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See course web page at www.bradherrick.com CHGN 124
Colorado School of Mines
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3 Ch l3 Equilibrium At equilibrium the rate at which products are produced from reactants equals the k rate at which reactants are produced from products Fit 5 Achieving chemical equilibrium for the reaction A B a The reaction of pure compound A with initial concentration Ao After a time the concentrations of A and B do not change The reason is that b the rates of the forward reaction kA and the reverse reaction kB become equal i Alo I IEqu ibrium achieved kfA iEquilibiium achieved 39 rates are equal I I A Rate kilBl Concentration E O O 3 i 2 g E 13 Variation in E I 3 concentrations in the approach 8 839 to equilibrium for N2 H2 5 g 2NH3 a The equilibrium is U U approached beginning with H and N2 in the ratio 31 b The K equilibrium is approached beginning with NH R uilibrium ex ression Rate law cannot be determined by sto1chiometry Equilibrium expressions are determined by stoichiometry For a general equilibrium a A b B S c C d D Note product over reactants I o The value of the equilibrium constant at any given temperature does not depend on the initial concentrations of reactants and products It also does not matter whether other substances are present as long as they do not react With a reactant or product The value of the equilibrium constant does vary With tem perature howeveri The common practice is to write equilibrium constants as di mensionless quantities For gases equilibrium constants can be expressed in terms of pressure rather than concentration For a general gasphase equilibrium a A b B S c C d D c d K p c p o k P 39d b p A p B Be Careful Kp is usually not equal to Kc Direction of chemical eguation and Q consider 2 03 g S 3 02 E K zlazl c 03 2 Kc 438 x1028 Note Formation of oxygen is heavily favored What ifwe wrote equilibrium as 3 02 g t 2 03 g 2 K O33 1 28 228 x103929 02 438 x 10 Note Kc still indicates that oxygen is heavily favored l Kc K c Heterogeneous Eguil39ib1ia Equilibria expressions depend on concentration Consider Ni 5 4 CO g SNiCO4 g What is the concentration of the solid Since moles of solid Ni per volume is constant comentmtion is constant The constant concentration is incorporated into the equilibrium constant Aura W K pNKCOL 39 E J W3 c P pc04 13 2 Heterogeneous equilibrium is equilibrium between different phases Solid components or liquid solvent components are usually ignored in writing equilibrium expressiongnw g Relationship between K and ED a y x 1 Recall for molarity For ideal gases pV nRT n p pA A gt ART VRTorRT 13quot Recall for general gasphase equilibria a A b B S c C d D l CJRT D pBb AlRTY B l l l 31 E v3 la Q U KP 39 KCRTf b Kc Where An coef cients of products coef cients0 16 reactants a LC 7 lt5 g LE CHATELIER S PRINCIPLE If a system in equilibrium is disturbed the system will adjust to reestablish equilibrium s i Effect of change in concentration if a reactant or product is added to a system at equilibrium the system will shift away from the added component to decrease the amount of the added component It a reactant or product is removed the system will shift toward the removed component to make more of the removed component A 4 l AEgtCD H39FBPS D 39 E J C l D Pr E 7 3 39quot Effect of 3 Chan e in ressure t 1 The addition of an Lneij gas increases the total pressure but has no effect on the concentrations or partial pressures of the reactants or products 2 Increasing pressure decreasing volume changes equilibrium to decrease total number of moles 39of gas 39 It is important to realize that although the changes we have discussed so far may alter the equilibrium position they do not alter the equilibrium constant Effect of a change in temperature The lalgegf is altegeg with a gigrggeintemgerature if energy is added to an exothermic system at equilibrium by heating it the shift will occur in the direction which will consume energy that is to the left Opposite would be true for an endothermic equation ie the reaction will act like concentration if heat is added it will shift away from the added energy sC hm REgtFCM 93 r r ax Effect of catal sis on uilibiium Catalyst has no effect on equilibrium Catalyst only affects speed of reaching equilibrium C 0 Reaction Sgotient Before equilibrium is reached state of reaction can be evaluated with reaction quotient For a general equilibrium a A b B S c C d D Q Cl ilDd Ala Bquot What is the difference between KC and Q K0 is only at equilibrium At a given temperature K0 is a constant Q is at any point in reaction At a given temperature Q can have any value If Q gt Kc amount of products is too much reaction will shift to reactant side If Q lt KC amount of reactant is too much reaction will shift to product side If Q KC system is at equilibrium 39 3 Dependence of equilibrium constants on temperature The equilibrium constant is a thermodynamic quantity and as such varies only with temperature in Kelvin This relationship is given by the Vania19ft Equation O k AH We R il T 4 E Relationship between A Goand the equilibrium constant K O The equilibrium constant can also be related to the change of free energy A G 36 RT 1n Kp V