1st Year Thermodynamic Lectures Dr Mark R. Wormald BIBLIOGRAPHY

Transcription

1 M.R.W. -- Thermodynamcs 5 1st Year Thermodynamc Lectures Dr Mark R. Wormald Soluton thermodynamcs :- Lecture 1. Behavour of deal solutons (and solvents). Defnton of dealty. Solvent n dlute solutons, collgatve propertes, osmotc pressure. Lecture 2. Behavour of non-deal solutons (and solutes). Defnton of non-dealty. Actvtes and actvty coeffcents. Lecture 3. Behavour of onc solutons I. Thermodynamcs of hydraton. Introducton to Debye- Hückel Theory. Lecture 4. Behavour of onc solutons II. Result of Debye-Hückel Theory, onc strength. More concentrated onc solutons. BIBLIOGRAPHY N.C. Prce, R.A. Dwek, R.G. Ratclffe and M.R. Wormald (2001) Prncples and Problems n Physcal Bochemstry for Bochemsts, pub. Oxford the standard Oxford 1st year bochemsts text. E.B. Smth, Basc Chemcal Thermodynamcs, pub. Oxford the standard Oxford 1 st year chemsts text, slghtly more advanced and very good. J. Robbns, Ions n soluton (2): an ntroducton to electrochemstry, pub. Oxford the standard Oxford 1 st year chemsts text on electrochemstry, agan slghtly more advanced and very good

2 M.R.W. -- Thermodynamcs 5 APPLICATION OF THERMODYNAMICS As we have seen, the deas / concepts of thermodynamcs are very general, they do not depend on the nature of the system. In order to apply thermodynamcs to chemcal systems we need to combne the thermodynamc equatons wth a model of the nteractons present n the system. If ths works then the model s an adequate descrpton of the system for the propertes we are nterested n. If ths does not work, then the model does not fully descrbe the mportant features and we need to refne our model. As we shall see, the complexty of the model requred depends on the property we are nterested n. For example; Osmotc pressure of a proten soluton Solublty of a proten n soluton requres a very smple model requres a much more complex model AN IDEAL GAS Physcal model: The molecules n an deal gas behave as a set of non-nteractng pont masses. The only nteractons that occur are collsons between the molecules and the walls of the vessel. The number and force of these collsons gve rse to the propertes pressure and temperature. Mathematcal model: PV = nrt where P s pressure, V s volume, n s number of moles present, T s temperature and R s the gas constant. Thermodynamc model: G = G + RT ln (P/P ) n Dervaton of G for an deal gas: G = H TS : H = U + PV : δ U = q + w : TδS = q : PδV = w Substtutng and takng a small ncrement gves; G = U + PV TS δg = δu + PδV + VδP TδS SδT δg = q + w w + VδP q SδT = VδP SδT

3 M.R.W. -- Thermodynamcs 5 At constant temperature, δt = 0 and so δg = VδP. PV = nrt : V = nrt/p δg = nrt/p δp Integraton gves G o G P 1 dg = nrt o dp P P o ( ) o G G = nrt ln(p) ln(p ) P = + o G G RT ln P o n A MIXTURE OF IDEAL GASES Physcal model: The molecules n a mxture of deal gases behave as a set of non-nteractng pont masses, the only nteractons that occur beng collsons. One molecule cannot tell what type of neghbour t has (they do not attract or repel) and so ts chemcal propertes are ndependent of the types of other speces present. Mathematcal model: P = P x : x = n /N total where P s total pressure, P s the partal pressure of component and x s the mole fracton of (number of moles of dvded by total number of moles). [ For nstance, f only 25% of the molecules are of type, then a quarter of the total collson wth the vessel wall wll nvolve molecules and thus molecules wll contrbute a quarter of the total pressure, P = ¼ P. ] Thermodynamc model: µ = µº + RT ln(p /Pº) AN IDEAL SOLUTION Physcal model: The molecules n a lqud must nteract (otherwse t would not be a lqud). In an deal soluton, all the molecules present nteract dentcally. One molecule cannot tell what type of neghbour t has (they attract and repel equally) and so ts chemcal propertes are ndependent of the types of other speces present. Mathematcal model: P = P * x where P * s the vapour pressure of the pure lqud, P s the partal vapour pressure of the component n soluton and x s the mole fracton of. Ths s Raoult s Law

