Chap.5
Thermodynamic
Properties
of
Homogeneous Mixture
(1) Chemical potential, μi : (為了解決數學上單位之問題)
目的:探討熱力學性質受溶液組成變化之影響
For pure substance: dU=TdS - PdV
→ d(nU)= Td(nS) - Pd(nV) …state change
For mixture: d(n U) = Td(nS) - Pd(n V) + ∑ μ i dn i …… (a)
i
同理:
d(n H) = Td(nS) - (n V)dP + ∑ μ i dn i
i
…… (b)
d(n A) = -(nS)dT - Pd(n V) + ∑ μ i dn i
…… (c)
d(n G ) = -(nS)dT + (n V)dP + ∑ μ i dn i
…… (d)
i
i
⎡ ∂ (n U) ⎤
由(a) → μ i = ⎢ ∂n ⎥
i ⎦ n S, n V , n
⎣
j≠ i
⎡ ∂ (n H) ⎤
由(b) → μi = ⎢ ∂n ⎥
i ⎦ n S, P, n
⎣
j≠i
⎡ ∂ (n A) ⎤
由(c) → μ i = ⎢ ∂n ⎥
i ⎦ T, n V , n
⎣
j≠i
⎡ ∂ (n G ) ⎤
由(d) →
μi = ⎢
⎥
⎣ ∂n i ⎦ T,P,n
= G i : partial molar Gibbs free
j≠i
energy of i
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Edited by Prof. Yung-Jung Hsu
(2) Partial molar property,
Mi :
偏微性質
M : 溶液總熱力學性質，M=PVTSHUGA。
M i : 物質 i 於純物質狀態下所表現之熱力學性質。
M i : 物質 i 於混合質狀態下所表現之熱力學性質。
Solution behaviors:
1. ideal solution
a.體積可加成，V=X1V1+X2V2。
b.混合過程不放熱，ΔH= 0(T=constant)
2. real solution
a. 體積不可加成，V≠X1V1+X2V2。(因混合前後分子間吸引
力改變)
b. 混合過程放熱，ΔH≠ 0(T≠ constant)
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For real solution: V≠X1V1+X2V2
要達到此線性組合，並非該物質單獨存在時所存在時所表現之熱
力學性質 M i ，而是該物質於溶液狀態下所展現之熱力學性質
M i ，則此線性組成關係才會成立
→ V = X 1V1 + X 2 V2
Define: partial molar property M i :
1 M i 之線性組合關係式 M = ∑ ni M i or M = ∑ X i M i
○
i
i
2 微分式: 因為 M = ∑ ni M i = n1 M 1 + n2 M 2
○
i
⎡ ∂M ⎤
⎛ ∂n ⎞
⎛ ∂n
= M 1 ⎜⎜ 1 ⎟⎟
+ M 2 ⎜⎜ 2
⎢
⎥
⎝ ∂n1 ⎠ T , P ,n2
⎣ ∂n1 ⎦ T , P ,n2
⎝ ∂n1
⎞
⎟⎟
= M1
⎠ T , P , n2
⎡ ∂M ⎤
= M2
⎢
⎥
∂
n
⎣ 2 ⎦ T , P , n1
⎡ ∂M ⎤
M
=
, M=VSHUGA
→ i ⎢ ∂n ⎥
⎣ i ⎦ T , P ,n j ≠ i
⎡ ∂G ⎤
= μi
If M=G 帶入， Gi = ⎢ ⎥
⎣ ∂ni ⎦ T , P ,n
j ≠i
∵ V = nV1 + nV2 , for real solution(1+2) as a function of n2 added into the
solution.