Consider A g B 9 Case 1 if A G lt0 for this reaction as written the reaction is spontaneous 39 1nKpgtOandK amp3isverylarge PA The equilibrium is shifted far to the right products 0 Case 2 if AG gt0 the reaction is non spontaneous ln KPlt0 and Kis very small The equilibrium is shifted far to the left reactants and very little products have formed l l Melts Electroch mistll SPONTANEOUS Peocesses The energy releasedin a spontaneous redox reaction can be used to perforlh elech39ical work This task is accomplished through a voltaic or galvanic cell a device in which the transfer of electrons takes place through an external pathway rather than directly between reactants cathode reduction occurs here Species undergoing reduction quotoxidizing agentquot receive electrons from the cathode Zns 2 my 2e Oxidation occurs at the anode rgduction occurs at the gathode The electrons ow spontaneously from the negative anode to the positive cathode The electrical circuit is completed by the movement of ions in solution Anions move toward the anode whereas cations move toward the cathode The cell compartments can be separated by either a porous glass barrier or by a salt bridge Cu2 aq 2e39 f9 Cus 39 Movement of cations 9 Movement of quotanions l anode oxidation occurs here Species undergoing oxidation quotreducing agentquot lose electrons here salt bridge or porous disk allows exchange of ions to keep electric neutrality While electroactive solutions remain separated 39 WhOde and reduction both begin with a consonant elecn39omotive force emf the driving force with which electrons are pulled through a wire volt V the 1mit of electrical potential It equals 1 jor ecoulomb electrons owfrom the anode of a voltaic cell to the cathode because of a difference in poten Elai energy The potential energy of electrons is higher in the anode than in the cathode and they spontaneously ow through an external circuit from the anode to the cathode The difference in potential energy per electrical charge the potential dz 39erence between two electrodes is measured in units of volts One volt V is the poten tial difference required to impart l I of energy to a charge of 1 coulomb C The potential difference between the two electrodes of a voltaic cell provides the drian force that pushes electrons through the external circuit Therefore we call thm39potenhal difference the electroinotive quotcausing electron motion force oremf 0139 cell potentiah 39 For any caireac tlon that proceeds spontaneously such as that in a voltaic cell the cell potential w l be minim The emf of a particular voltaic cell depends on the speci c reactions that occur at the cathode and anode the concentrations of reactants and products and the temperature which We will assume to be 25 C unless otherwise noted 0 all u n on RIG ff oxidation is loss reduction is gain 1 Porous barrier 5 To help remember these de nitions note that anode and oxidation both begin with a vowel and I IV Q Standard Reduction Potentials I o 4 39 I 39 To provide a basis for comparing the results of one experiment with another the following set of standardstate conditions for electrochemical measurements has been de ned 0 All solutions are 1 M o All gases have a partial pressure of 01 MPa 09869 atm Cell potentials measured under standard state conditions are represented by the symbol E0 The standard state cell potential E0 measures the strength of the driving force behind the chemical reaction 39To obtain a relatively large cell potential We have to react a strong reducing agent With a strong ox1dizmg agent Example Vo taic cell usingig gandard hydgggen electrodg Anode 2nd gt 2112 aq 2e 2Haq 22 gt H2g The experimental value for the standardstate cell potential for thereaction between zinc metal and ac1d 18 076 volts Zn 2 Haq Zn2aq H2g H2 gas 1 arm The cell potential for this reaction measures the relative reducing power of zinc metal compared with hydrogen gas But it doesn39t tell us anything about the absolute value of the reducmg power for either zinc metal or H2 D quot We therefore arbitrarily de ne the standardstate potential for the reduction of H ions to H2 gas as exactly zero volts quot I r 2 H 2 e gt H2 Pt black Iquot We will then use this reference point to calibrate the potential of any other half reaction hydrogen electrode SHE 39 The key to using this reference point is recognizing that the overall cell potential for a reaction must be the sum of the potentials for the oxidation and reduction halfreactions E0 iEoox Eored overall I The standard cell potential of a u 39 39 voltaic cell measures the 39 The half cell potentials for the voltaic cell 3 difference in the standard 39 39 reduction potentials of the l 39 cathode and the anode reactions zns Cu2aq gt Zn2aq Cus is 39 3 More Cathode i positive reduction M PM cathode i I I g E E ceu 034 076 vs a E 110 V a a n 39 in B cell 0 76 Anode Zn 9 Zn2 2e V 39 In a voltalc cell the cathode reaction is alWays the one that Mor39e has the more positive or less nega ve quot negative value for E ted Anode and made oxidation 39 439 E Sedreduction process Egedoxidation process cathode anode 39 5 39 A positive value q E indicates a spontaneous process and a negtioe value of E indicates a nonsgantaneous one Line Notation For Voltaic