4 M.R.W. -- Thermodynamcs 5 Thermodynamc model: o o [] µ (solu) = µ (l)+ RT ln( x ) = µ (l)+ RT ln [] o DILUTE SOLUTIONS In a pure lqud (whch s deal by defnton, all molecules nteract dentcally), a solvent molecule only nteracts wth other solvent molecules. In a very dlute soluton, a solvent molecule spends almost all the tme nteractng wth other solvent molecules and hence behaves the same as n a pure lqud. Addng another solute molecule; 1. decreases the concentraton of the solvent slghtly. 2. does not ncrease sgnfcantly the tme that a solvent molecule spends nteractng wth solute molecules. Ths meets the crtera for an deal soluton,.e. the chemcal propertes of a gven solvent molecule do not vary wth composton n dlute solutons. The chemcal potental of the solvent s gven by o [solv] µ solv = µ solv + RT ln [solv] o The presence of any solute wll lower the concentraton of the solvent, [solv] < [solv]º and ln([solv]/[solv]º) <

5 M.R.W. -- Thermodynamcs 5 Thus addton of solute wll lower the chemcal potental of the solvent,.e. the solvent wll be more stable n a dlute soluton than n the pure lqud. COLLIGATIVE PROPERTIES These are propertes of the system that depend only on the number of solute molecules present not on the type of solute molecules. There are three major collgatve propertes: Depresson of freezng pont. Elevaton of bolng pont. Solvent dffuson/osmotc pressure. Only the thrd of these, osmotc pressure, s very sgnfcant n bologcal systems. CHANGES IN FREEZING AND BOILING POINTS Sold Lqud Gas µ solv (s) = µ o solv(s) µ solv (l) = µ o solv(l) + RT ln([solv]/[solv]º) µ solv (g) = µ o solv(g) + RT ln(p solv /P o ) = µ o solv(g) at 1 atms G Sold Lqud Lqud + solute Gas Thus, the presence of the solute lowers the chemcal potental of the solvent n the lqud phase, but does not effect the chemcal potental n any other phase. The freezng pont wll be lowered and the bolng pont rased. T OSMOTIC PRESSURE Consder pure solvent (e.g. water) contaned nsde a bath of solvent + solute (e.g. water + NaCl) by a barrer permeable only to the solvent (e.g. a cell membrane)

6 M.R.W. -- Thermodynamcs 5 Insde Pure lqud µ In solv(l) = µ o solv(l) Outsde Soluton µ Out solv(l) = µ o solv(l) + RT ln([solv] Out /[solv]º) As [solv] Out < [solv]º, At equlbrum, µ In solv(l) > µ Out solv(l) µ In solv(l) = µ Out solv(l) Solvent transfer wll occur from n to out untl [solv] Out the same as [solv] In. Ths can never be reached, thus solvent transfer wll contnue untl there s no solvent remanng on the nsde. Consder the reverse stuaton, pure solvent + solute (e.g. water + proten) contaned nsde a bath of solvent (e.g. water) by a barrer permeable only to the solvent (e.g. a cell membrane). Insde Soluton µ In solv(l) = µ o solv(l) + RT ln([solv] In /[solv]º) Outsde Pure lqud µ Out solv(l) = µ o solv(l) As [solv] n < [solv]º, At equlbrum, µ In solv(l) < µ Out solv(l) µ In solv(l) = µ Out solv(l) Solvent transfer wll occur from out to n untl [solv] In = [solv] Out. Ths can never be reached, thus solvent transfer wll contnue untl there s no solvent remanng on the outsde or untl the membrane bursts. If the pressure s allowed to ncrease nsde, the pressure dfference ( P) between nsde and outsde (pushng solvent out) wll provde a counterbalancng force to the chemcal potental dfference (pushng solvent n). Ths requres a membrane strong enough to wthstand the nternal pressure (e.g. a cell wall)

7 M.R.W. -- Thermodynamcs 5 The change n chemcal potental wth pressure s gven by; δµ δp T where V s the partal molar volume of component. = V For a pressure dfference of P, the change n chemcal potental s then gven by; δµ = V P Thus; Insde Soluton µ In solv(l) = µ o solv(l) + RT ln([solv] In /[solv]º) + V solv P Outsde Pure lqud µ Out solv(l) = µ o solv(l) At equlbrum; thus; µ In solv(l) = µ Out solv(l) µ o solv(l) + RT ln([solv] In /[solv]º) + V solv P = µ o solv(l) Rearrangng gves; V solv P = - RT ln([solv] In /[solv]º) P s the osmotc pressure (sometmes represented as Π). Ths equaton can be smplfed usng; [solv] In /[solv]º = x solv = 1 - x solute 1 ln(x solv ) = - x solute and Thus; x solute / V solv = [solute] (molarty) P = RT.[solute]

