1 Chapter Objectives for Thermodynamics Mathematics Concepts Know the distinction between an ordinary derivative and a partial derivative. Know how to take the partial derivative of a function. Know how to find the partial derivative of a function using the technique of implicit differentiation. Know the interchangeability of mixed derivatives. Know how to invert a partial derivative. Know how to apply the chain rule to a partial derivative Understand the distinction between the chain rule and Euler’s chain rule. Know the applicability of Euler’s chain rule. Understand the distinction between a derivative and a differential. Know how to find the total differential of a function. Know how to test a differential for exactness. Understand how state functions are related to exact differentials. Know how to perform a line integral. Know how to approximate a function using a Taylor series. Know how to use a Taylor series to fit data to curve. Know how to use dimensional analysis to check expressional for dimensional consistency. Terms and Definitions Partial differentiation Implicit differentiation Euler’s chain rule Differential Derivative Total differential Key Equations First-tier 2z 2z xy yx Exact differential State function Line integral Taylor series Maclaurin series M x, y N x, y y x x y z z w y x w x y x x z y 1 y z x y z x z 1 y y x z x M x, y N x, y y x x y R R R R dR d d d d , , ,, , , ,, n 1 f x0 n f x x x0 n x n 0 n! 2 Introduction to Thermodynamics Concepts Be able to make the distinction between intensive and extensive thermodynamic quantities. Know the difference between the thermodynamic control and the kinetic control of a process. Know how to change the following systems in to each other: open, closed, isolated, adiabatic. Know the zeroth law of thermodynamics. Understand the consequences of the zeroth law of thermodynamics. Be able to compare and contrast the different temperature scales. Understand the necessity for absolute temperature scales. Understand how physical properties are used to measure temperature. Understand the theoretical basis behind the ideal gas thermometer. Know the fundamental definition for pressure. Know the distinction between the pressure units, bars and atmospheres. Be able to make conversions between different pressure units. Know the theoretical basis of the mercury barometer. Know how to use open-end and closed-end manometers. Know how Avogadro’s number is determined with x-ray diffraction data of a crystalline solid. Terms and Definitions Intensive Extensive Open Closed Isolated Adiabatic Ideal gas thermometer Pascal Bar Atmosphere(unit) Barometer Open-end manometer Closed-end manometer Avogadro’s number Key Equations First-tier K.E. T 1 mv 2 2 w Fdl F = ma Second-tier V V0 1 T lim 273.15 p 0 V Vt.p. Fx V x V Fx dx 3 Gases Concepts Know the basic assumptions of the kinetic theory of gases Understand the essentials of the argument of how to apply the kinetic theory of gases to calculate the pressure of a gas Know the relationship between the ideal gas constant and Boltzmann’s constant and under what conditions each is used Know how the root-mean-squared speed of an ensemble of gas molecules is calculated Understand how a one-dimensional velocity distribution is different than a three-dimensional velocity distribution Know how to calculate the average speed, most probable speed and RMS speed of an ensemble of gas molecules from the Maxwell-Boltzmann distribution function Be able to sketch the isotherm of an ideal gas on a p-V diagram Understand Boyle’s law at the molecular level Be able to sketch the isobar of an ideal gas on a V-T diagram Understand Charles’ law at the molecular level Understand Avogadro’s law at the molecular level Know the proportional relationships between pressure, volume, amount and temperature Be able to sketch the isochore of an ideal gas on a p-T diagram Understand how a thermodynamic system can be described using an equation of state Know how an ideal gas can be defined with microscopic assumptions, a macroscopic definition or an equation of state Know how equations of state can be written in terms of the molar volume Understand how real gases are described with different equations of state Understand the molecular interpretations of the “a” and “b” parameters in the van der Waals equation Know the two different ways that the virial equation of state can be written Know why a virial equation of state would be used in state of closed equation of state Understand the significance of the Boyle temperature for a real gas Know the shape of an isotherm for a real gas above, below and at its critical temperature Know the slope and curvature of an isotherm at the critical point is zero From a plot of compressibility versus pressure, know how to interpret microscopically deviations of compressibility from zero Know the process for converting a two-parameter closed equation of state into its reduced form Understand how the law of corresponding states is relating the nonideality of real gas to its critical point 4 Terms and Definitions Dalton’s law of partial pressure Boltzmann constant Probability density