8/18/2019 4 -- Formal Structure of Thermodynamics
1/29
MSEG 803Equilibria in Material Systems
4: Formal Structure of TD
Prof. ue!un "# $u
%u!ue!un&u'el.e'u
8/18/2019 4 -- Formal Structure of Thermodynamics
2/29
(st la) an' *n' la) in a sim+le system
(st la):
*n' la):
T%e functions U(S, V, N) an' S(U,V, N) are calle'
fundamental equations of a system. Eac% one of t%em
contains full information about a system.
Generally ener,y re+resentation
entro+y re+resentation
dU Q PdV δ = −
Q TdS δ =
dU TdS PdV = −1 P
dS dU dV T T
= +
i i
i
dU TdS y dx= + ∑1 i
i
i
ydS dU dx
T T = − ∑
8/18/2019 4 -- Formal Structure of Thermodynamics
3/29
Equations of state
T%e intensi-e -ariables in t%e fun'amental
equations )ritten as functions of t%e etensi-e
-ariables "for fie' mole numbers#:
Generally
dU TdS PdV = −
1 P dS dU dV
T T
= +
⇒
⇒
( , )T T S V = ( , ) P P S V − = −
1 1( , )S V
T T
= ( , ) P P
S V
T T
=
1 2( , ,..., ,...)i i i y y x x x=
8/18/2019 4 -- Formal Structure of Thermodynamics
4/29
/%emical +otential an' +artial molar quantities
/%emical +otential µ i for t%e com+onent i
uasi1static c%emical )or2
T%e +artial molar quantity x " x is an etensi-e function#
associate' )it% t%e com+onent i ")%en T P are constant#
+artial molar -olume
, ,...
i
i S V
U
N µ
∂= ÷∂ , ,...
i
i S V
S
N T
µ ∂= − ÷∂
c i i
i
W dN δ µ = ∑ i ii
dU TdS PdV dN µ = − + ∑
°
, , ( ) j
i
i T P N j i
x x
N ≠
∂= ÷∂
°
, , ( ) j
i
i T P N j i
V V
N ≠
∂= ÷∂
8/18/2019 4 -- Formal Structure of Thermodynamics
5/29
Euler relation
U an' S are bot% %omo,eneous first or'er
functions of etensi-e +arameters
λ is a constant
et λ = 1
Sim+le systems
1 2 1 2( , ,..., ,...) ( , ,..., ,...)i iU X X X U X X X λ λ λ λ =
1 2 1 2( , , ..., ,...) ( , ,..., ,...)i iU x x x U x x xλ λ λ λ
λ λ
∂ ∂=∂ ∂
1 2 1 2( , ,..., ,...) ( ) ( , ,..., ,...)
( ) ( )
i i ii
i ii i
U x x x x U x x xU x
x x
λ λ λ λ λ λ λ
λ λ λ
∂ ∂ ∂= × = ×
∂ ∂ ∂∑ ∑
1 2( , ,..., ,...)