8 M.R.W. -- Thermodynamcs 6 NON-IDEAL SOLUTIONS In an deal soluton, all the molecules present nteract dentcally and thus as the composton changes the chemcal propertes of any gven molecule do not change. Ths s obvously ncorrect n the vast majorty of cases. SOLUTE IN DILUTE SOLUTIONS The solvent n very dlute solutons can often be treated as deal because most solvent molecules are surrounded by other solvent molecules. The solute molecules are mostly surrounded by solvent molecules. At low concentratons the ntermolecular nteractons wll reman constant, regardless of the solute concentraton (untl solute molecules start nteractng wth each other). Ths meets the crtera for an deal soluton,.e. the chemcal propertes of a gven solute molecule do not vary wth composton n dlute solutons. However, the nature of the nteractons are not the same as n the pure lqud of the solute. We can deal wth ths by changng the standard state that we refer to. Standard state for a solute soluton of 1 M solute n whch the only sgnfcant nteractons are those between the solute and solvent molecules. The chemcal potental of an deal solute s then stll gven by µ = µ + RT ln([]/[] ) except that [] 1 M (not the concentraton of the solute n the pure lqud) µ chemcal potental of the solute at 1M concentraton assumng that the only nteractons are those wth solvent molecules (not the chemcal potental of the solute n the pure lqud) The assumpton that only nteractons between the solute and solvent molecules are sgnfcant s usually not vald at 1M concentraton, only at much, much lower concentraton. Thus µ º s not the chemcal potental of a 1 M soluton, but the chemcal potental of a very dlute soluton extrapolated to 1M. NON-IDEAL SOLUTIONS The equaton µ = µ + RT ln([]/[] ) works at very low concentratons of solute for both the solute and the solvent (assumng that we use the correct standard state). For hgher concentratons of solute ths equaton does not work. There are three approaches we can use at these concentratons: 1. Model the behavor emprcally (useful for extrapolatng to low concentratons). 2. Introduce a fudge factor to get the equaton to work. 3. Understand what s gong on so that we can correctly predct behavour (.e. get the correct equaton). VIRIAL EQUATIONS Many complex functons can be represented by a polynomal seres such as; f(x) = A + Bx + Cx 2 + Dx 3 + Ex 4 + Fx

9 M.R.W. -- Thermodynamcs 6 A, B, C, etc. are emprcally determned constants, called vral coeffcents. Ths s only a useful expanson f most of the hgher order coeffcents are zero. VIRIAL EQUATION FOR GASES The defnton of an deal gas s gven by the equaton PV = nrt. Most gases only obey ths equaton at low pressures. For non-deal gases, the equaton of state can be gven by; PV = 1 + B.P + C.P 2 + D.P nrt VIRIAL EQUATION FOR OSMOTIC PRESSURE For the solvent n an deal soluton, the osmotc pressure s gven by P = RT [solute] For the solvent n a non-deal soluton, the osmotc pressure can be wrtten as P = RT ([solute] + A[solute] 2 + B[solute] 3 + ) A, B, etc., are emprcal constants that have to be found by fttng to the expermental data. Usually B, C, etc. are all assumed to be zero. In ths case, a plot of P/RT[solute] versus [solute] should be a straght lne. CHEMICAL ACTIVITIES a - Actvty = Effectve concentraton Ths s defned by the equaton µ = µ o + RT ln (a ) whch s deemed to hold at all concentratons. The actvty coeffcent, γ, s defned as; ACTIVITY COEFFICIENTS γ a [] = = Effectve concentraton Real concentraton When the actvty coeffcent of a component s 1, a = x and the component shows deal behavour. Chemcal potentals are then gven by; o [] µ = µ + RT ln γ o [] All thermodynamc equatons only work f we use actvtes, not concentratons. For example, consder the reacton ATP(aq) + H 2 O ADP(aq) + P (aq) + H + (aq) The true equlbrum constant s gven by

10 M.R.W. -- Thermodynamcs 6 a a a γ γ γ [ADP] [P] [H ] K = = a a γ γ [ATP] [H O] + ADP P + ADP P + H H eq eq eq ATP HO 2 ATP HO 2 eq 2 eq The apparent equlbrum constant s gven by app K = [ADP] [P] [H ] + eq eq eq [ATP] [H O] eq 2 eq K s always constant, K app s not. K = γ γ γ ADP P + H app γ ATP γ HO 2 K DEMYSTIFYING ACTIVITY COEFFICIENTS The equaton can be rewrtten as µ = µ + RT In(γ []/[] ) µ = µ + RT ln([]/[] ) + RT ln(γ ) µ = µ (deal) + RT ln(γ ) Thus, RT ln(γ ) s the quanttatve value of the non-deal contrbuton to the chemcal potental. RT ln(γ ) measure of the H and S contrbutons of any ntermolecular nteractons not ncluded n the standard state ln (γ ) sucrose 0 glycne Ideal glycolamde NaCl CaCl 2 LaCl 3 []

