Prof. Dr. rer. nat. habl. S. Enders Faculty III for Process Scence Insttute of Chemcal Engneerng Department of Thermodynamcs Lecture Polymer Thermodynamcs 033 L 337 3. Chemcal Potental
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture state quanttes can be extensve or ntensve. 2 Extensve state quanttes wll double hs value f two equal systems wll be unted to one new system..e. mass m, volume V, energy U Intensve state quanttes wll eep hs value f two equal systems wll be unted to one new system..e. temperature T, pressure P, molar volume v, specfc volume v sp molar quanttes v = V n n = amount of substance densty [g/l] concentraton [mol/l] specfc quanttes v sp V = = m m = mass, M = molar mass v M
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 3 Mxtures are characterzed by composton. mole fracton: mole fracton of a component,, n a mxture s the relatve proporton of molecules belongng to the component to those n the mxture, by number of molecules. x n = x = n = weght (mass) fracton: weght fracton of a component,, n a mxture s the relatve proporton of weght belongng to the component to those n the mxture, by weght of molecules. relaton: n = w m M m = w = m =
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 4 Volume fracton: volume fracton of component,, of a mxture s defned as the quotent of the volume of the component,, and the whole volume. φ V = φ = V = ttenton: volume fractons depends on temperature and pressure. V m g n mol = ρ = V = ρ = ρ sp m m ρ m m relatons: 3 3 sp
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 5 We consder a mxture consstng of three gases (, and C). t equal values for pressure and temperature all three gases have accordng to the deal gas law the volumes V, V and V C. fter the mxng the whole volume has the value V. T,P = constant V V V C PV = nrt Removng the walls V V = V + V + V property of deal mxture C
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 6 V = V + V + V C propertes of deal mxture For deal mxtures at constant pressure and at constant temperature t s essental that the sum of the volumes of sngle gases, V, results n the volume of the mxture,v. deal mxture of gases mxture of deal gases Defnton of deal Gas: no ntermolecular forces and partcles have zero volume Defnton deal mxture: no statement about the ntermolecular forces between molecules of equal type and no statement about the partcle volume; the deal mxture can formed from deal or real gases Ideal gases buld up always an deal mxture. Real gases can form an deal mxture, dependng from ther nature.
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 7 Ideal gas: I =0 I = nteracton Ideal mxtures of gases: I =I =I Ideal gases buld up always an deal mxture. Real gases can form an deal mxture, dependng from ther nature.
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 8 V m V = V + V + V C extensve 3 v = V n 3 v m mol alternatve v sp, = V m v sp 3 m g nv = v n + v n + v n C C v n v n v n v = + + C n n n v = v x + v x + v x ntensve C C C
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 9 In a mxture of gases, each gas,, has a partal pressure, P, whch s the pressure whch the gas would have f t alone occuped the volume at constant temperature. defnton: P = xp P = P + P + P + = P C = Ths equaton holds true for deal and real mxtures of gases. Furthermore, n the case of deal mxtures of gases the followng equaton s vald: P = nrt V Dalton s Law
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 0 thought experment: sothermal expanson P = = P C mxng of, and C P = P + P + P C
