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)⎤ μ = 由(a) → i ⎢ ⎥ ∂n i ⎣ ⎦ nS,nV,n j≠i ⎡∂(nH)⎤ μ = 由(b) → i ⎢ ⎥ ∂ni ⎣ ⎦ nS,P,n j≠i ⎡∂(nA)⎤ μ = 由(c) → i ⎢ ⎥ ∂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 1 Edited by Prof. Yung-Jung Hsu (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) 2 Edited by Prof. Yung-Jung Hsu 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. 3 Edited by Prof. Yung-Jung Hsu (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 4 Edited by Prof. Yung-Jung Hsu (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 5 Edited by Prof. Yung-Jung Hsu 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 ≡ 0 , 上式變為Gi − Gi, = RT ln ai 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 6 Edited by Prof. Yung-Jung Hsu 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 7 Edited by Prof. Yung-Jung Hsu 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 8 Edited by Prof. Yung-Jung Hsu 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 9 Edited by Prof. Yung-Jung Hsu ※ 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 ) 10 Edited by Prof. Yung-Jung Hsu ∞ 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 11 Edited by Prof. Yung-Jung Hsu f 2 G2 − G1 = RT ln f1 If we take 1 as the pure state of a material and also view it as the fi standard state, 2 as the solution state, then G − G = RT ln = 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.