11 M.R.W. -- Thermodynamcs 7 IONIC SOLUTIONS AT INFINITE DILUTION Hydraton shell : layer of water molecules bound to an on. 1 st solvaton shell s bound drectly to the on, 2 nd hydraton shell s the next layer out, etc. Hydraton number :- The hydraton number s the number of water molecules on average bound to an on. These water molecules cannot be consdered as part of the bulk solvent because ther propertes are the not same as free water molecules. Hydrated rad :- The effectve radus of a hydrated on s gven by the sum of the onc radus and the solvaton shell. Because small ons have larger solvaton shells, they have larger hydrated rad (behave n soluton as though they were bgger) than large ons. In general an on wth a smaller radus and/or hgher charge wll have more solvaton shells a larger hydraton number a larger hydrated radus because they have stronger electrostatc nteractons wth the solvent. Enthalpy of hydraton : Gan of nteractons between solute ons and solvent molecules (monopole-dpole). Loss of nteractons between solvent molecules (dpole-dpole, hydrogen bonds). Ths leads to a large, negatve H (.e. favourable). Entropy of hydraton : Gan of entropy assocated wth the mxng of the solvent molecules and solute ons. Loss of entropy assocated wth the orderng of solvent molecules. Loss of entropy assocated wth the unsolvated solute ons. Ths leads to a small, and usually negatve S (.e. unfavourable). Note: f ons are n a crystal then there are extra H and S terms to consder. Ion H S G (kj mol -1 ) (J mol -1 K -1 ) (kj mol -1 ) Na K Mg Ca Cl Ar Fgures nclude the loss of entropy assocated wth the gas, as well as the entropy change of the solvent. The free energy term s domnated by H. IONIC SOLUTIONS AT FINITE DILUTION In the standard state of an onc solute, at nfnte dluton, each on s solvated by a sphere of water molecules. Ths wll stablse the ons through charge-dpole nteractons. At low concentratons of an onc solute, the solvated ons are surrounded by a cloud of oppostely charged solvated ons. Ths wll stablse the ons through charge-charge nteractons and thus lower ther chemcal potental

12 M.R.W. -- Thermodynamcs 7 DEBYE-HÜCKEL THEORY The object s to calculate the work requred to take one mole of solute ons from an deal soluton to a non-deal soluton, assumng that only solute-solute electrostatc nteractons are mportant. Calculatng the electrostatc nteractons n a dynamc system s more dffcult because the dstances between the ons change wth tme. Soluton: 1. Take lots of snapshots of the system at dfferent tmes. 2. Perform the calculaton on each structure. 3. Average the results. Ths wll work as long as we take enough snapshots to represent most of the possble states of the system (n buzz words fully explore the conformatonal and confguratonal space of the system). Ths s a very general approach used n most types of molecular modellng (called a Monte-Carlo calculaton). 1. Consder all ons relatve to a test on, j. Method :- 2. Take a random arrangement of all the other ons around ths test on. 3. Calculate the probablty of ths random arrangement occurrng (usng the Boltzmann equaton). 4. Calculate the electrcal potental at the test on as a result of ths random arrangement (usng the Posson equaton). 5. Repeat steps 2-4 for all possble arrangements of ons and sum all the weghted electrcal potentals. 6. Calculate the stablsaton of the test on due to the average electrcal potental. Assumptons :- 1. The solute s present n soluton as sngle solvated ons (there s no on par formaton). 2. The ons are regarded as pont charges. 3. Only coulombc nteractons are consdered. 4. The solvent s treated as a contnuous medum (ts molecular nature s gnored) unaffected by the presence of the ons. 5. The thermal energy of the soluton s much larger than the electrostatc nteractons. All these assumptons hold n dlute solutons. Basc equatons :- Boltzmann equaton - The number of ons per unt volume, n (r), found at a dstance r from the test on s gven by; n (r) = N.exp -U(r) kt where N s the total number of ons per unt volume, U(r) s ther potental energy, k s the Boltzmann constant and T s the temperature