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture ll thermodynamc quanttes of the frst law of thermodynamcs n an deal mxture are addtve. example: H h = n H = nh H = H + H + H + = H / n C = H H H HC H = + + + = n n n n n h = nh nh nh nh n n n n C C = + + + = = h= x h + x h + x h + = xh C C =
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 2 ll thermodynamc quanttes of the second law of thermodynamcs n an deal mxture are not addtve. Reason: For all mxtures (also deal mxtures) result an entropy of mxng. T,P = constant + V V 2 V=V +V 2 Δ S =Δ S +Δ S = S S + S S end start end start Mx deal gases: T,P = constant C P P nr T T T V v ds = dt + dv = dv = dv
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 3 Δ S =Δ S +Δ S = S S + S S end start end start Mx deal gas: T,P = constant S nr ds = dv ds = nr dv V start start V S end end end start V V + Vj S S = nr ln nr ln start = V V V + V V + V Δ MxS = nrln + nrln V V V = x V V = x V V V end
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 4 deal gas: T,P = constant V + V V + V Δ MxS = nrln + nrln V V V = x V V = x V x + x = x V + x V x V + x Δ MxS = nrln + nr V ln x V x V Δ S = n Rln x n Rln x = nr x ln x + x ln x ( ) ( ) ( ) ( ) ( ) Mx Δ S = nr x ln x entropy of mxng for deal mxture ( ) Mx =
Polymer Thermodynamcs 3. Chemcal Potental 3.. Ideal mxture 5 Δ S = nr x ln x ( ) Mx = Entropy of mxng for deal mxture G = H TS Δ G =Δ H TΔ S Mx Mx Mx Δ H = 0 for deal mxtures Δ G = T Δ S Mx Mx Mx Δ G = nrt x ln x ( ) Mx = F = U TS Δ F =Δ U TΔ S Gbbs energy of mxng for deal mxture Mx Mx Mx Δ U = 0 for deal mture Δ F = T Δ S Mx Mx Mx Δ F = nrt x ln x Helmholtz energy for deal mxture ( ) Mx =
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 6 Types of mxtures Homogenous mxtures havng deal or real behavor Heterogeneous mxtures conssts of two or more phases Interm znc sulfde surfactant
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture v 0 =58.7 ml/mol 7 60 water () + ethanol () 60 50 50 v [ml/mol] 40 30 deal real 40 30 real mxture T=25 C 20 20 v 0 =8. ml/mol 0,0 0,2 0,4 0,6 0,8,0 X Ethanol deal real deal E v = v0x + v0x v = v + v v E excess volume = mxng volume
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 8 60 60 50 50 v [ml/mol] 40 30 20 deal real 0,0 0,2 0,4 0,6 0,8,0 X Ethanol 40 30 20 v E [ml/mol] 0,0-0,5 -,0 -,5-2,0 0,0-0,5 -,0 -,5-2,0-2,5-2,5 0,0 0,2 0,4 0,6 0,8,0 X Ethanol mxng volume v E volume of mxture v real
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 9 v0 deal real deal E v = v0x + v0x v = v + v molar volume of component = pure-component volume real v = x v + x v v partal molar volume of component pure-component volume Partal molar volumes depend on the composton of real mxture.
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 20 Ideal gas: I =0 Ideal mxture I =I =I Real mxture I I I
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 2 v real v = x v + x v partal molar volume of component pure-component volume V = f( T, P, n, n ) 2 V V V V dv = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 V V = v = n n PT,, n 2 PT,, nj ì v Defnton of partal molar quanttes
V = Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture f( T, P, n, n ) 2 V V V V dv = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 V V = v = v n n PT,, n 2 PT,, nj ì Defnton of partal molar quanttes 22 dt = dp = 0 V V dv = dn + dn = v dn + v dn n dv = 2 2 2 PT,, n n2 PT,, n = 2 vdn
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 23 dt = dp = 0 dv = v dn V = n v ( ) dv = n dv + v dn = v dn = = = = = ndv = 0 Gbbs - Duhem equaton t sothermal sobarc condtons partal molar volumes depend on each other.