Cells Voltaic cells can be described by a line notation based on the following conventions o A single vertical line indicates a change in state or phase 0 Within a halfcell the reactants are listed before the products Concentrations of aqueous solutions are written in parentheses after the symbol for the ion or molecule 0 A double vertical line is used to indicate the junction between the halfcells porousdiskor saltbridgo o The line notation for the anode oxidation is written before the line notation for the cathode reduction The line notation for a standardstate Daniell cell is written as follows Zn 1 Zn21OM II Cu210M I Cu anode cathode oxidation reduction K Electrons ow from the anode to the cathode in a voltaic cell They ow from the electrode at which they are given off to the electrode at which they are consumed Reading from left to right this line notation therefore corresponds to the direction in which electrons ow B EM and FreeEetgrn AG is a measure of the span taneity of a process that occurs at constant temperature and pressure the emf E and the free39energy change AG are related by Equation AG nPE where AG free energy in Joules n moles of electrons exchanged in the redox reaction F the Faraday a censtant 96486 Coulombs per mole of electrons I E cell voltage JoulesCoulomb Note that AG and B have opposite signs For a spontaneous process AG is quot quot and E is quotquot39 C 1h Nernst Equation The dependence of the cell emf on concentration can be obtained from the de pendence of the freeenergy change on concentration AG is related to the standard freeenergylchange AGquot AG Ago RTln Q Q isithe reaction quotient Substituting AG nPE gt 391E nFE RTln Q 39 the Nemst equation i E E 13th s We can use this equation to nd the emf produced by a cell under nonstandard conditions or to determine the concentration of a reactant or product by mea suring the emf of the cell In general if the concentrations of reactants increase relative to those of products the cell reaction becomes more spontaneous and the emf increases As a voltaic cell operates reactants are converted into products which increases the value of Q and causes the emf to decrease 1 Concentration Cells as long as the concentrationsare different A concentration cell based on the NiZ Ni cell reaction a L Ni cathode Ni7 Obx1o3M Ni2 lOOM a concentration cell is a voltaic cell constructed using the same species in both the anode and cathode compartments E Cell EMF and Chemical Equilibrium n t As reactants are converted to products the value of Q increases so the value of E decreases lk l The cell emfeventually reaches E 0 Because AG nFE AG o w en E 0 when E O the cell reaction has reached equilibrium 9 Q K This useful equation tells us that the equilibrium constant for a redox reaction can be obtained from the value of the standard emf for the reaction F Batteries Abattery consists of one or more voltaic cells 7 When cells are con nected in series with the cathode of one attached to the anode of another the bat tery produces a voltage that is the sum of the emfs of the individual cells A primary cell must be discarded or recycled after its emf drops to zero A secondary cell can be recharged from an external power after its emf has dropped 0 Lead Acid battery rechargeable A 12 V automotive battery consists of six voltaic cells in series each producing 12 V The reactions that occur during discharge are as follows spongy Cathode 2 11392 g H304 aq 3 I39Faq 2 e gtPbSO4 S 2H20 1 lead Anode 395 H804quot aq n PbSO4s H aq 23 2 Pb02s Pb s 2 H80 aq 2 H aq 2Pbso4s 2 H20 1 The reactants lip and PbO2 between which electron transfer occurs serve as the electrodes To keep the electrodes m touching wood or glass ber spacers are placed between them Electrode connected Electrode connected to lead dlomda to spangg lead plates Sulfuric Banks of lead and acid lead dioxide plates 30 V and PbSO4 s have no effect on the emf of the lead storage battery helping the battery maintain a relatively constant emf during discharge The emf does vary somewhat with use because the concentration of H2804 varies with the extent of the cell discharge Because solids are excluded om the reaction quotient Q the relative amounts of Pbs Pb02s H One advantage of a lead battery is that it can be recharged using an external source of energy or generator driven by engine of the car 39 2 Pbso4 2 H20 9Pb02 Pb 2 H80 2 H 5nan o Alkaline battery non rechargeable The cell reaction can be approximately represented as follows Cathode 2 MnOz s 2 H20 1 25 2Mn0 OH 5 2 OH quot31 39 Separator Anode 39 Zn s 2 OH aq a Zn OI D2 S 25 Gasket CathOCE mo pins com luctor Anode Zn pins Nickel Cadmium rechargeable KOH electrolyte Until recently it was the most common rechargeable battery Cathode 2 Ni O OH 3 2 H20 1 2 6 quot92 Ni OH 2 s 2 OH aq V Anode Cd 5 2 OH aq Cd OH 2 s 26 As for the leadacid battery the solid reaction products adhere to the electrodes which permits the electrode reactions to be reversed during charging Drawbacks cadmium is a toxic heavy metal It presents a hazard for the environment and increases the weight of the batteries The newesgechargeable battery to receive large use is the Lithiumion battery Lithium is light and non toxic Fuels Cells galls A fuel cell is an