6.0 The Fundamental Equations

2

6.0 The Fundamental Equations
• The First and Second Laws of thermodynamics are both
relevant to the behavior of matter, and we can bring the
whole force of thermodynamics to bear on a problem by
setting up a formulation that combines them.
3

Combining the First and Second Laws
• The essential thermodynamic properties: T, U, S.
• For isolated system: (q = 0)
– Entropy as a good criterion for spontaneity and equilibrium.
• For close system (constant composition, but not constant q ) :
– Not a good criterion for constant (T, V ) or (T, P ).
– Need two more thermodynamic properties as criterion.
– Using Legendre transforms to generate two new functions.
– Helmholtz free energy (A ) as criterion for constant T and V.
– Gibbs free energy (G ) as criterion for constant T and p.
• For open system:
– Chemical potentials for each species of a system at equilibrium is the
same.
– Spontaneous mixing of two partially miscible liquids at specific T and
P.
4

5

6

6.1 The fundamental equation
• We have seen that the First Law of thermodynamics may be
written dU = dq + dw.
• For a reversible change in a closed system of constant
composition, and in the absence of any additional (nonexpansion)
work, we may set dw rev = −p dV and (from the
definition of entropy) dq rev =T dS, where p is the pressure of
the system and T its temperature.
• Therefore, for a reversible change in a closed system,
dU = T dS − p dV
7

6.1 The fundamental equation
• However, because dU is an exact differential, its value is
independent of path. Therefore, the same value of dU is
obtained whether the change is brought about irreversibly or
reversibly. Consequently, eqn
dU = T dS − p dV
applies to any change-reversible or irreversible- of a closed
system that does no additional (non-expansion) work. We
shall call this combination of the First and Second Laws the
fundamental equation.
• The fact that the fundamental equation applies to both
reversible and irreversible changes may be puzzling at first
sight. The reason is that only in the case of a reversible
change may T dS be identified with dq and −pdV with dw.
When the change is reversible, T dS > dq (the Clausius
inequality) and −p dV > dw. The sum of dw and dq remains
equal to the sum of T dS and −p dV, provided the
composition is constant. 8

6.1 Properties of the internal energy
• The eq. dU = T dS − p dV shows that the internal
energy of a closed system changes in a simple way
when either S or V is changed (dU ∝ dS and dU ∝ dV).
These simple proportionalities suggest that U should be
regarded as a function of S and V. We could regard U as
a function of other variables, such as S and p or T and V,
because they are all interrelated; but the simplicity of
the fundamental equation suggests that U(S,V ) is the
best choice.
• The mathematical consequence of U being a function of
U
U
S and V is that we dUcan express dS
an infinitesimal dV
change
S
V
V
S
dU in terms of changes dS and dV by
9

• The two partial derivatives are the slopes of the plots of
U against S and V, respectively.
• When this expression is compared to the
thermodynamic relation, we see that, for systems of
constant composition,
T
dU
d S
V
P
dU
dV
S
10

Combination of 1st and 2nd law for close system
– 1st law: dU = dq + dw.
– 2nd law: dS > dq irrev /T and dS = dq rev /T .
– If only PV-work involved, dw = - P dV
– combine 1st law and 2nd law for close system: dU = T dS - P
dV
⇨ It is the fundamental equation in a close system involving only
PV work, (an equation involves only state functions U, S, T, P,
V ). It applies to both reversible and irreversible processes
(state functions).
⇨ Just like -P and V are conjugate variables for work, dw = - P dV ,
now T and S are Ualso conjugate U
variables for heat: dq rev = T dS .
dU
S V
• Since internal energy
d d
S U (S,V
) Vis
an exact differential:
V
S
T
dU
d S
V
P
dU
dV
S
11

Fundamental Equations of Thermodynamics
• Derivative of an extensive property with respect to an extensive
property gives an intensive property.
• U (T,P ) or U (T,V ) can not be used to calculate all the
thermodynamic properties of system, only U (S,V ) will. And S
and V are called the natural variables of U
• If U = U (T,V ) =>
dU
U
V
dV
U
T
dT
• At constant P, (∂U/∂V ) T , (∂U/∂T ) V are not convenient forms.
T
V
1
α
V
x
T
P
V
T
P
U
T
P
U
T
P
U
V
π
T
T
αV C
12
V
T
V
P
U
T
V
π
T
U
V
T : internal pressure
T

Fundamental Equations of Thermodynamics
• To show how measurable properties T, V of a close system
(constant composition) can be related to thermodynamic quantities
like H :
For enthalpy H = U + PV,
dH = dU + P dV + V dP = (T dS – P dV ) + P dV + V dP
As H (S, P ), enthalpy changes with natural variable of S, P,
dH = T dS + V dP
H
S
T
Furthermore, dH as exact differential, the relations:
p
H
P
S
V
p
H
S
p
S
S
H
P
S
p
T
P
S
V
S
P
13