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(3) Gibbs-Duhem Equation
For any homogeneous fluid in a steady-state:
1. M i 線性組合: 因為 M = ∑ ni M i
i
For a binary A-B solution and consider the Gibbs free energy:
→ G = n A G A + nB GB
For one-mole of solution: G = X A G A + X B G B
→ d d G = G A dX A + G B dX B + X A d G A + X B d G B
-----(1 式)
2.函數分析: 因為 G=G(T, P, nA, nB)
全微分 →
⎛ ∂G ⎞
⎛ ∂G
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟⎟
dG = ⎜
dn A + ⎜⎜
⎟ dT + ⎜
⎟ dP + ⎜⎜
⎝ ∂T ⎠ P ,n
⎝ ∂P ⎠ T ,n
⎝ ∂n A ⎠ T , P ,nB
⎝ ∂n B
⎞
⎟⎟
dn B
⎠ T , P ,n A
⎛ ∂G ⎞
⎛ ∂G ⎞
=⎜
⎟ dT + ⎜
⎟ dP + G A dn A + G B dn B
⎝ ∂T ⎠ P ,n
⎝ ∂P ⎠ T ,n
For one-mole solution:
⎛ ∂G ⎞
⎛ ∂G ⎞
dG = ⎜
⎟ dT + ⎜
⎟ dP + G A X A + G B X B
⎝ ∂T ⎠ P , X
⎝ ∂P ⎠ T , X
-----(2 式)
(1 式)=(2 式)
⎛ ∂G ⎞
⎛ ∂G ⎞
dG = ⎜
⎟ dT + ⎜
⎟ dP − ( X A d G A + X B d G B ) = 0 ⎝ ∂T ⎠ P , X
⎝ ∂P ⎠ T , X
→ Gibbs‐Duhem Equation ∂M ⎞
⎛ ∂M ⎞
⎟ dT + ⎜
⎟ dP − ∑ X i d M i = 0
⎝ ∂T ⎠ P , X
⎝ ∂P ⎠ T , X
i
⎛
General form: d M = ⎜
(M=VSHUGA)
At T, P=constant, ∑ X i d M i = 0
i
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Edited by Prof. Yung-Jung Hsu
(4) Partial molar properties 之間的數學關係式
Ex1: H i = U i + P V i
Ex2: dG i = -Si dT + V i dP
→ Partial molar property 與一般熱力學性質具有相同數學關係式
(5) Fugacity(fi)，Activity(ai)
目的:解決純物質之 G 於某一情況下所發生之不合理現象。
For pure substance: dG= -SdT+VdP, at constant T → dG=VdP.
For ant pure gas i : dGi=VidP(any gas, constant T)
For ideal gas i : dG i =
RT
dP = RTdlnP (ideal gas, constant T)
P
Because any real gas at P=0 → ideal gas 代入上式
→ dG i
P =0
= RTdlnP P =0 (any gas at P=0, constant T) =RTdln0 (不合理)
→ for any gas at P=0, dG i ≠ RTdlnP
定義: Fugacity(fi)逸壓: dG i = RTdlnf i (constant T)
→ 為了代替原來真實壓力所定義出之假想壓力即為 Fugacity
Summary: ideal gas and constant T : dG i = RTdlnP
real gas and constant T : dG i = RTdlnf i
for ideal gas: f i = P,
for real gas: f i ≠ P,
fi
=1
P
fi
≠1
P
for mixture: dG i = RTdlnf i (因為壓力沒有 partial molar pressure)
→
2
2
∫ d G = ∫ RTd ln f
i
1
i
1
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2
ΔG i = G2 − G1 = RT ∫ d ln f i = RT ln
1
f2
f1
※ if take 1 as the pure state of a material and also view it as the
standard state, then Gi − Gi , pure = Gi − G 0 = RT ln
fi
f i0
fi
, 上式變為 Gi − Gi ,
= RT ln a i pure
f i0
Define: Activity, ai ≡
(6) Activity
1. Raoult’s Law:
Consider a binary A-B solution in equilibrium with their vapor at constant
T:
※Practically, this is only obeyed for very concentrated A with dilute B in
the A-B solution.