electrochemical cell inwhicb the reactants are supplied from outside rather thanfmming anintegral part of its construction therefore it is not called a battery self contained wCathode 4e 02g 2H20Z gt 40Haq Anode 2H2g 40Haq 4Hzoa 4equot 2H23 023 2H20l Hzinlet New 1y developed semipermeable membranes and catalysts allow for the operation of HzOz fuel cells at temperatures below 100 C Porous membrane 7 There are oH39rer39 examples of voltaic cells Th 39 39 139 I a p e pH meter that indicates the H of a N9 V substance is a voltaic cell that produces millivolt output related to the Hydrogen ion concentration The relationship between the cell potential output and the pH is given as follows PH 76 Emu 0592 G Corrosion Corrosion reactions are spontaneous redox re actions in which a metal is attacked by some substance in its environment and converted to an unwanted compound undesirable redox reactions Corrosion of Iron The rusting of iron requires both oxygen and water Other factors such as the pH of the solution the presence of salts contact with metals more dif cult to oxidize than iron and stress on the iron can accelerate rusting 02 xv Fe 9 Fe2 2e 05 2mg 1 4e39 40H Oxidation Reduction A point of strain in a steel object acts as an anodelwhere the iron is oxidized to ironOI ions and pits are formed Fe Fe Ze oxidation anode The electrons produced then ow through the nail to areas exposed tozo cathodes where oxygen is reduced to hydroxide ions OH 02 ZHZO 46quot gt 4OH reduction cathode I I At the same time the F e2 ions migrate through the moisture on the surface The overall reaction is obtained by balancing the electron transfer and adding the two halfreactions 2 These act as 2Fe gt Fez 2F oxidation mode 02 ZHZO 4839 gt 4OHquot reduction cathode ZFe 02 ZHZO 2Fe2 4OH net reaction The Fe2 ions can migrate from the anode through the solution toward the cathode region Where they combine with OH ions to form ironII hydroxide Iron is further oxidized by 02 to the 3 oxidation state 39 Thematerial we call rust is a complex hydratedform of ironII oxides and hydroxides with variable water composition it can be represented as Fe203 xIIZO The overall reaction if iron39lis39 2Fes Ozaq 39tz39OS Preventing the Corrosion of Iron 3939 a prote of P116 metal Y Galvanizing or coating steel with zinc a more active metal Applying a protective coating such as39pai nt 4 V39quot Galvanized iron which is iron coated with a layer of zinc uses the prin ciples of electrochemistry to protect the 16m corrosion even after the surface coat is broken Zns is easier to oxidize than is Fes 39 if the39 zinc coating is broken and the galvanized ironis 39 Xposed to oxygen and water the zincs ves as the anode and is corroded instead of the iron p iron serves as the cathode at which 02 is reduced I E Electrolysis NONSPONTANEOUS PROCESSES Chemical energy H glegtrolysis of n1oltepl compgylldi Na 2e gt2Nal lCathode l 39 gt 2e Anode l the electrolytic cell is part of the external circuit connected to the voltage source 6 9 the electrode of the electrolytic cell that is connected to the negative terminal of the voltage source is the the cell it receives electrons that are used to reduce a substance The 919990113 quot that are retrieved during the oxidation process at the source thus completing the circuit of the cell ctive lm a39nietal oxide to form naturally on the surface i r39 amp r 0 39 quotyr 9 M M gt Mg in J A an 2r gt Zn o z iv339 Fez39l39 23 gt Fe g Sn2 25quot gt Sn 0l4VI 39 cu2 2r Cu 0337 V S PD J corau yy Elect rochemical cell r Electrical energy Elect rolytic cell gm 5 5mm lI 5 Electrolysis of molten sodium chloride Clquot ions are oxidized to 029 at the anode and Na ions are reduced to Nal at the cathode 9 2Na2 2c1 z gt 2Nal 2123 cathode of anode travel to the positive terminal of the voltage f In electrolytic cells as in voltaic cells oxidation occurs at the anode and the 1 reduction occurs at the cathode The signs of the electrodes are reversed however 69 in the electrolytic cells the cathode is negative and the anode is positive The electrolysis of molten salts is an important industrial process for the production of active metals such as sodium and aluminum The Downs cell Ringshape Graphite iron cathode anode apparatus in which molten sodium chloride is commercially electrolyzed to produce SOdium metal and chlorine gas The liquid Na oats on the denser molten NaCl 3 Electrolysis of Aqueous Solutions 39 i The electrolysis of an aqueous solution is complicated by39the presence of water We must consider whether the water is oxidized to form 02 or reduced to form in addition to or instead of the ions of the salt 6c electrolysis of aqueous sodium sulfate V39 water is reduced in preference to Na at the cathode water is also preferentially oxidized relative to the sulfate ion SQ4 at the anode The net result is the electrolysis of water This occurs because H20 is more readily reduced 5 than Na and more readily oxidized than 804 The ions of NaZSO4 conduct the current through the solution but they take no part in the reaction Kquot 22H20 Ze gt H2 ZOH reduction cathode ZHZO gt 02 4H 4c oxidation anode 6HZO gt 2H2 02 4H 40H overall cell reaction k W g 41120 ZHZO 2H2 02 net reaction 39 ie39 Pt anode The el ol is of el a ueous Na 804 