Maxwell-Boltzmann distribution Average speed Most probable speed Root-mean-squared speed Barometric distribution law Boyle’s law Isotherm Isothermal compressibility Charles’ law Isobar Avogadro’s law Isochore Equation of state Covolume van der Waals equation Virial equation of state Boyle temperature Critical point Compressibility Reduced variables Law of corresponding states Key Equations First-tier 1 1 tr m v 2x v 2y v z2 mv 2 2 2 Fz dp z ma z dt 1 vrms Z N vi2 2 i N v 1 V V p T,n 2 F A tr 3 kT 2 v v G v dv v G v dv 2 0 p1V1 p2 V2 0 U 0 V T V1 V2 n1 n 2 V1 V2 T1 T2 p p RT a 2 Vb V PV RT Second-tier pz 2mvz 2E tr p 3V p p z,i Fz,i dt Fz,i 1 2 3kT 3RT v m M 1 2 RT B T C T D T 1 V V V2 V3 Vc 3b Tc 8a 27Rb pc a 27b 2 t 2 t1 p p0e W z,i F Fz,i m vz,i 2 lz Mgz RT RT 1 B T p C T p2 D T p3 p V 5 The First Law of Thermodynamics Concepts Know why the existence of state functions are crucial for the study of thermodynamics Know why thermodynamic quantities are state function and which are not Know why the reversibility condition is important for thermodynamic studies Know why the chemist’s definition of work, w p ex dV , is the same as the physicist’s definition of work w F d l Know the sign convention for work and heat Understand the microscopic interpretation of work and heat Understand why the first law of thermodynamics is a statement of the conservation of energy Know why making the distinction between exact and inexact differentials is important Understand why chemists will prefer using enthalpy over internal energy to calculate energy changes Know how to change a differential relationship (such as the definition of heat capacity) into an integral relationship Know how find a partial derivative from the total differential of a function Understand why Cp is greater than Cv. Be comfortable using molar quantities (such as molar volume) rather than the extensive quantities Know how to calculate isobaric, isochoric, isothermal and adiabatic heat Know why using the definition for the extant of reaction is help in studying the thermodynamics of chemical reactions Know the definitions for the standard state pressure, temperature and concentration Know the definition of the biological standard state Know the conceptual basis for Hess’ law Know how to use Hess’ law to find the enthalpy (or other state function) for a chemical reaction Know the conceptual basis for using the “products minus reactants” of formation enthalpies to find the enthalpy of reaction Be able to compare and contrast constant-pressure calorimetry with constant-volume calorimetry Know the products of combustion for an organic molecule Know the general process for measuring the heat of combustion using a bomb calorimeter Know how to use the enthalpy of combustion of a substance to find its enthalpy of formation Know how to convert an internal energy change to an enthalpy change and vice versa for a gasphase chemical reaction Know how to use heat capacity data to calculate energy changes at nonstandard temperatures Know how to use bond enthalpies to approximate the reaction enthalpy for a chemical reaction Know why the use bond enthalpies can only give approximate reaction enthalpies Be able to calculate work under isobaric, isochoric, isothermal and adiabatic conditions Understand the subtleties distinguishing the Joule coefficient and Joule-Thomson coefficient Understand how the Joule-Thomson coefficient is measured classically and presently Understand the significance of the Joule-Thomson coefficient for refrigeration Be able to calculate internal energy changes for isobaric, isochoric, isothermal and adiabatic conditions Be able to calculate enthalpy changes for isobaric, isochoric, isothermal and adiabatic conditions 6 Know the definitions for the six measureable partial derivatives to this point in the class Know that the internal energy can be considered to be a function of temperature and volume Know that the enthalpy can be considered to be a function of temperature and pressure Terms and Definitions State function Reversibility Theorem of maximum work Work Heat Thermal expansivity Enthalpy Specific heat Heat capacity Extant of reaction Formation reaction Bomb calorimeter Kirchoff’s law Joule coefficient Joule-Thomson coefficient Key Equations First-tier p = pex w p ex dV U q v H q p T J V U U q w q mCT dU đq + đw H Cp T p U Cv T v 1 V V T p H = U + PV n i n i,0 i T JT p H Second-tier U U U V T p T V V T T p 1 U Cp CV nR 1 p V T Cp CV R 7 The Second Law of Thermodynamics Concepts Understand Kelvin’s statement of the 2nd law that no cyclic process can completely convert heat to work Understand Clausius’ statement of the 2nd law that no cyclic process can completely heat and cool without the use of some work Understand the 2nd law in terms of the entropy of the universe always increasing for any spontaneous process Understand how the relationship between volume and temperature is derived for an adiabatic process Be able to sketch an isotherm and adiabat on a p-V diagram to distinguish between the two process Know the thermodynamic definition of entropy Be able to sketch a Carnot cycle on a p-V diagram Know how