2. Henry’s Law
If, however, the bond energies A-A and B-B are not equal, the
evaporation rate (~vapor pressure) of A and B should be modified.
e.g. PA = X A PA0
re' ( A)
= X A K A (Henry’s Law)
re ( A)
, where re' ( A) is evaporation rate of A in A-B
re ( A) is evaporation rate of pure A
KA is Henry’s law constant
PA=XAKA
→
(Henry’s Law)
PB=XBKB
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3. Vapor-Pressure V.S. composition plot for A-B:
4. Correlation with activity:γ
Consider a species i in the solution in equilibrium with its vapor at
constant T:
※ If the vapor of i is ideal gas → f i = Pi (vapor pressure of i )
, and f i , pure = PA0 (standard vapor pressure)
∴ ai =
Pi
Pi 0
a. If the component i exhibits Raoultian behavior (ideal solution)
→ Pi = X i Pi 0
As a result, ai =
Pi
X i Pi 0
=
= X i (Raoult’s law and ideal
Pi 0
Pi 0
solution)
→
fi
= X i ⇒ fi = X i fi0
0
fi
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For example: Fe-Cr alloy at 1600℃
b. If the component i obeys Henry’s law:
→ Pi = X i K i
As a result, ai =
Pi
X K
= i 0 i = X i K i (Henry’s law and ideal
0
Pi
Pi
solution)
For example: Fe-Ni binary alloy at 1600℃
c. Relationship between Henry’s law and Raoult’s Law
In a binary system, if one component obeys Henry’s law
→ another component must follow Raoult’s law
(7) To determine
Mi
Consider G in the binary A-B solution system
∵ G = f (T , P, n A , n B )
⎯全微分
⎯⎯→
⎛ ∂G ⎞
⎛ ∂G
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟⎟
dG = ⎜
dn A + ⎜⎜
⎟ dT + ⎜
⎟ dP + ⎜⎜
⎝ ∂T ⎠ p ,n
⎝ ∂P ⎠T ,n
⎝ ∂n A ⎠ T , P ,n A
⎝ ∂n B
⎞
⎟⎟
dn B
⎠ T , P ,n A
At constant T and P → dG = G A dn A + GB dnB
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For one mole of solution → dG = G A dn A + GB dnB
⎛ ∂G ⎞
⎟⎟
對 X B 微分→ ⎜⎜
⎝ ∂X B ⎠T , P
⎛ ∂G
⎜⎜
⎝ ∂X B
⎛ ∂X ⎞
⎛ ∂X
= G A ⎜⎜ A ⎟⎟ + G B ⎜⎜ B
⎝ ∂X B ⎠T , P
⎝ ∂X B
⎞
⎟⎟
⎠T ,P
⎞
1 式)
⎟⎟ = G B − G A -----(○
⎠T , P
2 式)
又 G = X A G A + X B GB -----(○
2 → G + X A ⎛⎜⎜ dG ⎞⎟⎟
1 x X A )+○
(○
⎝ dX B ⎠T , P
⎛ dG
⎝ dX B
∴ GB = G + X A ⎜⎜
= X A GB + X B GB = GB
⎛ dG ⎞
⎞
⎟⎟
⎟⎟ = G − X A ⎜⎜
⎝ dX A ⎠ T , P
⎠T ,P
⎛ dG ⎞
⎛ dG
⎟⎟ = G − X A ⎜⎜
⎝ dX A ⎠ T , P
⎝ dX B
同理 G A = G + X B ⎜⎜
⎞
⎟⎟
⎠T ,P
※ Method1: 代數法
∴for binary A-B solution system
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Edited by Prof. Yung-Jung Hsu
※ Method2: 幾何法
1→ I A = I B + d M
由○
3
----○
2→ I B = M − X A d M
由○
4
----○
dX A
dX A
→IA = M − X A
dM dM
dM
+
= M + (1 − X A )
dX A dX A
dX A
= M + XB
dM
dM
= M − XB
dX A
dX B
與 Method1 比較→ I A = M A , I B = M B
∴ M A , M B at X A,i 之幾何法求解
(i) 作 M V.S. X A 圖 at X A = X A.i 之切線
(ii) 與 X A = 0 之交點為 I B , X A = 1 之交點為 I A
(iii) M A = I A , M B = I B