pr dduces H2g at V ge cathodezand 02 at the anode Bromthymol blue indicator has been Pt cathode added to the solution This indicator I turns blue in the basic solution I L V near the cathode where OH is 1 produced and yellow in the acidic solution near the anode where H is formed c 4 20120 Ze H2g ZOH ZHZO 02g 4H 42 Reduction Oxidation cx Electrolysis of g gueog sodiug chloride the cheaper method of producing Cl gas amp The electrolysis 39of NaClaq leads to a somewhat unexpected result At the cathode H20 is reduced to H2 as above The possible reactions at the anode are the oxidation of H200 as above or the oxidation of Clquotaq 2C1 aq gt C12g 2e 32d 136 V Based on the 2 values we would expect H20 to be oxidized in preference to C1 Experiments show however that Cl39 is oxidized rather than H20 This countermnn ve result occurs because of the kinetics of the electrode process in essence even though the oxidation of H20 is thermodmamically favored the ag tivation energy for the oxidation of Cl is lower so it is kinetically favored ZCI39 9 C12 Zequot I oxidation anode ZHZO 28 gt 20H H2 reduction cathode ZHZO 2Cl ZOH H2 312 overall cell reaction as net ionic equation 2Na gt ZNa spectator ions v b v ZHZO ZNaCl gt ZNaOH H2 C12 overall cell reaction as formula unit equation Electrol is of aqueous NaCl solution Although several reactions occur at both the anode and the cathode the net result is the production of H2g and NaOH at the cathode and Clzg at the anode A few drops of phenolphthalein indicator were added to the soluu39on The solution turns pink at the cathode where OH ions are formed 39 21120 22 H2g 20 2C1 gt C12g Ze Reduction Oxidation This electrolytic process is industrially signi cant because the reactants are plen tiful and the products H2 C12 and NaOH are important commercial substances J 395 39 u Eegolysis Application Production of at forms of elements 0quot Aluminum om HallHeroult process Separan39on of sodium and chlorine Down39s cell Purify copper for wiring Electroglating Jewelry 14 K gold plated Bumpers on cars Chromium plated electroplatinginvolves using electrolysis to deposit a thin layer of one metal onranother metal in order to improve beauty or resistance to corrosion Cathode steel strip N12aq 2equot gt Nis 5391 O28 v Anode nickel strip Nis gt Nizaq 2e Eged 028V II If we look at the overall reaction it appears that nothing has been accomplished Nickel dissolves from the anode to form Nizaq At the cathode Ni2oq is reduced and forms a nickel quotplatequot on lthe cathode 39 lampere1m or 1A1Cs second Number of aquot Product HalfReaction in HalfReaction electrode Amount Produced Agaq equot gt Ags 1 39 Ag cathode 1 mol 107868 g 2Haq Ze gt H2g 2 H2 cathode mol 1008 g Cu2aq 29quot gt Cus r doo Jon 2 Cu cathode imol 31773 g 39Au3aq 32 Aus 3 Au cathode mol 65656 g V 2c1 39 9 C12g Ze I WARM 2 c12 anode gmoi 35453 g 112 Lm 2H20 gt 02g 4Haq 4F 4 Oz anode kmol 8000 g 560 LSTP The steps relating the quantity of electrical charge used in electrolysis to the amounts of substances oxidized or reduced A 39 39 l 39 1 Moles 3909 39 33K L reduceder 39 reduced39 ssse 39 Electrical Work for spontaneous process AG is a measure of the maximum useful work wm that can be extracted AG mm quotYlFE I The cell emf E is a positive number for a voltaic cell so mm will be a negative r number for a yoltaic cell Remember that work done by the system on its sur roundings is indicated by a negative sign for w 39 from the process 3 In an electrolyg39c cell we use an external source of energy to bring about a nonspontaneous electrochemical process In this case AG for the cell reaction is positive and Emu is negalive n the number of moles of electrons forced into the system the external potential total electrical charge supplied to the system by the external source of electricity nXF watt W is a unit of electrical power 1 W 1 Js a I39oule a wattsecond Electrical Work 39 3600 s 1Is 6 1kWh 1000W1hr 1hr 1W 36 X 10 we can calculate the maximum work obtainable from the voltaic cells and the work required to bring about desired electrolysis reactions CHGN 124 Supplementary Material DIFFUSION To this point we have assumed the ratelimiting step in a chemical reaction results from collisions between reacting molecules While this is often an excellent assumption it is not always true Frequently atoms or molecules moving from a region of high to low concentration controls the ratelimiting step in a reaction In other words moving against a concentration gradient This motion is referred to as diffusion We have all experienced diffusion For example when a bottle of perfume is opened it takes only a few seconds before someone across the room is aware of the smell The molecules of the perfume vapor have diffused across the room and reacted with molecules in the olfactory lobes of the observer s nose This reaction is fast compared to the time needed for the perfume molecules to cross the room and so in this process the ratelimiting step is the diffusion step A more technologically interesting example is offered by the processing of silicon chips Before the chips in your calculator and computers will function properly they must be processed to create what is called a pn junction Essentially this involves allowing a