( )
ii i i
i ii
U x x xU x y x
x
∂= × = ×
∂∑ ∑
i i
i
U TS PV N µ = − + ∑ 1 i ii
P S U V N
T T T
µ = + − ∑
8/18/2019 4 -- Formal Structure of Thermodynamics
6/29
Gibbs1Du%em relation
(st la) of TD:
in sim+le systems
5n a sin,le com+onent sim+le system:
i i
i
U TS PV N µ = − + ∑ ( ) ( ) ( )i ii
dU d TS d PV d N µ = − + ∑⇒
i i
i
dU TdS PdV dN µ = − + ∑
( ) ( ) ( )i i i ii i
TdS PdV dN d TS d PV d N µ µ − + = − +∑ ∑
0i i
i
SdT VdP N d µ − + =∑1
( ) ( ) ( ) 0iii
P Ud Vd N d
T T T
µ + − =∑
d sdT vdP µ = − +
8/18/2019 4 -- Formal Structure of Thermodynamics
7/29
Summary of t%e formal structure of TD
T%e fun'amental equation by itself contains full
information about t%e system
6 of con!u,ate -ariables: N
6 of ,enerali7e' )or2 terms: N - 1 6 of -ariables: 2N
6 of in'e+en'ent -ariables "t%ermo'ynamic 'e,ree of
free'om#: N - 1
6 of equations of state: N n in'i-i'ual equation of state 'oes not com+letely
c%aracteri7e t%e system
ll equations of state to,et%er contain full information about
t%e system "Euler relation#
8/18/2019 4 -- Formal Structure of Thermodynamics
8/29
Eam+le: i'eal ,as
9ot a fun'amental equation
9ot an eq. of state in t%e ener,y re+resentation
Gibbs1Du%am eq. in t%e
entro+y re+resentation
/ombine all 3 equations of state:
0 0
0 0
( ) ln( ) ln( ) P V u v
s c s c RT u v
µ = − = + +
PV NRT =
V U Nc T =
1 V c
T u=
P R
T v= 1( ) ( ) ( )
P d u d v d
T T T
µ = × + ×
( ) ( )V du dv
c Ru v
= − −
1( ) ( ) ( ) P S U V N T T T
µ = + −
8/18/2019 4 -- Formal Structure of Thermodynamics
9/29
Ener,y minimum +rinci+le
Entro+y maimum +rinci+le: in an isolate' system
equilibrium is reac%e' if S is maimi7e' %en dU = 0 "isolate' system# S is maimi7e' in equilibrium
Ener,y minimum +rinci+le: for a ,i-en -alue of total
entro+y of a system equilibrium is reac%e' if U isminimi7e' %en dS = 0 U is minimi7e' in equilibrium
0U
S x
∂ = ÷∂
2
2 0
U
S x
∂ ÷
∂
8/18/2019 4 -- Formal Structure of Thermodynamics
10/29
Ener,y minimum +rinci+le
t state S ta2es t%e maimum -alue if U is ta2en as a constant;
similarly U ta2es t%e minimum -alue if S is ta2en as a constant.
8/18/2019 4 -- Formal Structure of Thermodynamics
11/29
Ener,y minimum +rinci+le
Start )it%
See /allen section
8/18/2019 4 -- Formal Structure of Thermodynamics
12/29
T 1 T 2
Eam+le (: equilibrium in an isolate'
system after remo-al of an a'iabatic
+artition "i.e. only allo)s %eat flo)
bet)een sub1systems#
is a constant/onstraint:
t%ermal equilibrium
9o) instea' of t%e enclosure con'ition "dU = 0 # let=s start from
t%e ne) constraint t%at dStot = dS1 + dS2 = 0
t%ermal equilibrium: t%e same equilibrium state results>
1 21 2 1 2 1
1 2 1 2
1 1( ) 0tot
S S dS dS dS dU dU Q
U U T T δ
∂ ∂= + = × + × = × − =
÷ ÷∂ ∂
1 2tot U U U = + ⇒ 2 1Q Qδ δ = −
1 2T T =⇒
1 21 2 1 2 1 1 2
1 2
( ) 0tot U U
dU dU dU dS dS dS T T S S
∂ ∂= + = × + × = × − = ÷ ÷∂ ∂
1 2
T T =⇒
8/18/2019 4 -- Formal Structure of Thermodynamics
13/29
Sim+le mec%anical systems
Entro+y remains constant in a
+urely mec%anical system
)%ere
k is t%e s+rin, constant
( )U mgx F x dx= − + ∫ F kx=
0dU mgdx Fdx= − + =
mg F kx= =⇒
8/18/2019 4 -- Formal Structure of Thermodynamics
14/29
e,en're transformations
?ot% S an' U are functions of etensi-e -ariables; %o)e-er in+ractical e+eriments ty+ically t%e controlle' -ariables
"constraints# are t%e intensi-e ones>
e,en're transformations: fun'amental relations e+resse'
as functions of intensi-e -ariables
e,en're transformations +reser-e t%e informational content
e,en're transform of a fun'amental equation is also a
fun'amental equation
8/18/2019 4 -- Formal Structure of Thermodynamics
15/29
Ent%al+y H(S, P, N) = U + PV = TS + µ N
Partial e,en're transform of U : re+laces t%e etensi-e +arameter V )it% t%e intensi-e +arameter P
For a com+osite system in mec%anical contact )it% a +ressure
reser-oir t%e equilibrium state minimi7es t%e ent%al+y o-er t%e mani1
fol' of states of constant +ressure "equal to t%at of t%e reser-oir#.
