a pure substance, we know how to measure TVPUS.
2. From TVPUS we can calculate the chemical potential m.
3. Chemical potential requires the absolute entropy.
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Incompressible pure substance
Gibbs equation:
Incompressibility:
Integration:
dm = -sdT + vdP
v = constant
m (T,P ) = m (T,P0 ) + ( P - P0 ) v
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Ideal gas
Gibbs equation:
Law of Ideal gas:
dm = -sdT + vdP
Pv = RT
Integration:
m (T ,P ) = m (T ,P0 ) + RT log
Look up values:
m (T,P0 ) = u (T ) -Ts (T ,P0 )
P
P0
P0 = 1 atm
15
Ideal-gas mixture
chemical potential of
component A
in an ideal-gas mixture
chemical potential of
pure ideal gas A
at 1 atm
m A (T,PA ) = m 0A (T ) + RT log PA
partial pressure of
component A
The chemical potential of a component in an ideal-gas mixture is the
same as the chemical potential of the component in the pure gas,
provided we use the partial pressure of the component.
16
Chemical potential of water in moist air
relates to relative humidity
Model the moist air as an ideal-gas mixture
(
m T ,PH
2O
) = m (T ,P
H2O, sat
) + RT log P
PH O
2
H2O, sat
RH =
(T )
PH O
2
( )
PH O, sat T
2
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plan
•
•
•
•
•
Definition of chemical potential
Examples of chemical potential
Equilibrium of two systems
Equilibrium of a chemical reaction
Equilibrium of phases
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Two systems exchanging energy, space and molecules
diathermal, moving, permeable to components A and B
U’, V’, NA’, NB’
S’(U’, V’, NA’, NB’)
open system (‘)
U’’, V’’, NA’’, NB’’
S’’(U’’, V’’, NA’’, NB’’)
isolated system
open system (‘’)
Isolated system conserves energy, space, and matter over time:
dU’ + dU’’ = 0.
dV’ + dV’’ = 0
dNA’ + dNA’’ = 0
dNB’ + dNB’’ = 0
Isolated system not in equilibrium generates entropy over time: dS’ + dS’’ > 0
Isolated system in equilibrium keeps entropy constant over time: dS’ + dS’’ = 0
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Equilibrium of two systems
diathermal, moving, permeable to components A and B
U’, V’, NA’, NB’
S’(U’, V’, NA’, NB’)
open system (‘)
U’’, V’’, NA’’, NB’’
S’’(U’’, V’’, NA’’, NB’’)
isolated system
open system (‘’)
æ1
ö æ 1
ö
m¢
m¢
m ¢¢
m ¢¢
P¢
P ¢¢
¢ ÷÷ + çç dU ¢¢ + dV ¢¢ - A dN ¢¢A - B dN ¢¢A ÷÷
d S ¢ + S ¢¢ = çç dU ¢ + dV ¢ - A dN ¢A - B dN B
T¢
T¢
T¢
T ¢¢
T ¢¢
T ¢¢
èT¢
ø è T ¢¢
ø
æ m ¢ m ¢¢ ö
æ m ¢ m ¢¢ ö
æ1
æ P ¢ P ¢¢ ö
1 ö
¢
= ç - ÷ dU ¢ + ç - ÷ dV ¢ - çç A - A ÷÷ dN ¢A - çç A - A ÷÷ dN B
è T ¢ T ¢¢ ø
è T ¢ T ¢¢ ø
è T ¢ T ¢¢ ø
è T ¢ T ¢¢ ø
(
)
Thermal equilibrium: T ¢ = T ¢¢
Mechanical equilibrium: P¢ = P¢¢
Chemical equilibrium of component A: m¢A = m¢¢A
Chemical equilibrium of component B: m¢B = m¢¢B
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Measuring chemical potential of a
component in a system
chemical potential of water in the wine
Chemical potential affects everything. Everything measures chemical potential
weights
A membrane
permeable to H2O only
gas
pure H2O
m(T,P)
open system
liquid
fire
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Hygrometer
humidity sensors
First inventor: Johann Heinrich Lambert (1755)
Humidity affects everything. Everything is a hygrometer.
Today’s opportunity: The Internet of things.