the Carnot cycle is used to proof that entropy is state function How the change of entropy of an ideal gas in the Carnot cycle is zero How the change of entropy for any substance in the Carnot cycle is zero How the change of entropy for any substance in any cycle is zero Know why the efficiency of a heat engine approaches 100% only if it is cooled to absolute zero Understand the statistical view of entropy especially in terms of dispersal of energy states Know how to calculate the entropy change of a substance for given change of temperature Know how to calculate the entropy change of a phase transition Understand why standard state entropies are not zero (in contrast to standard state enthalpies) Know how to calculate the standard entropy change of a reaction from a stoichiometric sum of standard entropies Understand why Trouton’s rule is approximately valid Be able to make a precise statement of the third law of thermodynamics Know the Nernst heat theorem Understand the consequences of the third law of thermodynamics Know the application of the Einstein-Debye law Understand how to calculate a standard state entropy Understand the derivation of the equation dA 0 for constant temperature and volume conditions Understand the derivation of the equation dG 0 for constant temperature and pressure conditions Understand how free energy predicts the spontaneity of physical or chemical process Know that the value of free energy yields the maximum amount of available work for a process Know how to find the Gibbsian relationships from the differential of internal energy, enthalpy, Helmholtz free energy and Gibbs free energy Understand conceptually the origin of the Gibbsian relationships Know how to find the Maxwell relationships from the differential of internal energy, enthalpy, Helmholtz free energy and Gibbs free energy Understand conceptually the origin of the Maxwell relationships Understand the relationship between fugacity and pressure Know how a thermodynamic equilibrium constant is defined in terms of fugacities Understand conceptually how fugacity coefficients can be found from experimental compressibility data Know the derivation for the pressure dependence of Gibbs free energy 8 Know how the coefficient of performance for a refrigerator is related to the efficiency of a heat engine Know the power requirement for refrigeration grow quadratically with cooling requirement Know that the performance of a refrigerator decreases as temperature decreases Understand the basic processes behind adiabatic demagnetization when it is used to ultralow temperatures Understand how evaporative cooling works Understand how a heat pump is similar to an air conditioner Terms and Definitions Spontaneous process Heat capacity ratio Adiabat Carnot cycle Efficiency Trouton’s rule Nernst heat theorem Einstein-Debye law Helmholtz free energy Gibbs free energy Maxwell’s relations Fugacity Thermodynamic equilibrium constant Fugacity coefficient Gibbs-Helmholtz equation Coefficient of performance Adiabatic Demagnetization Heat pump Key Equations First-tier dq dS rev T Sf dS Si Tf Ti Cp dT T dU T dS p dV Cp Cv 1 2 p1V p 2 V Srxn iSi0 w qh A = U – TS dH T dS V dp Tf Si Ti dS Cv dT T G = H – TS dA p dV SdT U p V S H T S p A p V T A S T V G V p T H G 2 T T p T R Nk Sf i U T S V T p V S S V T 1 c Th T V S p p S f p dG SdT V dp H V p S G S T p p S T V V T p G f Gi nRT ln f pi c qc w S V T p p T f K f p0 f D p0 c C p0 f B p0 a A d b 9 Second-tier T2 V1 T1 V2 1 qh qc Th Tc S k ln V S nR ln f Vi C v aT 3 f i Z 1 dp ln p Chemical Equilibrium Concepts Know the definition of chemical potential as partial molar Gibbs free energy Be able to rewrite the differentials of internal energy, enthalpy, Helmholtz free energy and Gibbs free energy to the functional dependence of the amount of substance Know how to derive the pressure dependence of the chemical potential for a gas Understand how the definition of the reaction quotient (and equilibrium constant) arises from the Gibbs free energy of reaction Understand the difference between the Gibbs free energy of reaction and the standard Gibbs free energy of reaction Know how the definition of the reaction quotient is affected for heterogeneous equilibria Understand the meaning of the van’t Hoff equation in its integral and differential forms Know the Gibbs free energy of reaction is related to the extent of reaction Know the condition for equilibrium in terms of chemical potential for a chemical reaction Understand why a reaction’s equilibrium constant changes with temperature Know the basis for finding the pressure dependence of the equilibrium constant Terms and Definitions Chemical potential Heterogeneous equilibria Gibbs free energy of reaction van’t Hoff equation Key Equations First-tier G U H A 1 n1 T,p,n 2 ,n 3 , n1 S,V,n 2 ,n 3 , n1 S,p,n 2 ,n 3 , n1 V,T,n 2 ,n 3 , p f i RT ln f pi G rxn G 0rxn RT ln Q G 0rxn RT ln K G G rxn p,T ln K H 1 R T p f p f p C D Kp a f A p 0 f B p 0 b 0 c 0 d Second-tier K H0 1 1 ln 2 R T2 T1 K1 K V0 ln 2 dp RT K1 0 10 Phase Transitions Concepts Know the role of chemical potential in a phase transition Understand how a phase