Ent%al+y c%an,e in an isobaric +rocess is equal to %eat ta2en in or
,i-en out from t%e sim+le system
Differential: ( )dH d U PV TdS VdP dN µ = + = + + ∑
8/18/2019 4 -- Formal Structure of Thermodynamics
16/29
Ent%al+y minimi7ation +rinci+le
/onsi'er a com+osite system )%ere all sub1systems
are in contact )it% a common +ressure reser-oir
t%rou,% )alls non1restricti-e )it% res+ect to -olume
++ly U minimum +rinci+le to reser-oir @ system:
T%e system is in mec%anical equilibrium )it% t%e
reser-oir:
0r r r r tot sys sys sys sysdU dU dU dU P dV dU P dV = + = − = + =
r
P P =⇒ ( ) 0tot sys sys sys sys sys sys sysdU dU P dV d U P V dH = + = + = =
2 2 2( , ) ( ) 0r sys sys sys sys sys sysd U S V d U P V d H = + = >
8/18/2019 4 -- Formal Structure of Thermodynamics
17/29
$elm%olt7 +otential F(T, V, N) = U - TS = - PV + µ N
Partial e,en're transform of U : re+laces t%e etensi-e
+arameter S )it% t%e intensi-e +arameter T
For a com+osite system in t%ermal contact )it% a
t%ermal reser-oir t%e equilibrium state minimi7es t%e
$elm%olt7 +otential o-er t%e manifol' of states ofconstant tem+erature "equal to t%at of t%e reser-oir#.
Differential: ( )dF d U TS SdT PdV dN µ = − = − + ∑
8/18/2019 4 -- Formal Structure of Thermodynamics
18/29
$elm%olt7 free ener,y
System in t%ermal contact )it% a
%eat reser-oir
T%e )or2 'eli-ere' in a re-ersible
+rocess by a system in contact )it%
a t%ermal reser-oir is equal to t%e
'ecrease in t%e $elm%olt7 +otential
of t%e system
$elm%olt7 Afree ener,yB: a-ailable
)or2 at constant tem+erature
System
δ Q
δ W
State → ?: dF
$eat
reser-oir at T
( )
r r W dU Q dU T dS
d U TS dF
δ δ = − = +
= − =
8/18/2019 4 -- Formal Structure of Thermodynamics
19/29
or2 +erforme' by a system in contact )it%
%eat reser-oir cylin'er contains an internal +iston on eac% si'e of
)%ic% is one mole of a monatomic i'eal ,as. T%e cylin'er
)alls are 'iat%ermal an' t%e system is immerse' in a
%eat reser-oir at tem+erature 0C/. T%e initial -olumes oft%e t)o ,aseous subsystems "on eit%er si'e of t%e
+iston# are (0 an' ( res+ecti-ely. T%e +iston is no)
mo-e' re-ersibly so t%at t%e final -olumes are an' <
res+ecti-ely. $o) muc% )or2 is 'eli-ere' Solution (: 'irect inte,ration of PdV "isot%ermal +rocess#
Solution *: ∆W = ∆F
8/18/2019 4 -- Formal Structure of Thermodynamics
20/29
$elm%olt7 +otential minimi7ation +rinci+le
/onsi'er a com+osite system )%ere all sub1systems
are in t%ermal contact )it% a common %eat reser-oir
t%rou,% )alls non1restricti-e )it% res+ect to %eat flo)
++ly U minimum +rinci+le to reser-oir @ system:
T%e system is in t%ermal equilibrium )it% t%e reser-oir:
0r r r r tot sys sys sys sysdU dU dU dU T dS dU T dS = + = + = − =
r T T = ⇒ ( ) 0tot sys sys sys sysdU d U T S dF = − = =2 2 2( ) ( , ) 0 sys sys sys sys sys sys sysd F d U T S d U S V = − = >
8/18/2019 4 -- Formal Structure of Thermodynamics
21/29
Gibbs +otential G(T, P, N) = U - TS + PV = µ N
e,en're transform of U : re+laces bot% S an' V )it% t%eintensi-e +arameters T an' P
For a com+osite system in contact )it% a t%ermal
reser-oir an' a +ressure reser-oir t%e equilibrium state
minimi7es t%e Gibbs +otential o-er t%e manifol' of statesof constant tem+erature an' +ressure.