Fundamental Equations of Thermodynamics
For Helmholtz free energy A = U - TS,
dA = dU – T dS – S dT = (T dS – P dV) – T dS – S dT
As free energy A (V, T ), changes with natural variable of V, T,
dA = - S dT – P dV
A
T
S
Furthermore, dA as exact differential, we have
V
A
V
T
P
V
A
T
V
T
T
A
V
T
V
S
V
T
P
T
V
• Now the internal pressure T :
π
T
U
V
T
S
T
V
T
P
P
T
T
V
P
14

Fundamental Equations of Thermodynamics
For Gibbs free energy G = U + PV – TS = H - TS,
dG = dH – T dS – S dT = (T dS + V dP ) – T dS – S dT
As free energy, G (P,T ) changes with natural variable of P, T,
dG = - S dT + V dP
G
T
Furthermore, dG as exact differential, we have
p
S
G
P
T
V
P
G
T
p
T
T
G
P
T
p
S
P
T
V
T
P
15

Fundamental Equations of Thermodynamics
• 在 均 相 系 統 的 外 延 性 質 狀 態 函 數
• U (S,V ), H (S,P ), A (T,V ), G (T,P ) 可 表 為 :
dU = +T dS - P dV
dA = - S dT - P dV
dH = +T dS + V dP
dG = - S dT + V dP
+T dS
dU
dH
-P dV
dA
dG
+V dP
-S dT
16

Fundamental Equations of Thermodynamics
• Furthermore, from U (S,V ), H (S,P ), A (T,V ), G (T,P ) ) we
have the four relations:
U
S
V
H
S
p
T
H
P
S
G
P
T
V
U
V
S
A
V
T
P
A
T
V
G
T
p
S
17

Legendre transforms:
A linear change of variables that involves subtracting the product of conjugate
variables from an extensive property of a system.
For any state function f (X,Y), since df (X,Y) is exact differential;
f f
df dX dY m dX n dY
X Y
Y
X
now df = m dX+n dY with
f f
m n
m n X Y
Y
X
also
Y
X X Y
f
Now we define a new state function: gm,Y
f X,Y
X
X Y
g f mX
dg df d mX m dX n dY m dX X dm X dm
n
dY
For state function g (m,Y ): dg
g g
with - X and n
m Y
Y
g
m
dm
g
Y
Also dg is exact differential ====>
m
Y
f
Y
18
X
m
dY
X
X
Y
dm
m
n dY
n
m
Y

均 相 封 閉 系 統 的 熱 力 學 性 質 轉 換 :
•
dU
-P
dV
T
dS
m P
m T
U
V
U
S
dH
T
dS
V
dP
V
S
H U PV U
A U T
S U
U
V
U
S
V
S
V
S
df
m dX
n
dY
g f mX
dg
X
dm
n
dY
dH
V
dP
T
dS
dA
S
dT
P dV
m T
m V
H
S
P
H
P
S
G H T
S H
U H V P H
H
S
H
P
P
S
S
P
dG
dU
-S
dT
V
-P
dV
T
dP
dS
19

均 相 封 閉 系 統 的 熱 力 學 性 質 轉 換 :
•
dA
m
m
dG
S
dT
P dV
P
S
A
V
A
T
T
V
-S
dT
V
dP
G A P V A
U A S T A -
A
V
A
T
V
T
T
V
df
m dX
n dY
g f mX
dg
X
dm
n
dY
dG
V
dP
- S dT
dU
T
dS
- P dV
m S
G
T
P
H G ST G
G
T
P
T
dH
T
dS
V
dP
m V
G
P
T
A G V P G
G
P
T
P
dA
P
dV
S dT
20

Fundamental Equations of Thermodynamics
• 由 均 相 封 閉 系 統 的 狀 態 方 程 式 得 到 的 性 質 :
dU
-P
dV
T
dS
T
U
S
T
V
d
H
S
H T
dS
V
P
dP
U
P -
-
V
S
A
V
T
P
dU dH
dA
dG
V
V
H
P
S
G
P
T
S
A
S
T
V
G
T
P
dA
S
dT
P dV
dG
-S
dT
V
dP
21