transition temperature can be found from the temperature dependences of the chemical potentials of the two phases Know how chemical potential is related to molar entropy Know how boiling points and freezing points change with pressure Know the basis for the Clapeyron equation Know what assumptions are used to modify the Clapeyron equation to become the ClausiusClapeyron equation Understand the meaning of Clausius-Clapeyron equation in its differential and integrated forms Be able to distinguish between 1st order phase transitions, 2nd order phase transitions and lambda transition Be able to give an example for each type of phase transition Understand why kinetics can be important issue is phase transitions Terms and Definitions 1st order phase transition 2nd order phase transition Lambda transition Phase Phase transition Clausius equation Clausius-Clapeyron equation Key Equations First-tier d S dT dp S H dT V TV H vap 1 1 p ln 2 R T2 T1 p1 Second-tier p 2 p1 H T2 T1 V T1 H vap d ln p R 1 d T 11 Mixtures and Solutions Concepts Know the distinction between molar quantities and partial molar quantities Know the total volume of a mixture can be found from partial molar volumes of the components Know why partial molar volume can be negative Know how all partial molar quantities are defined Know the definition of chemical potential Understand how chemical potentials are related to each other via the Gibbs-Duhem equation Understand how the activity of liquid component is related to the chemical potential of its vapor Know the molecular level conditions for an ideal solution Know how to define and ideal solution thermodynamically Understand the thermodynamic basis for the molecular level conditions for an ideal solution Understanding the basis for finding the Gibbs free energy and entropy of mixing Understand the enthalpy of mixing for an ideal solution is zero Understand the molecular level reasons why the enthalpy of mixing for an ideal solution is zero Understand the basis for finding the freezing point depression and boiling point elevation of a solution Understand the basis for finding the osmotic pressure of an ideal solution Understand how the focus of colligative properties changes from the solvent to the solute Know the van’t Hoff factor helps distinguish strong electrolytes, weak electrolytes and nonelectrolytes Know how colligative properties can be used to measure the activity of solvents in a solution Terms and Definitions Partial molar volume Gibbs-Duhem equation Activity Ebullioscopic constant Cryoscopic constant van’t Hoff factor Key Equations First-tier G 1 n1 T,p,n 2 ,n3 , V V dV dn1 dn 2 n1 n 2 n 2 n1 A l *A l ln a A RT G mix G G * aA pA p*A p A x A p*A 1 s 1* l RT ln x1 n1d1 n 2d2 0 VA VA* H A H*A 1 T, patm , x1 1* T, patm Second-tier G mix nRT x i ln x i i Smix nR x i ln x i i M1RTm*2 Tm c2 k f c2 H fus 12 Phase Diagrams Concepts Know the derivation of the Gibbs phase rule Know how to apply the Gibbs phase rule to a two-component liquid-vapor phase diagram Know how to use a tie line to find the composition of vapor and the composition of liquid in the two phase region of a two-component liquid-vapor phase diagram for a specific temperature and total composition Know how to use the lever rule to determine the relative amounts of vapor and liquid in a twocomponent liquid-vapor phase diagram for a specific temperature and total composition Understand how different volatilities between two liquids in a mixture affects the composition of the vapor above the liquid Understand how the two-component liquid-vapor phase diagram aid in the understanding of a fractional distillation Know the difference between fractional distillation and steam distillation Know when steam distillation is a preferred separation process Be able to identify the azeotropic composition of a mixture on a liquid-vapor phase diagram Know why identification of azeotropes is important in the practical application of distillation as a separation process Know the regions in a two-component liquid-vapor phase diagram with an immiscible liquid phase and how to apply a tie line to find compositions and the level rule to find relative amounts Know the regions in a two-component liquid-liquid phase diagram and how to apply a tie line to find compositions and the level rule to find relative amounts Know the regions in a two-component liquid-solid phase diagram and how to apply a tie line to find compositions and the level rule to find relative amounts Understand the advantage of knowing a solid mixture’s eutectic point Be able to reproduce two-component liquid-solid phase diagram from a set of cooling curves Understand the role that kinetics plays in creating solids that are not in equilibrium with their liquid phases Be able to identify solid-state compound formation via a congruent melting point on a twocomponent liquid-solid phase diagram Be able to state the reaction for a specific congruent melting point Be able to identify a peritectic point on a phase diagram Be able to state the reaction for incongruent melting around a peritectic point 13 Terms and Definitions Gibbs phase rule Isopleth Dew point