This Physical Chemistry lecture uses graphs from the following textbooks:

4. This Physical Chemistry lecture uses graphs from the following textbooks: P.W. Atkins, Physical Chemistry, 6. ed., Oxford University Press, Oxford 1998 G. Wedler, Lehrbuch der Physikalischen Chemie, 4. ed., Wiley-VCH, Weinheim 1997

5. PHYSICAL CHEMISTRY: An Introduction static phenomena dynamic phenomena equilibrium in macroscopic systems THERMODYNAMICS ELECTROCHEMISTRY change of concentration as a function of time (macroscopic) KINETICS (ELECTROCHEMISTRY) macroscopic phenomena STATISTICAL THEORY OF MATTER stationary states of particles (atoms, molecules, electrons, nuclei) e.g. during translation, rotation, vibration STRUCTURE OF MATTER CHEMICAL BOND • bond breakage and formation • transitions between quantum states • STRUCTURE OF MATTER • (microscopic) KINETICS • CHEMICAL BOND microscopic phenomena

6. Matter: Substance, intensive and extensive properties, molarity and molality Substance • A substance is a distinct, pure form of matter. • The amount of a substance, n, in a sample is reported in terms of the unit called a mole (mol). In 1 mol are NA=6.0221023 objects (atoms, molecules, ions, or other specified entities). NA is the Avogadro constant. Extensive and intensive properties  An extensive property is a property that depends on the amount of substance in the sample. Examples: mass, volume… • An intensive property is a property that is independent on the amount of substance in the sample. Examples: temperature, pressure, mass density…  A molar property Xm is the value of an extensive property X divided by the amount of substance, n: Xm=X/n. A molar property is intensive. It is usually denoted by the index m, or by the use of small letters. The one exemption of this notation is the molar mass, which is denoted simply M.  A specific property Xs is the value of an extensive property X divided by the mass m of the substance: Xs=X/m. A specific property is intensive, and usually denoted by the index s. Measures of concentration: molarity and molality  The molar concentration(‘molarity’) of a solute in a solution refers to the amount of substance of the solute divided by the volume of the solution. Molar concentration is usually expressed in moles per litre (mol L-1 or mol dm-3). A molar concentration of x mol L-1 is widely called ‘x molar’ and denoted x M. • The term molalityrefers to the amount of substance of the solute divided by the mass of the solvent used to prepare the solution. Its units are typically moles of solute per kilogram of solvent (mol kg-1).

7. Some fundamental terms: System and surroundings: For the purposes of Physical Chemistry, the universe is divided into two parts, the system and its surroundings.  The system is the part of the world, in which we have special interest.  The surroundings is where we make our measurements. The type of system depends on the characteristics of the boundary which divides it from the surroundings: (a) An open system can exchange matter and energy with its surroundings. (b) A closed system can exchange energy with its surroundings, but it cannot exchange matter. (c) An isolated system can exchange neither energy nor matter with its surroundings. Except for the open system, which has no walls at all, the walls in the two other have certain characteristics, and are given special names:  A diathermic (closed) system is one that allows energy to escape as heat through its boundary if there is a difference in temperature between the system and its surroundings. It has diathermic walls.  An adiabatic (isolated) system is one that does not permit the passage of energy as heat through its boundary even if there is a temperature difference between the system and its surroundings. It has adiabatic walls.

8. H2O (water) 25°C 1 bar H2 + ½ O2 25°C 1 bar stable unstable metastable stable metastable Homogeneous system: The macroscopic properties are identical in all parts of the system. Heterogeneous system: The macroscopic properties jump at the phase boundaries. Phase: Homogeneous part of a (possibly) heterogeneous system. Equilibrium condition:  The macroscopic properties do not change without external influence.  The system returns to equilibrium after a transient perturbation.  In general exists only a single true equilibrium state. Equilibria in Mechanics: Equilibria in Thermodynamics:

9. The concept of “Temperature”:  Temperature is a thermodynamic quantity, and not known in mechanics.  The concept of temperature springs from the observation that a change in physical state (for example, a change of volume) may occur when two objects are in contact with one another (as when a red-hot metal is plunged into water): If, upon contact of A and B, a change in any physical property of these systems is found, we know that they have not been in thermal equilibrium. A B A B + The Zeroth Law of thermodynamics: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, than C is also in thermal equilibrium with A. All these systems have a common property: the same temperature. Energy flows as heat from a region at a higher temperature to one at a lower temperature if the two are in contact through a diathermic wall, as in (a) and (c). However, if the two regions have identical temperatures, there is no net transfer of energy as heat even though the two regions are separated by a diathermic wall (b). The latter condition corresponds to the two regions being at thermal equilibrium.