∴ M A = f (X A )
B.C. :
X A ≅ 1.0 , M A = M A (pure)
XA = 0, MA = MB
∞
(infinite dilution)
M B = f (X A)
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B.C. :
X A ≅ 1.0 , M B = M B
∞
(infinite dilution)
X B = 0 , M B = M B (pure)
(8) Property of mixing
※在相同狀態下(constant T and P )，溶液混和前後其熱力學性質
之差異。
1. Gibbs free energy change:
Consider a binary A-B solution, the Gibbs free energy change
before and after the formation of solution:
ΔG M = G M = G final − Ginitial
= GsolutionA− B − G pureA& pureB
GsolutionA− B = n A G A + n B G B , G pureA& pureB = n A G A + n B G B
G A : Partial molar free energy of solution, G A : molar free energy
of pure A
∴ ΔG M = n A (G A − G A ) + n B (GB − GB ) ------(3 式)
A
Definition: Δ M A = M A − M A pure A 變成 solution 後 A 之性質變化
…... relative partial molar property 相對部分性質
A
e.g. ΔG A = G A − G A ,
M
∴ ΔG M = n A ΔG A − nB ΔGB
M
For solution at constant T:
2
2
∫ d G = ∫ RTd ln f
i
1
i
1
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Edited by Prof. Yung-Jung Hsu
G2 − G1 = RT ln
f2
f1
If we take 1 as the pure state of a material and also view it as the
standard state, 2 as the solution state, then Gi − Gi pure = RT ln
fi
= RT ln ai
fi0
M
∴ G A − G A = RT ln a A = ΔG A 帶入(3 式)
GB − G B = RT ln a B = ΔG B
M
帶入(3 式)
→ ΔG M = n A ( RT ln a A ) + nB ( RT ln a B )
For one mole of solution → ΔG M = RT ( X A ln a A + X B ln a B ) …… any
solution
For an ideal solution: a A = X A , a B = X B → ΔG M = RT ( X A ln X A + X B ln X B )
2. Volume change:
ΔV M = V final − Vinitial = VsolutionA− B − V pureA& pureB
= ( n A V A + n B VB ) − ( n A V A + n B VB )
= n A (V A − V A ) + n B (VB − VB )
= n A ΔV A
M
+ n B ΔVB
M
For one mole of solution → ΔVM = X A (V A − V A ) + X B (VB − VB ) …… any
solution
⎛ ∂G ⎞
∵ d G A = − S A dT + V A dP ⇒ V A = ⎜⎜ A ⎟⎟ ------(4 式)
⎝ ∂P ⎠T
⎛ ∂G ⎞
& d G A = − S A dT + V A dP ⇒ V A = ⎜⎜ A ⎟⎟ ------(5 式)
⎝ ∂P ⎠T
⎡ ∂ (G A − G A ) ⎤
⎥ ,where G A − G A = RT ln a A
∂P
⎢⎣
⎥⎦ T
(4 式)-(5 式) → V A − V A = ⎢
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⎡ ∂ (G B − G B ) ⎤
VB − VB = ⎢
⎥ ,where G B − G B = RT ln a B
∂P
⎢⎣
⎥⎦ T
同理
For an ideal solution: (G A − G A ) id = RT ln X A 帶入上式
& (GB − GB ) id = RT ln X B 帶入上式
→ (V A − V A ) id = ⎛⎜
RT ln X A ⎞
M
⎟ = 0 帶回 ΔV 中
∂
P
⎝
⎠T
& (VB − VB ) id = ⎛⎜
RT ln X B ⎞
M
⎟ = 0 帶回 ΔV 中
∂
P
⎝
⎠T
⇒ ΔV M = X A (V A − V A ) id + X B (VB − VB ) id = 0 …… ideal solution
3. Entropy change:
ΔS M = X A ( S A − S A ) + X B ( S B − S B ) …… any solution
ΔS M ,id = − R( X A ln X A + X B ln X B ) = − R ∑ X i ln X i …… ideal solution
i
4. Enthalpy change:
ΔH M = X A ( H A − H A ) + X B ( H B − H B ) …… any solution