curve Bubble point curve Tie line Lever rule Fractional distillation Theoretical plate Azeotrope Upper consulate temperature Lower consulate temperature Liquidus Solidus Eutectic point Eutectic composition Eutectic temperature Cooling curve Congruent melting Congruent melting point Peritectic point Peritectic reaction Incongruent melting Key Equations First-tier f cp2 Second-tier p*A yA x A * * pB 1 x A x A p A xA p y A p*B * A y A p*B p*A 14 Electrolytic Solutions Concepts Understand the relationships between electrical forces, electrical energies, electric potentials and electric fields Know conceptually what the unit of Faraday represents Know conceptually what the dielectric constant describes Know the distinction between conductance and conductitvity and why the distinction is important Know why the distinction between conductivity and molar conductivity is important for electrolytic solutions Know Kohlrausch’s law of independent migration for ions and its relevance in explaining the molar conductivity of electrolytic solutions Know how the molar conductivity of weak electrolytic solutions varies with concentration Know how the molar conductivity of strong electrolytic solutions varies with concentration Know how Arrhenius explained the molar conductivity of weak electrolytic solution with the degree of dissociation parameter Know how the Arrhenius theory of weak electrolytes fits with Ostwald’s dilution law Know how the Arrhenius theory of weak electrolytes fits with the law of mass action Know why the Arrhenius theory of weak electrolytes fails to explain the molar conductivites of strong electrolytic solutions Understand how the Debye-Huckel theory attempted to describe the charge distribution of ions around an ion in solution via a balance between thermal energies and electrical attraction energies Understand the goal behind the analysis to find the charge distribution of ions in solution Understand how the thickness of the ionic atmosphere affects the description of the interaction between charges in solution Understand how the nonideal behavior of an electrolytic solution is related to the Gibbs free energy of an ideal solution Understand that the definition of standard state of a solution affects its standard Gibbs free energy Know the assumptions made to simplify the extended Debye-Hückel to the Debye-Hückel limiting law Know the range of applicability for the extended Debye-Hückel law, the Debye-Hückel limiting law and the Davies equation Know the standard state for ions in solution Know the standard state for ions is used to find the measured values for other ions in solution Know why the use of mean ionic activity coefficients is necessary Know how to calculate the mean ionic activity coefficient for an ion in solution 15 Terms and Definitions Faraday’s constant Electric potential Electric field Dielectric constant Conductance Conductivity Molar conductivity Electrolyte Nonelectrolyte Degree of dissociation Ostwald’s dilution law Law of mass action Debye-Hückel theory Ionic strength Thickness of ionic atmosphere “Distance of closest approach” Extended Debye-Hückel law Debye-Hückel limiting law Davies equation Mean ionic activity coefficient Key Equations First-tier F 1 Q1Q 2 4 0 r 2 E G I GV F 1 Q1 Q 2 40 r 2 l A c W 0 1 Q1Q 2 4 0 r I 1 N i Zi2 2 i W 1 Q1 Q 2 4 0 r log j Z2j B I a b a b Second-tier c 0 K 1 0 2 (a 0 x)(b0 x) K c0 x (d0 x) j Zi eNi Zi eNi e i i Z2j e 2 ln j 2 0 RT 1 a Zi e j kT kT 1 0 2 8e I a RT ln i w dQ I log j Z2j B cI 1 I 16 Electrochemistry Concepts Know the difference between oxidation and reduction Know the difference between oxidizing agent and reducing agent Know the relationship between the conservation of charge and a balanced redox reaction Be able to write an electrochemical reaction in line notation Know the major application for a galvanic reaction Be able to make a sketch that illustrates the essential features of a galvanic cell Understand the relationship between the work done in an electrochemical reaction and its cell potential Know how cell potential is related to the Gibbs free energy of a reaction Know how to use the Nernst equation to calculate the concentration dependence of the cell potential Be able to use the Nernst equation to calculate the cell potential in a concentration cell Be able to use the Nernst equation to find the equilibrium constant for an electrochemical reaction Know how to calculate the standard cell potential for an electrochemical reaction using standard reduction potentials Know the essential differences between a galvanic cell and an electrolytic cell Know some applications for electrolytic reactions Be able to make a sketch that illustrates the essential features of a galvanic cell Know the connections between time, current and mass for an electrolytic reaction Terms and Definitions Oxidation Reduction Oxidizing agent Reducing agent Cathode Anode Redox couple Galvanic reaction (cell) Daniell cell Liquid junction Concentration cell Nernst equation Standard hydrogen electrode Standard reduction potential Electrolytic reaction (cell) Key Equations First-tier we = -eE r G FE E E0 RT ln Q F Q=It