Differential: ( )dG d U TS PV SdT VdP dN µ = − + = + + ∑
8/18/2019 4 -- Formal Structure of Thermodynamics
22/29
Gibbs free ener,y an' c%emical +otential
Sim+le systems:
Sin,le com+onent systems:
Multi1com+onent systems:
molar Gibbs +otential
+artial molarGibbs +otential
/onsi'er a c%emical reaction:
c%emical equilibrium con'ition
i i
i
G U TS PV N µ = − + = ∑G
N µ =
i i
i
G x N
µ = ∑0i i
i
v A∑ ƒ °i
i
dN
const d N dv = = °
0i i i ii i
dG SdT VdP dN v d N µ µ = + + = =∑ ∑⇒0i i
i
v µ =∑
8/18/2019 4 -- Formal Structure of Thermodynamics
23/29
First or'er +%ase transition
t T m 0 C/ an' ( atm liqui' )ater an' ice can
coeist dGwate-i!e = d(H - TS) = 0 at T m 0 C/ ( atm
∆Swate-i!e = ∆H wate-i!e "T m ∆U wate-i!e "T m at T m 0 C/ (atm
T%e 'iscontinuity of H an' U are c%aracteristic of
first or'er +%ase transition
t T H 0 C/ an' ( atm ice s+ontaneously melts dGwate-i!e = d(H - TS) # 0 at T H 0 C/ ( atm
∆Swate-i!e # ∆H wate-i!e "T = ∆Qwate-i!e "T : irre-ersible
+rocess
Th d i E t
8/18/2019 4 -- Formal Structure of Thermodynamics
24/29
Constraints Thermodynamic
potentialExtremumprinciple
Example
U V constant
dU = 0, dV = 0
S (U, V, N) = U"T + PV"T -
µ N"T
dS = 1"T$dU + P"T$dV - µ "T$dN
S ma
dS = 0, d
2
S % 0
5solate' systems
S V constantdS = 0, dV = 0
U (S, V, N) = TS - PV + µ N
dU = TdS - PdV + µ dN
U mindU = 0, d 2 U # 0
Sim+le mec%anicalsystems consistin, of
ri,i' bo'ies
S P constantdS = 0, dP = 0
H (S, P, N) = TS + µ N
dH = TdS + VdP + µ dN
H mindH = 0, d 2 H # 0
Systems in contact )it%a +ressure reser-oir'urin, a re-ersiblea'iabatic +rocess
T V constantdT = 0, dV = 0
F (S, V, N) = - PV + µ N dF = - SdT - PdV + µ dN
F mindF = 0, d 2 F # 0
Ieactions in a ri,i''iat%ermal container at
room tem+erature
T P constantdT = 0, dP = 0
G (T, P, N) = µ N
dG = - SdT + VdP + µ dN
G mindG = 0, d 2 G # 0
E+eriments +erforme'at room tem+erature an'
atmos+%eric +ressure
8/18/2019 4 -- Formal Structure of Thermodynamics
25/29
Generali7e' Massieu functions
e,en're transforms of entro+y S
Maimum +rinci+les of Massieu functions a++ly
8/18/2019 4 -- Formal Structure of Thermodynamics
26/29
General case
e,en're transform re+laces a -ariable )it% its con!u,ate