dopant like boron to diffuse into a silicon wafer to a specified depth Figure 1 To make a pn junction one side of a silicon wafer is exposed to hot boron gas The boron atoms diffuse into the silicon The process is stopped When the boron atoms have diffused to a known depth by removing the born atmosphere and cooling the wafer In this and many more processes the depth to Which the diffusing species penetrates must be controlled exactly Such control requires a detailed understanding the rate laws governing diffusion And like all reaction rates the key to understanding is the mechanism and its associated activation energy Diffusion mechanisms Diffusion involves the movement of atoms or molecules from one stable position to another In moving between these stable positions the diffusing species must pass through an energy barrier characterized by some activation energy Ea The magnitude of Ea varies dramatically depending on the size of the diffusing species and the material through which it is diffusing Diffusion through a gas or liquid 00 00 oo 000 o In a gas or a liquid over time there is usually sufficient room between gas and liquid molecules to accommodate the motion of diffusing molecules and correspondingly the activation energy for diffusion is low Diffusion in gas and liquids takes place at moderate temperatures as can be seen when one adds cream to coffee The activation energy for diffusion of small molecules through liquids is on the order of 10 kJoulesmole Interstitial diffusion in a solid 00 00 0 00 000 5 0 000 0 0 The atoms of solids are much less mobile than those of liquids and consequently are less able to accommodate the motion of diffusing species However some small atoms which are located in the interstices between solid atoms and are consequently called interstitials have only moderate activation energies I like to think of interstitial diffusion like a motorcycle moving through congested traffic And like a motorcycle small atoms such as boron carbon hydrogen and helium diffuse comparatively rapidly through many solids Activation energies for interstitial diffusion are on the order of 50 to 100 kJoulesmole Diffusion in a crystalline solid On the other hand large atoms and molecules are constrained and unable to move through crystalline solids as the activation energy required for such jumps is huge on the order of thousands of kJoulesmole However measurement of jump rates in solids suggests that large atoms move more readily than is expected Diffusion via vacancies O O Oecago 0 go o O O O Ovacancyo 0 O O O 0 To explain this unexpected observation it has been proposed that large atoms diffuse through crystalline solids via naturally occurring defects called vacancies The activation energies associated With vacancyassisted diffusion are just a little larger than those associated With interstitial diffusion 200 to 400 kJoulesmole making vacancy diffusion a significant process at elevated temperatures Diffusion in general displays first order kinetics ie rate k Sd Where k is the rate constant and Sd is the concentration of the diffusing species In the case of vacancy assisted diffusion the rate for the overall process obeys second order kinetics ie rate k Sd V Where here V is the concentration of vacancies However because vacancy concentration is a constant at a given temperature and a given solid composition the apparent rate law is first order In this mechanism vacancies are acting as catalysts and anything that increases the concentration of vacancies Will increase the diffusion rate This is a subject to Which we Will return at the end of the semester Like chemical reactions diffusion is a thermally activated process and obeys an Arrheniustype equation EaRT D D0 6 Where D is called the diffusion constant or diffusivity and is expressed with units of area per second Do is the equivalent ofA in our previously discussion of temperature dependence of the rate constant Selected values of D are given in Table 1 Note the variation in the diffusivity with the mechanism of diffusion TABLE 1 Diffusing Substance m M Dcm3 s12 Au vacancy diffusion Cu 400 5 X 1013 C intrestitial diffusion Fe 950 10 7 Methanol liquid diffusion H20 18 14 X 10 5 H2 gas diffusion Air 0 0611 Integrated Rate law As in our previous discussion of rate laws it was frequently convenient to eXpress the concentration of products as a function of time in the socalled integrated rate law In the study of diffusion this is also desirable However now we have two variables Once again consider the interstitial diffusion of boron atoms into silicon Assume that there is some constant partial pressure of boron gas above the silicon wafer Call it Bo Over time the concentration of boron interstitials at some depth into the wafer will change as shown in the following diagram Bl 3 t1 t2 t3 Depth into wafer In the case of diffusion we need to know not only the concentration as a function of time but also the concentration as a function of distance from the source The instantaneous form of our rate law can be expressed as a differential equation called Fick s Second Law and D 923 9t 9x2 Here as before Sd is the concentration of the diffusing species tis time and x is the depth of penetration The solution of this differential