ΔH M ,id = 0 …… ideal solution
※ Summary: For an ideal solution, the entropy change upon mixing
ΔG M ,id = RT ∑ X i ln X i < 0 (不可逆自發)
i
ΔS M ,id = − R ∑ X i ln X i > 0 (不可逆自發)
i
ΔH M ,id = 0 (不放熱亦不吸熱)
ΔV M ,id = 0 (體積可加成)
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(9) Activity coefficient
γi:
判斷 ideal solution/ real solution 之
差異
1. Excess property: 剩餘性質
∴ G XS ≡ ΔG M − ΔG M ,id = RT ( X A ln
2. Activity coefficient: γ i =
aA
a
+ X B ln B )
XA
XB
ai
, activity coefficient of i in the solution:
Xi
∴ G XS = RT ( X A ln γ A + X B ln γ B ) = X A ( RT ln γ A ) + X B ( RT ln γ B ) ------(6
式)
∵ M = ∑ X i G AXS , where G AXS is excess molar property (剩餘部分
性質)
For a binary A-B solution: G XS = X A G AXS + X B G BXS ------(7 式)
Combine(6 式)&(7 式) → G AXS = RT ln γ A , G BXS = RT ln γ B
⇒ ∴ ln γ i =
GiXS
RT
3. γ i for ideal /real solution:
For ideal solution: ai = X i ⇒ γ i =
For real solution: ai ≠ X i ⇒ γ i =
14
ai
=1
Xi
ai
≠ 1, & γ i = f ( X i )
Xi
Edited by Prof. Yung-Jung Hsu
4. Experimental determination of ai & γ i :
a. by Gibbs-Duhem equation at constant T&P: ( ai V.S. X i )
→
∑ X dM
→
∑ X dG
→
∑ X RTd ln f
i
i
i
= 0 , M=G 代入
i
= 0 , 又 d Gi = RTd ln f i 代入
i
ai =
i
= 0 , RT 為常數 ∴ ∑ X i d ln f i = 0 …… (8 式)
fi
⇒ f i = ai f i o 代入(8 式)
o
fi
→ ∑ X i d ln ai f i o = 0
→ ∑ X i (d ln ai + d ln f i o ) = 0 , ( f i o =constant, ∴ d ln f i o = 0 )
→ ∑ X i d ln ai = 0
For a binary solution system: X A d ln a A + X B d ln a B = 0
→ d ln a A = −
XB
d ln a B
XA
上式積分, from pure A to A solution:
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X A=X A
∫ d ln a
A
=
X A=X A
∫
X A =1
右式= ln a A
左式= −
X A =1
X A=X A
X A=X A
∫
(
X A =1
−
− ln a A
XB
d ln a B
XA
X A =1
= ln a A
X A=X A
(∵ a A
X A =1
= 1, ln a A
X A =1
= 0)
XB
)d ln a B
XA
∴原式 ⇒ ln a A
X A=X A
=−
X A=X A
∫
(
X A =1
XB
)d ln a B
XA
(i) From data, we get a B V.S. X B data
(ii)Rearrange the data in the form (− ln a B ) V.S. (
(iii)Plot (− ln a B ) V.S. (
XB
)
XA
XB
)
XA
→ We can get (ln a A ) V.S. X A from the area,
then we get a B V.S. X B data
b. by Gibbs-Duhem equation at constant T&P: ( γ i V.S. X i )
→
∑ X dM
→
∑ X dG
i
i
i
XS
= 0 , M = G XS 代入
= 0, 又 dG
XS
= RTd ln γ i 代入
∴ RT ∑ X i d ln γ i = 0 ⇒ ∑ X i d ln γ i = 0
For a binary solution system: X A d ln γ A + X B d ln γ B = 0
同理 ⇒ ln γ A
X A=X A
=−
X A=X A
∫
X A =1
(
XB
)d ln γ B
XA
(i) From data, we get γ B V.S. X B data
(ii)Rearrange the data in the form (− ln γ B ) V.S. (
16
XB
)
XA
Edited by Prof. Yung-Jung Hsu
(iii)Plot (− ln γ B ) V.S. (
XB
)
XA
→ We can get (ln γ A ) V.S. X A from the area,
then we get γ B V.S. X B data
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