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 24 0 0 = = = = dt dp n dv bnary mxture made from and ndv + ndv = 0 ndv = ndv xdv = xdv = ( x) dv x dv dv = ( x ) dx dx TP, TP, x dv dv + ( x ) = 0 dx dx TP, TP,
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 25 v 30 28 26 v real =v deal +v E v deal =x v 0 +x v 0 30 28 26 v [ml/mol] 24 22 20 v E 24 22 20 8 v real =x v +x v 6 6 0,0 0,2 0,4 0,6 0,8,0 8 v x
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 26 30 30 25 v 0 v v 0 25 v [ml/mol] 20 5 v 20 5 0 0 0,0 0,2 0,4 0,6 0,8,0 x For pure components vald: v =v 0
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 27 dv dv dt = dp = 0 v = x v x + ( x ) = 0 v= x v + x v = x v + x v = dx dx TP, TP, ( ) v v v = x + v + ( x ) v = v v x x x TP, TP, TP, v x TP, = v v
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 28 30 Data nalyss example v E = x x 28 expermental data v [ml/mol] 26 24 22 v E v deal 20 0,0 0,2 0,4 0,6 0,8,0 x v E [ml/mol],0 0,5 0,0-0,5 -,0 -,5-2,0-2,5 v E =f(x )=x x * -3,0 0,0 0,2 0,4 0,6 0,8,0 x
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 29 real deal E v = v + v = xv0 + xv0 + xx = xv + x v x v = real TP, xv + xv + xx x x 0 0 ( 2 ) = v v + x 0 0 v v v = + + v x x x real x v x TP, TP, TP, real v v v = x + x + v v = v v x x TP, x TP, TP, 0 v Gbbs-Duhem equaton v
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture 30 real real v v v0 v0 ( 2x) x x TP, TP, ( 2 ) v v v ( 2 ) 0 0 0 0 = + = v v v v + x = = v v + x + v v ( 2 ) = v v + x + v 0 0 v = v + x v = v + x 2 2 0 0 = xv + xv xv + xx,0 x 0 0 v E [ml/mol] 0,5 0,0-0,5 -,0 -,5-2,0-2,5-3,0 0,0 0,2 0,4 0,6 0,8,0 x v E =f(x )=x x *
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture data analyss alternatve possblty startng pont (defnton): V n PT,, nj ì = v 3 nv v n v v = = n + v = n + v n n P, T, n n P, T, n n P, T, n P, T, n nv v n v v = = n + v = n + v n n P, T, n n P, T, n n P, T, n P, T, n = =
Polymer Thermodynamcs 3. Chemcal Potental 3.2. Volume of real mxture data analyss alternatve possblty 32 v v v = n + v and v = n + v n PT,, n n PT,, n PT,, n real deal E our example v = v + v = xv0 + xv0 + xx real n n nn v = v0 + v0 + 2 n n n v n n n n n = v v n n n n ( ) ( ) 2 0 2 0 3 n n n n n v = n v 2 0 v 2 0 3 + n n n v = v + x 2 0 n n n n v + v + n n n 0 0 2
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 33 Generalzaton for any state functon Z Z = f( T, P, n, n ) 2 Z Z Z Z dz = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 Z n PT,, nj ì = z The partal molar quantty of component,, s the partal dervaton of an extensve state functon accordng the amount of mole at constant pressure, constant temperature and constant amount of mole of all other components.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 34 Generalzaton to any state functon Z H = f( T, P, n, n ) 2 H H H H dh = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 H n PT,, nj ì = h The partal molar quantty of component,, s the partal dervaton of an extensve state functon accordng the amount of mole at constant pressure, constant temperature and constant amount of mole of all other components.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 35 ll thermodynamc mxng quanttes of the frst law of thermodynamcs are not zero for real mxtures..e. mxng enthalpy Δ Mx H calorc effects that occur durng the producton of real mxtures.e. dluton of sulfurc acd usng water leads to temperature change Δ H = H H Mx end start H = n h + n h H = n h + n h start 0 0 end ( ) + ( ) Δ H = nh + nh nh nh = n h h n h h Mx 0 0 0 0 Δ H = n h + n h = H E E E Mx Vdeo H E can be measured.e. H 2 SO 4 95 J/mol
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 36 G = f( T, P, n, n ) 2 bnary mxture G G G G dg = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 Z n PT,, nj ì = z G n PT,, nj ì = μ The chemcal potental of component, μ, s the partal dervaton of the extensve state functon free enthalpy accordng the amount of mole of the consdered component at constant pressure, temperature und amount of mole of all other components n the mxture. Hence the chemcal potental s the partal molar quantty of the free enthalpy.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 37 G = f( T, P, n, n ) 2 bnary mxture G G G G dg = dt + dp + dn + dn T P n n 2 Pn,, n Tn,, n PT,, n 2 PT,, n 2 2 2 G G = S = V T P Pn,, n Tn,, n 2 2 dg = SdT + VdP + μ dn + μ dn 2 2 Generalzaton of mxtures contanng -components dg = SdT + VdP + μdn = Gbbs fundamental equaton Ths equaton contans all thermodynamc nformaton of mxtures.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 38 F = f( T, V, n, n ) 2 bnary mxture F F F F df = dt + dv + dn + dn T V n n 2 Vn,, n Tn,, n TVn,, 2 TVn,, 2 2 2 F F = S = P T V Vn,, n Tn,, n 2 2 df = SdT PdV + μ dn + μ dn 2 2 Generalzaton of mxtures contanng -components. df = SdT PdV + μdn = Gbbs fundamental equaton Ths equaton contans all thermodynamc nformaton of mxtures.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 39 U = f( V, S, n, n ) 2 bnary mxture U U U U du = dv + ds + dn + dn V S n n 2 Sn,, n Vn,, n V, S, n 2 V, S, n 2 2 2 U U = P = T V S Sn,, n Vn,, n 2 2 du = PdV + TdS + μ dn + μ dn 2 2 Generalzaton of mxtures contanng -components. du = TdS PdV + μdn = Gbbs fundamental equaton Ths equaton contans all thermodynamc nformaton of mxtures.