equation which will give us the counterpart of the integrated rate law depends on the initial distribution of diffusing species a point source will diffuse differently than a planar source as in the example of boron atoms diffusing into a silicon wafer While a general solution to Fick s Second Law will take us too far a field we will give the solution to the problem of planar diffusion from a surface with a constant concentration of the diffusing species at the surface Sdov0 where the subscript 00 denotes the concentration initially t 0 and on the surface 6 0 In this case the concentration at time tand distance x from the surface is given by the relation 5 S l erf i d Ix d 00 The error function erf is a common function occurring often in statistics just as the sine or cosine functions appear in geometry Like all functions it delivers a number depending on the value of the argument Many calculators calculate the value of the error function If you do not have such a calculator the tabulated value for the error function are given in Table 2 below Using this function we can calculate the concentration of a diffusing species at a given time if we know the corresponding diffusion constant Try using this function to solve the problems posted on the web site and included at the end of these notes Table 2 The error function 2 a z 2 a z 0 00000 085 07707 0025 00282 090 07970 005 00564 095 08209 010 01125 10 08427 015 01680 11 08802 020 02227 12 09103 025 02763 13 09340 030 03286 14 09523 035 03794 15 09661 040 04234 16 09763 045 04755 17 09838 050 05205 18 09891 055 05633 19 09928 060 06039 20 09953 065 06420 22 09981 070 06778 24 09993 Exercise For The ldle Mind To increase its corrosion resistance chromium is diffused into steel at 980 C If during diffusion the surface concentration cs of chromium remains constant at 100 how long Will it take in days to achieve a Cr concentration of 18 at a depth of 0002 cm below the steel surface D0 054 cm2sec Ea 286 kJmole It is desired to diffuse indium into pure silicon such that the indium concentration at a depth of 3 X 10394 cm from the surface Will be onehalf of the surface concentration How long should the silicon be in contact With indium at 1600K in order to accomplish this diffusion D 8 X 103912 cm2 s A sample of steel Which has been carburized carbon atoms have been allowed to diffuse into the surface at 930 C for ten 10 hours has a carburized depth of 004 cm How long must this same steel be carburized at the same temperature to produce a carburized depth 008 cm Ch 11 Solutions lConcentration units of Solutions a percent by mass v By de nition 6 Normality solute mass of solute X 100 mass of solution M number of moles of solute liters of solution m number of moles of solute Kg of solvent number of moles of solute no moles solute no moles solvent X solute X solvent number of moles of solvent Nomoles solute no moles solvent X solute X solvent 2 1 equivalents of solute 4 liters of solution at equivalent of solute mass of solute equivalent weight equivalent weight molecular mass divided by number of ionizable hydrogen of an acid 2 Solubility number of hydroxides of a base number of protons gained by a base electrons transferred in a redox reaction de nitions solubility amount of solute needed to form a saturated solution in a given quantity of solvent aturated solution that contains the maximum amount of solute Sin riaturated solution contains more solute than the saturation amount Solubility g solute 100 g H20 300 260 220 180 140 100 60 20 811 C 12H2 l NaBr f KBI 39 N 590 a 1 7 C32SO43 0 20 4O 6O 80 100 Temperature C The temperature dependence of solubility for various solids Factors affecting solubility pressure effect Heggy s law for gases the amount of gas dissolved in a solution is directly a proportional to the pressure above the solution EL i Pa 5 a b tamperaturgeifegt usually a solid dissolves more rapidly with increased temperature note V i d Gases are usually different they show decreased solubility as the temperature Increases Ex thermal pollution structureeffectz like dissolves like Polar liquids tend to dissolve readily in polar solvents the solubility of M increases with increasing molecular mass The attractive force between the gas and solvent molecules are mainly London dispersion forces which increase with the size and molecular mass of the gas molecules When the gas reacts chemically with the solvent a much higher gas solubility results ex Chlorine reacts with water and is used as a bactericide in municipal water supplies Silicontetxachloride is a colourless volatile liquid it is soluble in water This is in contrast with carbon tetrachloride This solubility is due to a hydrolysis reaction which happens because the atomic radius is such that the water molecules can attack the silicon atom of silicon tetrachloride Since carbon has a smaller atomic radius than silicon the chlorine atoms e etely shield the carbon from attack 2 H20 gt Si02 SiCl4 4 HCl Hydrogen bonding between solute and solvent may also lead to higher solubility Ex ethanol is miscible in water The number of carbon atoms in an alcohol affects its solubility in water As the length of the carbon chain increases the OH group becomes an ever smaller part of the