Fundamental Equations of Thermodynamics
• For close system, we derive the Maxwell relations for state
functions U=U (S,V ), H=H (S,P ), A=A (T,V ), G=G (T,P ) :
Internal energy dU = T dS - P dV.
U = H - PV = A + TS = G - PV + TS
Enthalpy dH = T dS + V dP
H = U + PV = A + PV + TS = G + TS
Helmholtz free energy dA = - P dV - S dT
A = U -TS = H -PV-TS = G -PV
Gibbs free energy dG = V dP - S dT.
G = U + PV - TS = H - TS = A + PV
T
V
T
P
P
T
S
S
V
V
T
p
P
S
V
S
S
V
T
p
V
S
P
T
22

Fundamental Equations of Thermodynamics
• From U (S,V ), H (S,P ), A (T,V ), G (T,P ) we have
H = U + PV = A + PV + TS = G + TS
U = H - PV = A + TS = G - PV + TS
G = U + PV - TS = H - TS = A + PV
A = U - TS = H - PV - TS = G - PV
23

Chemical Potential
• In 1876 Gibbs introduced the concept of the chemical potential μ
to the fundamental equations in order to discuss phase
equilibrium and reaction equilibrium of of a species. For U:
dU = T dS – P dV + μ 1 dn 1 + μ 2 dn 2 + …
• If dn i moles of species i are added to system at constant S and V,
there is a change in internal energy in terms of μ 1 dn 1 .
• If a system contains N s different species, U is a function of its
natural variables S, V and {n i }:
dU = T dS –P dV +
N S
i 1
μ d
i
n i
• The natural variables of U are all extensive:
N S
U
U
dU
dS
dV
S
V
, n
V
S
, n
i 1
i
i
U
ni
S , V , n
j i
dn
i
24

Fundamental Equations of Thermodynamics
• Fundamental Equation: An expression show how the internal
energy U (S,V ) changes with natural variable of S, V
• dU = T dS - P dV + μ
i
dn i where dU is an exact
differential.
i 1
• Several important expressions can be derived from this relation,
including the Maxwell relations.
• The relations:
•
T
U
S
show how measurable properties T, P of a open system can be
related to thermodynamic quantities like U. Furthermore,
U
V
S
V
V ,
S
n i
N S
P
U
S
V
S
V
25
U
V
S ,
n i
T
V
S ,
i
U
n
i
P
S
S , V ,
n
V ,
i n i
n j i

Chemical Potential
dU = T dS - P dV +
N S
i 1
μ d
i
n i
U
S
V
U
U
dS
dV
d
V
n
n
,
i
S , ni
i S
, V , n
j i
n
i
• 若 系 統 性 質 可 依 其 自 然 變 數 的 函 數 決 定 , 則 任 ㄧ 內 涵 性 質 (T, P, μ i ) 均 可 為
自 身 的 兩 種 外 延 性 質 的 比 值 . 此 三 式 可 稱 為 系 統 的 狀 態 函 數 . 對 內 能 的 馬
克 斯 威 關 係 式 表 為 :
T
V
T
n
i
S ,
n
V ,
S,V,
i n i
n
V, n
i
i
P
S
μi
S
P
n
i
μ
i
n
j
S,V,
S,V,n
n
S, n
i
i
j i
μi
V
μ
j
n
i
S,V,n
i j
26

Chemical Potential
• If we substitute the second law in the form dS ≥ δq/T and
若 將 熱 力 學 第 二 定 律 的 形 式 換 成 不 同 的 表 示 : dS ≥ δq/T 與
ƌw=-P ext dV+
dU
T
dS
P
N S
i 1
ext
μ i
dn i
dV
N S
i 1
dn
i
, 則 內 能 變 化 量 dU
• The internal energy remains constant if the infinitesimal change
occurs at equilibrium under constant entropy, volume, and {n i }
在 恆 體 積 , 固 定 成 份 與 恆 熵 值 下 的 平 衡 , 其 內 能 維 持 不 變 .
(dU ) S,V,{n ≤ 0 .
j}
• The criterion for spontaneous change in the system involving PVwork
and specified amounts of species., U must be approaching
the minimum at constant S, V, and {n i }:
在 恆 體 積 , 固 定 成 份 與 恆 熵 值 下 的 自 發 變 化 , 其 內 能 變 化 趨 向 於 最 低 值 .
• The integrated form as:
N S
均 相 開 放 系 統 的 內 能 以 積 分 形 式 表 示 為 : U T
S P V
27
i
i 1
in i

Fundamental Equations of Thermodynamics
• 由 均 相 開 放 系 統 的 狀 態 方 程 式 得 到 的 性 質 :
U
A
P -
-
V
V
S
T
, n
T ,
i n i
P
U
S
T
U
A
V
S
A
S
T
V
H
S
, n
P ,
i n i
H
G
V
V
G
T
, n
P ,
i n i
H
P
S
G
P
, n
T ,
i n i
28