Polymer Thermodynamcs 3. Chemcal Potental 3.3. Generalzaton 40 H = f( P, S, n, n ) 2 bnary mxture H H H H dh = dp + ds + dn + dn P S n n 2 Sn,, n Pn,, n PSn,, 2 PSn,, 2 2 2 H H = V = T P S Sn,, n Pn,, n 2 2 dh = VdP + TdS + μ dn + μ dn 2 2 Generalzaton of mxtures contanng -components. dh = TdS + VdP + μdn = Gbbs fundamental equaton Ths equaton contans all thermodynamc nformaton of mxtures.
dg = SdT + VdP + μdn Polymer Thermodynamcs 3. Chemcal Potental = 3.3. Generalzaton () ()+(2) Vn, Pn, 4 F G = = S T T df = SdT PdV + μdn = (2) (2)+(3) U F = = P V V Sn, Tn, = + μ (3) (3)+(4) S = Vn, S Pn, du TdS PdV dn U H = = T dh = TdS + VdP + μdn = (4) ()+(4) H G = = P P Sn, Tn, V U H F G = = = = n n n n SVn,, SPn,, TVn,, TPn,, j j j j μ
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum a) thermc equlbrum: T I = T II =... = T constant temperature b) mechancal equlbrum: P I = P II =... = P constant pressure c) materal equlbrum: μ I = μ II =... = μ n The chemcal potental of the component,, s equal n all present phases. 42 Entropy S ds=0 state functon free Enthalpy G equlbrum dg=0 state varable state varable
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 43 startng pont: du = TdS PdV + μdn = P ds = du + dv μdn = T T T = 0 The total system at constant volume conssts of 2 subsystems ( and ). system system n,,,,, V U S T P n, V, U, S, T, P The total system s located n an adabatc contaner. ds = ds + ds du = du and dv = dv and dn = dn
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum P ds = du + dv μdn = 0 T T T = wth ds = ds + ds P P 0 = + + + T T T T T T du du dv dv μ dn μ dn = = = = = wth du du and dv dv and dn dn P P μ μ 0 = + T T T T T T thermc equlbrum dv = dn = 0 du dv dn = 0 = du T T T = T 44
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 45 P P μ μ 0 = + T T T T T T mechancal du dv dn = equlbrum du dn P P 0 = dv wth T T P = T T materal equlbrum du μ μ 0 = dn wth T = T = T T = = = = dv 0 0 = P μ = μ
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 46 du = TdS PdV + μdn = () The drect applcaton of equaton () leads n ths case to problems n calculaton of dn. varable transformaton va ntegraton U = TS PV + μn = exact dfferental du = TdS + SdT PdV VdP + μ dn + n dμ = = comparson of both equaton subtracton of eq. () from eq. (2) (2) 0 = SdT VdP + ndμ = General Gbbs-Duhem equaton
dependence on temperature Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum dg = SdT + VdP + μdn = 47 law of Schwarz: z z G G = = r q q r n T T n r q qr P, n PT, PT, Pn, G G μ = μ = n T n T PT, PT, Pn, G G S = S = = s T n T n Pn, Pn, Pn, PT, PT, μ T = P s
dependence on temperature Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum alternatve ( μ / T ) dg = SdT + VdP + μdn μ T μ μ T P st = = 2 2 T T T P = 48 G = H TS μ = h Ts Ts μ = h ( μ / T ) T P = h T 2