•
•
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•
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Bimaterial strip
Hair-tension hygrometer
Wet-bulb and dry-bulb
Dew-point hygrometer
Capacitor
Resistor
Thermal conductivity
Weight
https://en.wikipedia.org/wiki/Hygrometer
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Sensors for chemical potentials of
various components
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•
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Humidity sensor
pH sensor
Oxygen sensor
CO2 sensor
Electronic nose
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Henry’s law (1803)
N2 in air
PN2
N2 dissolved in water
yN2
(
)(
PN in air = Henry's constant yN in water
2
2
)
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Solubility
N2 in air
PN2
N2 dissolved in rubber
mole/volume
(r
N2 in rubber
) = (sulubility)( P
N2 in air
)
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plan
•
•
•
•
•
Definition of chemical potential
Examples of chemical potential
Equilibrium of two systems
Equilibrium of a chemical reaction
Equilibrium of phases
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Reaction
piston
weights
Fix T,P
Change U,V
NA, NB,
NC, ND,
Isolated system (IS)
reaction
chamber
Qout
thermal reservoir, T
A chemical reaction conserves the
number of atoms in each species n A + n B Û n C + n D
A
B
C
D
(ni are stoichiometric coefficients):
Increment of the number of each
dN A = -n Ade , dN B = -n Bde , dNC = n Bde , dN B = n Dde
component (e is the degree of reaction):
Conservation of energy: Qout +U + PV = constant
Entropy is additive:
Q
SIS = out + S U ,V , N A , N B , N C , N D
T
(
)
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Equilibrium of a reaction
Conservation of atoms: dN A = -n Ade , dN B = -n Bde , dNC = n Bde , dN B = n Dde
Conservation of energy: Qout +U + PV = constant
Entropy is additive:
Q
SIS = out + S U ,V , N A , N B , N C , N D
T
(
Condition of equilibrium:
dSIS = 0
Calculus and definitions:
dSIS =
dQout
T
+
)
dU PdV m AdN A m BdN B mC dN C m DdN D
+
T
T
T
T
T
T
Chemical equilibrium: -m An A - m Bn B + mCnC + m Dn D = 0
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Ideal-gas reaction
Chemical reaction: n A A + n B B Û n C C + n D D
Condition of equilibrium: -m An A - m Bn B + mCnC + m Dn D = 0
Chemical potential of a component in
an ideal- gas mixture:
Define
mi (T,P ) = mi0 (T ) + RT log Pi
n
Condition of equilibrium:
( )
n
PC C PDD
n A nB
PA PB
Equilibrium constant:
( )
( )
( )
( )
0
DG0 T º n C mC0 T + n Dm D
T - n Am 0A T - n Bm 0B T
= KP
é
0
DG
T
K P T = expêê
RT
ë
( )
( ) ùú
ú
û
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van’t Hoff equation
Equilibrium constant:
Algebra:
Calculus:
é
0
DG
T
K P T = expêê
RT
ë
( ) ùú
( )
( )
log K P T = -
dT
( )
( )
DG 0 T
( ) =-
d log K P T
ú
û
RT
(
( ))
d DG 0 T
RTdT
+
( )
DG0 T
RT 2
( )
Recall:
dDG0 T = -DS 0 T dT
Recall:
DG0 T = DH 0 T -TDS 0 T
van’t Hoff equation:
( )
( )
( )
( ) = DH 0 (T )
d log K P T
dT
RT 2
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Simultaneous reactions
Reaction 1
Reaction 2
1
H2O Û H2 + O2 ,
2
1
H2O Û H2 +OH,
2
PH P 1/2
2
O2
PH O
2
= K P1
P 1/2 POH
H2
PH O
2
= K P2
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plan
•
•
•
•
•
Definition of chemical potential
Examples of chemical potential
Equilibrium of two systems
Equilibrium of a chemical reaction
Equilibrium of phases
32
Equilibrium of two phases
U’, V’, NA’, NB’
S’(U’, V’, NA’, NB’)
phase (‘)
U’’, V’’, NA’’, NB’’
S’’(U’’, V’’, NA’’, NB’’)
isolated system
phase (‘’)
Thermal equilibrium: T ¢ = T ¢¢
Mechanical equilibrium: P¢ = P¢¢
Chemical equilibrium of component A: m¢A = m¢¢A
Chemical equilibrium of component B: m¢B = m¢¢B
In equilibrium, the two phases have the same temperature, the
same pressure, and the same chemical potential of each
component. A total of 2 + C equations. C = number of components.
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The Gibbs phase rule
IV = 2 + C - PH
IV = number of independent variables
C = number of components
PH = number of phases in equilibrium
C components: 1, 2,…, C
PH phases: (‘), (‘’),…
Composition of phase (‘): y’1, y’2,…y’C-1
Composition of phase (‘’): y’1, y’2,…y’C-1
…
All phases have the same T and the same P. 2 variables
Total number of number fractions: PH(C-1)
Chemical potential of each component is the same in all phases: (PH-1)C equations
IV = 2 + PH(C-1) –(PH-1)C
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The Gibbs phase rule
Pure substance, C = 1
Gibbs rule: IV = 3 - PH
IV = number of independent variables
PH = number of phases in equilibrium.
Number of phases in equailibrium
PH
IV
Single phase
1
2
Two phases in equilibrium (two-phase boundary)
2
1
Three phases in equilibrium (triple point)
3
0
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Two-component (binary) system, C = 2
Gibbs rule: IV + PH = 4
IV = number of independent variables
PH = number of phases in equilibrium.
Set pressure at a fixed pressure
A diagram of two variables, T and yB
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Three phases in equilibrium in a binary
mixture: eutectic point
https://en.wikipedia.org/wiki/Eutectic_system
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Water-salt phase diagram
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Three-component (tertiary) system, C = 3
Gibbs rule: IV + PH = 5
Set pressure at a fixed value
Set temperature at a fixed value
IV = number of independent variables
PH = number of phases in equilibrium.
Gibbs triangle: Each point in the triangle
represents a composition, , yB and yC
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Stainless steel phase diagram at 900 degrees Celsius (ASM 1-27)
Summary
• Chemical potential is a child of entropy and a
component.
• Chemical potential (of a species of molecules) in
a pure substance coincides with the Gibbs
function per molecule.
• Chemical potential of a component in an idealgas mixture is the same as that of the pure
component, provided we use partial pressure of
the component.
• Use chemical potential to analyze equilibrium of
systems, reactions, and phases.
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