molecule and the molecule becomes less and less polar like an hydrocarbon The solubility of the alcohol decreases correspondingly On the other hand if the number of the OH groups along the C chain increases more hydrogen bonding takes place and the solubility of the alcohol increases size of the particles In a solution the size of the particles is small molecular size In an heterogenous mixture sand water the particles are large and heavy and separate under the in uence of gravity Between these extremes is the colloid ex hemoglobin the dispersion of the particles is Uniform but the particles are large enough to scatter the light This is called the Tyndall effect Most colloightherefore appear cloudy or opaque 3 Colligative Properties Physical properties which depend only on the NUMBER of solute particles in solution and not the kind of particle These particles can be molecules or ions They include Lowering of vapor pressure which causes boiling point elevation freezing point depression Membrane osmotic pressure a vapor pressure lowering of a liquid Vapor pressure of liquids always decrease when nonvolatile solutes molecules or ions are dissolved in them After dissolution there are fewer solvent molecules 1 at the surface to vaporize the vapor pressure is decreased betert dissolaiiam Raoults law the ngr pressure of a solvent in a solution decreases as the mole fraction of the solvent decreases o LP solvent X solvent 39 P solvent 1 Xsolvent 171016 acuon 0f SOlVent x mm vapor pressure of pure solvent P Mm va or ressure of solvent in solution SO O The is de ned as solvent P solvent P solvent O 0 Substituting into the equation of Raoult Al soivemr P solvent X Psolvem I Xsolvent P olvent Remember Xsolvent Xsolute 1 Xsolute 1 Xsolvent 0 Then solvent Xsolute 39 Psolvent The lowering of the vapor pressure of a solvent by adding a non volatile solute has two very useful consequences i the solution has a higher boiling point than the pure solvent it takes a higher temperature for the pressure to build up inside the bubble and be equal or greater than the atmospheric pressure ii the solution has a lower freezing melting point than the pure solvent it is more dif cult for the solvent molecules to attract each other and form the solid phase kquot The difference hetvveen the boiling freezing points of the pure solvent and the solution ATM A Tr depends only on the number of particles dissolved the solution The particles can be molecules or ions 39 39 39 x Boiling point elevation A Tb 39 Kb m i Freezing point depressicin ATF Kfm i ATb solution BP solvent Kb molal boiling point elevation constant Cm ATf FP pure solvent FP solution Kt molal freezing point depression constant Cm m molality i van t Hoff factor number of particles that result from the dissociation of one formula unit or molecule of solute I 39 I Ex K3PO4QK 2 51904393 L a 61 electrolyte 1 b membrane osmotic pressure Osmosis spontaneous process by which solvent molecules pass through a semipenneable membrane from a solution of low concentration of solute to a sel tion with a high concentration of solute High solute s low solute N concentration 30 39 concentration 5 e0 e low solute high solute concentration concentration The pressure exerted under this condition is called osmotic pressure These pressures are very powerful ex lM nonelectrolyte in water exerts a pressure of 224 atm The osmotic pressure is proportional to the molarity of the solution 11 MRT iMRT J EltPacTEpVAL2EI where T osmotic pressure in atrn or torr M molarity in molL R 0082 Latm mol K or 624 Ltorr molK i The osmotic pressure is used routinely to estimate the molecular weights of very large molecules polymers biological macromolecules The m simglgjalue is usually less due to ion pairing more inrportant in concentrated solutions 5 39 s 4 Non ideal behavior 39 I 1 1 Ideal solution solution of very similar liquids Nearly ideal behavior is often observed when the solutesolute solventsolvent and solutesolvent interactions are very similar The enthalpy of solution is very close to zero P W Xa Pa X Pb Raoult s law V Both liquids contribute to the total vapor pressure Vapor pressure of pure B Examplepf total pressure predicted by Raoult s law a Ailblfipgrprewr 133me t a W e of CH3 2 0 m sojubbn a 6i 39 39 5 P g M g x 99 4 gig Benzene Toluene X 73 o Non ideal solution if the solvent has a specral a imty for the solute such as hydrogen bonding both will have a lower tendency to escape than expected The enthalpy of solution is large and negative exothermic more energy is released when the liquids are mixed than the energy required for them to expand Vapor pressure quoti f 1 ti 39f Example of total pressure smaller than calculated 0 830 a l 5 i CH 1 141ng 3COquot39H 0 exec CH3 5 5 f g Acetone water i l 4 M gt 39 lt XB o Ifthe solvent and solute do not interact effectively polar non polar liquids both will have a higher tendency to escape than expected The enthalpy of solution is positive endothermic More energy is required to expand the h liquids than is released when the liquids are mixed V a 44 V aw so Example of total pressure larger than calculated 239 a 39 WWW H trbi H H Ft trt t f H 39 39 HH HHHHHH C Ethanol Hexane 1 39
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