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 49 dependence on pressure dg = SdT + VdP + μdn = z z G G = = r q q r n P P n r q qr T, n PT, PT, Tn, G G μ = μ = n P n P TP, PT, Tn, Tn, G G V = V = = P n P n Tn, Tn, PT, TP, v μ P = T v
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 50 0 = sdt vdp + xdμ = general Gbbs-Duhem equaton applcaton to pure substance: 0 = sdt vdp + dμ dμ = sdt + vdp ntegraton at T=const. from reference pressure P 0 to system pressure P μ ( PT, ) P P RT dμ = vdp = dp P 0 0 0 μ ( P, T) P P standard term transton term 0 P 0 μ( PT, ) μ( P, T) = RTl n μ( PT, ) μ( P, T) RTln 0 = + P deal gas P P 0
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 5 0 = sdt vdp + xdμ = applcaton to pure substances: 0 = sdt vdp + d μ dμ = SdT + VdP μ ( PT, ) dμ = P 0 0 μ ( P, T) P for deal gas vdp μ The chemcal potental of real gas depends on the thermc equaton of state v(t,p). = + P μ( PT, ) μ( P, T) RT ln 0 ( PT, ) μ( P, T) RTln P 0 for real gas: f = fugacty 0 lm f = P P 0 general Gbbs-Duhem equaton ntegraton at T=const. from reference pressure P 0 to system pressure P f P = + 0
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 52 for deal gas μ = + P μ( PT, ) μ( P, T) RT ln 0 ( PT, ) μ( P, T) RTln P 0 for real gas: f = Fugacty 0 (Lews) μ f P = + 0 P f P P 0 ( P, T ) = μ( P, T ) + RT ln + RT ln 0 ϕ = f P deal gas ϕ fugacty coeffcent P 0 real gas lmϕ = f μ( PT, ) = μideal ( PT, ) + RTln = μideal ( PT, ) + RTlnϕ P
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 53 pure real gas μ P f P P 0 ( P, T ) = μ( P, T ) + RT ln + RT ln 0 f = ϕp real gas n a mxture: deal gas f = ϕ P = ϕ xp real gas ϕ fugacty coeffcent 0 P ϕxp μ( PT,, x) = μ0( P, T) + RTln + RTln 0 P P 0 (,, ) 0(, ) ln P μ PT x = μ P T + RT RTln 0 + ( x) + RTln( ϕ) P deal gas deal mxture real mxture
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum chemcal potental of components n lqud mxtures defnton for deal mxtures: 0 μ ( T, P, x ) = μ ( T, P) + RTlnx () 54 The standard potental μ 0 (T,P) s the chemcal potental of the pure component,, at system temperature, T, and system pressure, P. standard state: pure substance Features of deal mxtures: ll mxng quanttes (v E, h E ) of the frst law of thermodynamcs are zero. The chemcal potental can be calculated usng eq. (). Real mxtures: ll mxng quanttes (v E, h E ) of the frst law of thermodynamcs are not zero. Hence, the chemcal potental of eq. () needs a correcton. μ ( T, P, x ) = μ ( T, P) + RTlna a = actvty of substance 0 a = xγ γ = actvty coeffcent of substance
Polymer Thermodynamcs 3. Chemcal Potental 3.4. Phase equlbrum 55 μ ( T, P, x ) = μ ( T, P) + RTlnx defnton: 0 The standard potental μ 0 (T,P) s the chemcal potental of the pure component,, at system temperature, T, and system pressure, P. standard state: pure substance applcaton to gas mxture partal pressure P = xp P= P = PV for deal gases: = nrt P μ( T, P) = μ0( T, P) + RTln P