Fundamental Equations of Thermodynamics
• 均 相 開 放 系 統 的 內 涵 性 質 的 狀 態 方 程 式 :
dU = T dS - P dV
U
P -
V
T
T
U
S
V
H
S
P
S
,
n i
,
n i
,
n i
dH
T
dS
V
dP
N S
i 1
N S
i 1
in i
i
dn i
H U PV
U
A U T
S U
G H T
S H
U
V
U
S
S ,
V ,
H
S
P ,
n i
n i
n i
V
S
S
P
U
A
T
S
H
G
V
V
H
P
S
,
n i
U H V P H
H
P
S ,
n i
P
29

Fundamental Equations of Thermodynamics
• 均 相 開 放 系 統 的 內 涵 性 質 的 狀 態 方 程 式 :
dA = -S dT - P dV
A
P
V
dG
-S
dT
V
dP
T
A
S
T
V
G
S -
T
P
,
,
n i
n i
,
n i
N
S
i 1
N S
i 1
i
dn i
G A P V A
in i
H G ST G
A
V
A
U A S T A -
T
V ,
G
T
T ,
P ,
n i
n i
n i
T
V
T
P
U
A
T
S
H
G
V
V
G
P
T
,
n i
A G V P G
G
P
T ,
n i
P
30

Fundamental Equations of Thermodynamics
• From U (S,V ), H (S,P ), A (T,V ), G (T,P ) we have
H U PV
G TS
N S
TS i n i
i 1
G
N S
i 1
i
n i
U G
PV
TS
TS
PV
N S
G H - TS
i 1
U-TS
PV
in i
A PV
N S
i 1
in i
A U - TS
N S
PV i
n i
i 1
G PV
31

Fundamental Equations of Thermodynamics
• 均 相 開 放 系 統 在 平 衡 下 有 固 定 的 內 涵 性 質 (T, P, μ i ) :
U TS
PV
H TS
A
N
PV
N S
i 1
S
i 1
N
G i
n i
G
i
n
i
i
N
n
S
i 1
T , P ,
S
i 1
i
i
n
n j i
n
i
i
G
G PV TS
G P
P
G
G TS
G T
T
i
G PV
U
i
n
i
P , n
G
G P
P
S , V ,
n j i
i
T , n
H
i
n
i
i
S , P ,
n j i
T ,
n
P , n
i
T
A
i
n
i
G
T
T , V ,
n j i
i
32

Fundamental Equations of Thermodynamics
Example1 Calculation of molar thermodynamic properties for an ideal gas.
Since the molar Gibbs energy of an ideal gas is given by
G
P
G RT ln
P
Derive the corresponding expressions for V , U , H , S , and A.
Ans: Using equation,
G
RT G
G
P
V S
P
P
R ln S
T
T
T
P
P
P
P
A G PV A RT ln H G T
S G T S H
P
U G T S PV G T
S PV H -PV
H RT U
G
S -
and U G T
S -RT
where
T
P
Note that the internal energy U and enthalpy H of an ideal gas are
independent of pressure and volume.
-R ln
P
P
33

Fundamental Equations of Thermodynamics
• 均 相 開 放 系 統 的 平 衡 或 自 發 的 要 件 與 熱 力 學 基 本 關 係 :
dU
T
dS
P dV
dH
T
dS
V
dP
N
N
S
i 1
S
i 1
dA
S
dT
P dV
dG
S
dT
V
dP
U
ni
H
ni
N
S
i 1
N
S
i 1
S , V , n
S , P , n
A
ni
G
ni
j i
j i
d
d
T , V , n
T , P , n
j i
j i
n
n
i
d
i
n
dn
i
i
dU
0
S , V , n
dH 0
( P P )
i
S , P , n
ext
dA 0
T , V , n
dG
0
( P P , T T
)
T , P , n
ext surr
i
i
i
34

Fundamental Equations of Thermodynamics
For Irreversible
Processes
For Reversible
Processes
( dS ) V,U,{nj} > 0 ( dS ) V,U ,{nj} = 0
( dU ) V,S,{nj} < 0 ( dU ) V,S,{nj} = 0
( dH ) P,S,{nj} < 0 ( dH ) P,S,{nj} = 0
( dA ) T,V,{nj} < 0 ( dA ) T,V,{nj} = 0
( dG ) T,P,{nj} < 0 ( dG ) T,SP,{nj} = 0
Table1 Criteria for Irreversibility and Reversibility for Processes
Involving No Work or Only Pressure-Volume Work.
35

Fundamental Equations of Thermodynamics
Fig1 When a system undergoes
spontaneous change at constant T
and P, the Gibbs energy
decreases until equilibrium is
reached.
36

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40

41

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43

44

45

46

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50

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