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(nU) = Td(nS) - Pd(nV) + ∑ μidn i (a) i d(nH) = Td(nS) - (nV)dP + μ dn 同理 ∑ i i : i …… (b)

d(nA) = -(nS)dT - Pd(nV) + ∑ μidn i i …… (c)

d(nG) = -(nS)dT + (nV)dP + ∑ μidn i i …… (d)

⎡∂(nU)⎤ 由(a) → μi = ⎢ ⎥ ∂n i ⎣ ⎦ nS,nV,n j≠i

⎡∂(nH)⎤ 由(b) → μi = ⎢ ⎥ ∂ni ⎣ ⎦ nS,P,n j≠i

⎡∂(nA)⎤ 由(c) → μi = ⎢ ⎥ ∂ni ⎣ ⎦T,nV,n j≠i

⎡∂(nG)⎤ μi = 由(d) → ⎢ ⎥ = G i : partial molar Gibbs free ∂ni ⎣ ⎦T,P,n j≠i energy of i

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(2) Partial molar property, M i : 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 2V2

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 ⎞ ⎜ 1 ⎟ ⎜ 2 ⎟ ⎢ ⎥ = M 1 ⎜ ⎟ + M 2 ⎜ ⎟ = M 1 ∂n1 ∂n1 ∂n1 ⎣ ⎦ T ,P,n2 ⎝ ⎠T ,P,n2 ⎝ ⎠T ,P,n2

⎡∂M ⎤ ⎢ ⎥ = M 2 ∂n2 ⎣ ⎦T ,P,n1 ⎡∂M ⎤ → M i = ⎢ ⎥ , M=VSHUGA ∂ni ⎣ ⎦T ,P,n j≠i

⎡∂G ⎤ If M=G帶入，Gi = ⎢ ⎥ = μi ∂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 = nA GA + nB GB

For one-mole of solution: G = X A GA + X B GB

→ d dG = GAdX A + GB dX B + X AdGA + X B dGB -----(1 式)

2.函數分析: 因為 G=G(T, P, nA, nB)

全微分 →

⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎜ ⎟ ⎜ ⎟ dG = ⎜ ⎟ dT + ⎜ ⎟ dP + ⎜ ⎟ dnA + ⎜ ⎟ dnB ⎝ ∂T ⎠ P,n ⎝ ∂P ⎠T ,n ∂nA ∂nB ⎝ ⎠T ,P,nB ⎝ ⎠T ,P,nA

⎛ ∂G ⎞ ⎛ ∂G ⎞ = ⎜ ⎟ dT + ⎜ ⎟ dP + GAdnA + GB dnB ⎝ ∂T ⎠ P,n ⎝ ∂P ⎠T ,n For one-mole solution: ⎛ ∂G ⎞ ⎛ ∂G ⎞ dG = ⎜ ⎟ dT + ⎜ ⎟ dP + GA X A + GB X B -----(2 式) ⎝ ∂T ⎠ P,X ⎝ ∂P ⎠T ,X (1式)=(2 式)

⎛ ∂G ⎞ ⎛ ∂G ⎞ dG = ⎜ ⎟ dT + ⎜ ⎟ dP − (X AdGA + X B dGB ) = 0 ⎝ ∂T ⎠ P,X ⎝ ∂P ⎠T ,X → Gibbs‐Duhem Equation

⎛ ∂M ⎞ ⎛ ∂M ⎞ General form: d M = ⎜ ⎟ dT + ⎜ ⎟ dP − ∑ X i d M i = 0 ⎝ ∂T ⎠ P,X ⎝ ∂P ⎠T ,X i (M=VSHUGA)

At T, P=constant,∑ X i d M i = 0 i

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(4) Partial molar properties 之間的數學關係式

Ex1: H i = U i + PV 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) RT For ideal gas i : dG = dP = RTdlnP (ideal gas, constant T) i P Because any real gas at P=0 → ideal gas 代入上式

dG = RTdlnP 不合理 → i P=0 P=0 (any gas at P=0, constant T) =RTdln0 ( )

→ 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

f for ideal gas: f = P, i = 1 i P f for real gas: f ≠ P, i ≠ 1 i P

for mixture: dG i = RTdlnf i (因為壓力沒有 partial molar pressure)

2 2 dG = RTd ln f → ∫ i ∫ i 1 1

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2 f 2 ΔG i = G − G = RT d ln f = RT ln 2 1 ∫ i 1 f1 ※ if take 1 as the pure state of a material and also view it as the

0 fi standard state, then Gi − Gi, = Gi − G = RT ln pure 0 f i

f i Define: Activity, ai ≡ , 上式變為Gi − Gi, = RT ln ai 0 pure f i

(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. ' 0 re (A) e.g. PA = X A PA = 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 → fi = Pi (vapor pressure of i )

0 , and fi, pure = PA (standard vapor pressure)