13 M.R.W. -- Thermodynamcs 7 Posson equaton - The electrcal potental at a dstance r from the test on, Φ(r), for a sphercally symmetrc system s gven by; r 1 2 δ r δr 2 δ Φ(r) δr ( ) = - 1 r ε ρ where ε s the permttvty of the medum and ρ(r) s charge densty at a dstance r from the test on. Potental energy - The potental energy, U(r), of an on wth charge z.e at a dstance r from the test on s gven by; U(r) = z.e. Φ (r) where e s the charge of an electron and Φ(r) s the electrcal potental at a dstance r from the test on. Charge densty - The charge densty (net charge per unt volume), ρ(r), at a dstance r from the test on s gven by; ρ(r) = n (r).z.e

14 M.R.W. -- Thermodynamcs 8 SOLUTION OF THE DEBYE-HÜCKEL THEORY The fnal result of the Posson-Boltzmann equaton s the smple Debye-Hückel equaton; log ( γ ) = - A.z 2 10 I where A s a constant, ndependent of the speces present n soluton, gven by; 3 3 e 10L A = 0.5 (n aqueous soluton) kt ( π) ( ε ) and I s the onc strength of the soluton, dependent on composton, gven by; Ionc Strength = I = 1 [j]z 2 j 2 j Mean actvty coeffcents :- In practce, t s not possble to measure γ j for a sngle on as there are always counter ons present. Thus, we defne a mean actvty coeffcent γ ± as; gvng; + + ( + ) 1 z z = z z ± γ γ γ log( γ ) = -A z z I ± + - Ths s known as the Debye-Hückel lmtng law. 1. Ions are always stablsed by the presence of other ons n soluton (log 10 (γ ) < 0, so γ I < 1). 2. Hgher charged ons are stablsed more than lower charged ons by other ons n soluton. 3. Ths stablsaton depends on the concentraton and charges of the other ons, not ther chemcal dentty. 4. There should be a lnear relatonshp between log 10 (γ ) and I, the gradent of the lne dependng on the charges of the ons. All these predctons are found expermentally to hold at low onc strength, but only at low onc strength

15 M.R.W. -- Thermodynamcs 8 log 10 γ ± NaCl ZnCl2 Ideal behavour 1:1 2: : ½ I (M ) NON-ZERO ION SIZES One approxmaton of the Debye-Hückel theory s that ons are pont charges. They can thus approach nfntely closely. Ths approxmaton does not matter at very hgh dluton where the ons are, on average, far apart, but causes errors at lower dlutons. Ths mnmum dstance of approach (a) can be ncluded n the Debye-Hückel calculaton, gvng the full Debye-Hückel equaton; Az 2 log( γ ) = B a I I where B s another constant, ndependent of the soluton, and a s the sum of the onc rad (r + + r - ). IONIC SOLUTIONS AT HIGHER IONIC STRENGTHS In the standard state of an onc solute, at nfnte dluton, each on s solvated by a sphere of water molecules. At low concentratons of an onc solute, the ons are surrounded by a cloud of oppostely charged ons, stablsng the soluton. At hgh concentratons of an onc solute, there wll not be enough water molecules around to solvate all the ons present, thus there wll be a destablsng effect on the soluton. At very hgh concentratons of an onc solute n water, on par formaton (A + + B - AB) becomes mportant, changng both the type of speces present and the onc strength of the soluton. EQUILIBRIUM CONSTANTS IN SOLUTION K = products reactants [x ] γ [x ] γ γ products = K γ reactants m The true equlbrum constant, K (defned n terms of actvtes), s a constant, the apparent equlbrum constant, K m (defned n terms of concentratons), s not. SALTING IN AND SALTING OUT OF PROTEINS K [a S Proten(aq) ] Proten(s) Proten(aq) : K s = = γ[proten(aq)] [a ] Proten(s) K s s a constant, thus as γ changes the actual concentraton n soluton, [Proten(aq)], must change. γ < 1 Increased solublty Saltng n

16 M.R.W. -- Thermodynamcs 8 γ > 1 Decreased solublty Saltng out Protens often have a hgh charge so the effect of onc strength on γ s large. Dfferent protens have dfferent charges and so respond to changes n onc strength n dfferent ways. CHARGED MEMBRANES The equvalent model of the Debye-Hückel theory for membranes s the Gouy-Chapman theory. Ion concentratons close to membranes -- The electrostatc effects of a membrane are strong enough to alter the actual concentratons of ons near to the membrane. Ions of opposte charges are attracted to the membrane, ons of smlar charges are repelled. These nteractons can be strong enough to result n a layer of ons bound drectly to the membrane surface. Ion actvtes close to membranes -- The presence of the charged membrane wll effect the chemcal potentals of ons nearby. Ths s drectly analogous to the Debye-Hückel theory, the membrane contrbutng to the effectve onc strength of the soluton