For a t%ermo'ynamic system its TD function )ill be t%e
e,en're transform )%ere t%e -ariables are constraine'
/ontrolle' -ariables: ε && , σ xx , σ ''
is t%e TD +otential t%at
is minimi7e' in equilibrium
?eam
)all )all
7
y
0, ,
ii iii x y z dU TdS dN V d µ σ ε == + + ∑
0 0
0 0 0
( ) xx xx yy yy
zz zz xx xx yy yy
d d U TS V V
SdT dN V d V d V d
φ σ ε σ ε
µ σ ε ε σ ε σ
= − − −
= − + + − −
( , , , , ) xx yy zz T N φ σ σ ε
8/18/2019 4 -- Formal Structure of Thermodynamics
27/29
Deri-in, equilibrium con'itions
T, N 1 T ,N 2 P P
Equilibrium in a system surroun'e'by 'iat%ermal im+ermeable )alls in
contact )it% a +ressure reser-oir after
remo-al of an im+ermeable +artition
"i.e. allo)s mass flo) bet)een sub1
systems#.are constants/onstraints:
is a constant
c%emical equilibrium
Gibbs +otential minimi7ation:
,T P
1 2tot N N N = + ⇒ 2 1dN dN = −
1 21 2
1 2
1 1 2( ) 0
tot
G GdG dN dN
N N
dN µ µ
∂ ∂= × + × ÷ ÷∂ ∂
= × − =
⇒ 1 2 µ µ =
8/18/2019 4 -- Formal Structure of Thermodynamics
28/29
/ou+le' equilibrium@
@
@@
@
@
@
@
V e
bo of an electrically con'ucti-e solutioncontainin, a +ositi-ely c%ar,e' ion "+e#
s+ecies is se+arate' into t)o +arts by an
im+ermeable electrically insulatin, internal
+artition. -olta,e ∆V is a++lie' across
+ -
t%e t)o +arts. 5f t%e +artition becomes +ermeable to t%e ion but still remains
insulatin, )%at is t%e equilibrium con'ition )it% res+ect to t%e ion "assumin,constant tem+erature an' +ressure#
Fun'amental equation:
/onstraints:
dG SdT VdP dN Vd! µ = − + + +
1 1 1 11 2 1 2
1 2 1 2
1 1 2 1 ,1 , 2 1 1 2 1( ) ( ) ( )
tot
" " "
G G G GdG dN dN d! d!
N N ! !
dN d! V V dN d! V µ µ µ µ ∆
∂ ∂ ∂ ∂= × + × + × + ×
÷ ÷ ÷ ÷∂ ∂ ∂ ∂ = × − + × − = × − + ×
1 2dN dN = −
1 2d! d!= − 1 1d! #" dN = ×
1
( ) 0tot "
dG dN #" V ∆µ ∆= × + = ⇒ 1 ,1 2 ,2" " #"V #"V µ µ + = +
8/18/2019 4 -- Formal Structure of Thermodynamics
29/29
Electroc%emical +otential
Describes t%e equilibrium con'ition of c%ar,e'
c%emical s+ecies "ions electrons#
/%emical +otential: Electroc%emical +otential:
)%ere is t%e -alence numberof t%e ion "'imensionless# e is
t%e elementary c%ar,e an' V
is t%e local electrical +otential
Eam+le: ion 'iffusion across
cell membrane
" #"V µ µ = + µ
Top Related