Pi ∴ ai = 0 Pi a. If the component i exhibits Raoultian behavior (ideal solution)

0 → Pi = X i Pi

0 Pi X i Pi As a result, ai = 0 = 0 = X i (Raoult’s law and ideal Pi Pi solution)

fi 0 → 0 = X i ⇒ fi = X i fi fi

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For example: Fe-Cr alloy at 1600℃

b. If the component i obeys Henry’s law:

→ Pi = X i Ki

Pi X i Ki As a result, ai = 0 = 0 = X i Ki (Henry’s law and ideal 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 M i Consider G in the binary A-B solution system

∵G = f (T, P,nA ,nB )

全微分 ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎜ ⎟ ⎜ ⎟ ⎯⎯⎯→ dG = ⎜ ⎟ dT + ⎜ ⎟ dP + ⎜ ⎟ dnA + ⎜ ⎟ dnB ⎝ ∂T ⎠ p,n ⎝ ∂P ⎠T ,n ∂nA ∂nB ⎝ ⎠T ,P,nA ⎝ ⎠T ,P,nA

At constant T and P → dG = GAdnA + GB dnB

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For one mole of solution → dG = GAdnA + GB dnB

⎛ ∂G ⎞ ⎛ ∂X ⎞ ⎛ ∂X ⎞ 對 X 微分→ ⎜ ⎟ = G ⎜ A ⎟ + G ⎜ B ⎟ B ⎜ ∂X ⎟ A ⎜ ∂X ⎟ B ⎜ ∂X ⎟ ⎝ B ⎠T ,P ⎝ B ⎠T ,P ⎝ B ⎠T ,P

⎛ ∂G ⎞ ⎜ ⎟ = G − G -----(○1式) ⎜ ∂X ⎟ B A ⎝ B ⎠T ,P

又G = X A GA + X B GB -----(○2式)

⎛ dG ⎞ (○1x X )+○2→ G + X ⎜ ⎟ = X G + X G = G A A ⎜ dX ⎟ A B B B B ⎝ B ⎠T ,P

⎛ dG ⎞ ⎛ dG ⎞ ∴G = G + X ⎜ ⎟ = G − X ⎜ ⎟ B A ⎜ dX ⎟ A ⎜ dX ⎟ ⎝ B ⎠T ,P ⎝ A ⎠T ,P

⎛ dG ⎞ ⎛ dG ⎞ 同理G = G + X ⎜ ⎟ = G − X ⎜ ⎟ A B ⎜ dX ⎟ A ⎜ dX ⎟ ⎝ A ⎠T ,P ⎝ B ⎠T ,P

※ Method1: 代數法

∴for binary A-B solution system

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※ Method2: 幾何法

d M 由○1→ I A = I B + ----○3 dX A d M 由○2→ I B = M − X A ----○4 dX A d M d M d M → I A = M − X A + = M + (1− X A ) dX A dX A dX A d M d M = M + X B = M − X B 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)

∞ X A = 0 , M A = M B (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:

M ΔG = GM = G final − Ginitial

= GsolutionA−B − GpureA& pureB

GsolutionA−B = nA GA + nB GB , G pureA& pureB = nA GA + nB GB

GA : Partial molar free energy of solution, GA : molar free energy of pure A

M ∴ ΔG = nA (GA − GA ) + nB (GB − GB ) ------(3 式)

A Definition: ΔM 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. ΔGA = GA − GA ,

M M M ∴ ΔG = nAΔGA − nB ΔGB For solution at constant T:

2 2 dG = RTd ln f ∫∫i i 1 1

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f 2 G2 − G1 = RT ln f1 If we take 1 as the pure state of a material and also view it as the f standard state, 2 as the solution state, then G − G = RT ln i = RT ln a i i pure 0 i fi

M ∴ GA − GA = RT ln aA = ΔGA 帶入(3 式)

M GB − GB = RT ln aB = ΔGB 帶入(3 式)

M → ΔG = nA (RT ln a A ) + nB (RT ln aB )

M For one mole of solution → ΔG = RT(X A ln a A + X B ln aB ) …… any solution

M For an ideal solution: a A = X A , aB = X B → ΔG = RT(X A ln X A + X B ln X B )

2. Volume change:

M ΔV = V final −Vinitial = VsolutionA−B −VpureA& pureB

= (nAVA + nBVB ) − (nAVA + nBVB )

= nA (VA −VA ) + nB (VB −VB ) M M = nAΔVA + nB ΔVB

For one mole of solution → ΔVM = X A (VA −VA ) + X B (VB −VB ) …… any solution

⎛ ∂G ⎞ ∵ dG = −S dT +V dP ⇒ V = ⎜ A ⎟ ------(4 式) A A A A ⎜ ∂P ⎟ ⎝ ⎠T

⎛ ∂G ⎞ & dG = −S dT +V dP ⇒ V = ⎜ A ⎟ ------(5 式) A A A A ⎜ ∂P ⎟ ⎝ ⎠T

⎡∂(GA − GA )⎤ (4式)-(5 式) → VA −VA = ⎢ ⎥ ,where GA − GA = RT ln aA ⎢ ∂P ⎥ ⎣ ⎦T

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⎡∂(GB − GB )⎤ 同理 VB −VB = ⎢ ⎥ ,where GB − GB = RT ln aB ⎢ ∂P ⎥ ⎣ ⎦T

id For an ideal solution: (GA − GA ) = RT ln X A 帶入上式

id &(GB − GB ) = RT ln X B 帶入上式

id ⎛ RT ln X A ⎞ M → (VA −VA ) = ⎜ ⎟ = 0 帶回 ΔV 中 ⎝ ∂P ⎠T

id ⎛ RT ln X B ⎞ M &(VB −VB ) = ⎜ ⎟ = 0帶回 ΔV 中 ⎝ ∂P ⎠T

M id id ⇒ ΔV = X A (VA −VA ) + X B (VB −VB ) = 0 …… ideal solution 3. Entropy change:

M ΔS = X A (S A − S A ) + X B (S B − S B ) …… any solution

M ,id ΔS = −R(X A ln X A + X B ln X B ) = −R∑ X i ln X i …… ideal solution i 4. Enthalpy change:

M ΔH = 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

M ,id ΔG = RT ∑ X i ln X i < 0 (不可逆自發) i

M ,id ΔS = −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 γ i : 判斷 ideal solution/ real solution 之

差異

1. Excess property: 剩餘性質

XS M M ,id aA aB ∴G ≡ ΔG − ΔG = RT(X A ln + X B ln ) X A X B

ai 2. Activity coefficient: γ i = , activity coefficient of i in the solution: X i

XS ∴G = RT(X A lnγ A + X B lnγ B ) = X A (RT lnγ A ) + X B (RT lnγ B ) ------(6

式)

XS XS ∵ M = ∑ X i GA , where GA is excess molar property (剩餘部分

性質)

XS XS XS For a binary A-B solution: G = X A GA + X B GB ------(7 式)

XS XS Combine(6 式)&(7 式) →GA = RT lnγ A , GB = RT lnγ B

G XS ⇒ ∴lnγ = i i RT

3. γ i for ideal /real solution:

ai For ideal solution: ai = X i ⇒ γ i = =1 X i

ai For real solution: ai ≠ X i ⇒ γ i = ≠ 1, &γ i = f (X i ) X i

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4. Experimental determination of ai &γ i :

a. by Gibbs-Duhem equation at constant T&P: ( ai V.S . X i )

→ ∑ X i d M i = 0 , M=G 代入

→ ∑ X i dGi = 0, 又 dGi = RTd ln f i 代入

→ ∑ X i RTd ln fi = 0, RT 為常數 ∴∑ X i d ln f i = 0 …… (8 式)

f i o ai = o ⇒ f i = ai f i 代入(8 式) fi

o →∑ X i d ln ai f i = 0

o o o →∑ X i (d ln ai + d ln f i ) = 0 , ( fi =constant, ∴ d ln f i = 0 )

→∑ X i d ln ai = 0

For a binary solution system: X Ad ln a A + X B d ln aB = 0

X B → d ln a A = − d ln aB X A 上式積分, from pure A to A solution:

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X A = X A X A = X A X d ln a = − B d ln a ∫ A ∫ X B X A =1 X A =1 A

右式=ln a A − ln a A = ln a A (∵ a A = 1,ln a A = 0 ) X A = X A X A =1 X A = X A X A =1 X A =1

X A =X A X 左式= − ( B )d ln a ∫ X B X A =1 A

X A =X A X B ∴原式⇒ ln a A = − ( )d ln aB X A = X A ∫ X X A =1 A

(i) From data, we get aB V.S . X B data

X B (ii)Rearrange the data in the form (−ln aB ) V.S . ( ) X A

X B (iii)Plot (−ln aB ) V. S . ( ) X A

→ We can get (ln a A ) V. S . X A from the area,

then we get aB V.S . X B data

b. by Gibbs-Duhem equation at constant T&P: (γ i V.S . X i )

XS → ∑ X i d M i = 0 , M = G 代入

XS XS → ∑ X i dG = 0, 又 dG = RTd lnγ i 代入

∴ RT ∑ X i d lnγ i = 0 ⇒ ∑ X i d lnγ i = 0

For a binary solution system: X Ad lnγ A + X B d lnγ B = 0

X A =X A X B 同理⇒ lnγ A = − ( )d lnγ B X A = X A ∫ X X A =1 A

(i) From data, we get γ B V.S . X B data

X B (ii)Rearrange the data in the form (−lnγ B ) V.S . ( ) X A

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X B (iii)Plot (−lnγ B ) V.S . ( ) X A

→ We can get (lnγ A ) V. S . X A from the area,

then we get γ B V.S . X B data

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