10. Assumption: Linear relation between the Celsius temperature  and an observable quantity x, like the length of a Hg column, the pressure p of a gas at constant volume V, or the volume V of the gas for constant pressure p: Left: The variation of the volume of a fixed amount of gas with the temperature constant. Note that in each case they extrapolate to zero volume at -273.15 C. Right: The pressure also varies linearly with the temperature, and extrapolates to zero at T= 0 (-273.15 C). For the pressure p, this transforms to: Observation: For all (ideal) gases one finds  Introduction of the thermodynamic temperature scale (in ‘Kelvin’): and The thermodynamic temperature scale: In the early days of thermometry (and still in laboratory practice today), temperatures were related to the length of a column of liquid (e.g. Mercury, Hg), and the difference in lengths shown when the thermometer was first in contact with melting ice and then with boiling water was divided into 100 steps called ‘degrees’, the lower point being labelled 0. This procedure led to the Celsius scale of temperature with the two reference points at 0 °C and 100 °C, respectively.

11. Work, heat, and energy:  The fundamental physical propertyin thermodynamics is work: work is done when an object is moved against an opposing force. (Examples: change of the height of a weight, expansion of a gas that pushes a piston and raises the weight, or a chemical reaction which e.g. drives an electrical current)  The energy of a system is its capacity to do work. When work is done on an otherwise isolated system (e.g. by compressing a gas or winding a spring), its energy is increased. When a system does work (e.g. by moving a piston or unwinding the spring), its energy is reduced.  When the energy of a system is changed as a consequence of a temperature difference between it and the surroundings, the energy has been transferred as heat. When, for example, a heater is immersed in a beaker with water (the system), the capacity of the water to do work increases because hot water can be used to do more work than cold water. Heat transfer requires diathermic walls.  A process that releases energy as heat is called exothermic, a process that absorbs energy as heat endothermic. (a) When an endothermic process occurs in an adiabatic system, the temperature falls; (b) if the process is exothermic, then the temperature rises. (c) When an endothermic process occurs in a diathermic container, energy enters as heat from the surroundings, and the system remains at the same temperature; (d) if the process is exothermic, then energy leaves as heat, and the process is isothermal.

12. When energy is transferred to the surroundings as heat, the transfer stimulates disordered motion of the atoms in the surroundings. Transfer of energy from the surroundings to the system makes use of disordered motion (thermal motion) in the surroundings. When a system does work, it stimulates orderly motion in the surroundings. For instance, the atoms shown here may be part of a weight that is being raised. The ordered motion of the atoms in a falling weight does work on the system. Work, heat, and energy (continued): Molecular interpretation • In molecular terms, heat is the transfer of energy that makes use of chaotic molecular motion (thermal motion). • In contrast, work is the transfer of energy that makes use of organized motion. • The distinction between work and heat is made in the surroundings.

13. State functions and state variables STATEMENT • If only two intensive properties of a phase of a pure substance are known, all intensive properties of this phase of the substance are known, or • If three properties of a phase of a pure substance are known, all properties of this phase of the substance are known. example: - p and T as independent variables means: Vm (=v) = f(p,T), i.e. the resulting molar volume is pinned down, or - p, T, n as independent variables means: V = f(p,T,n) • The resulting function is termed a state function. • The variables which describe the system state, are termed - state variables, and are related to each other via the - state functions.

14. 1 2 3 The thermal equation of state and the perfect gas equation Thermal equation of state: • The thermal equation of state combines volumeV, temperatureT, pressurep, and the amount of substancen: V = f(p,T,n)orVm = v = f(p,T) The “perfect gas” (or “ideal gas”): • mass points without expansion • no interactions between the particles • a real gas, an actual gas, behaves more and more like a perfect gas the lower the pressure, and the higher the temperature Some empirical gas laws: V = f(T) for p=const.: “isobars” p = f(T) for V=const.: “isochors” p = f(V) for T=const.: “isotherms” V = const.  ( + 273.15°C) = const.’  T p = const.  ( + 273.15°C) = const.’  T p  V = const. (Boyle-Mariotte; 1664/1672) (Charles, 1798; Gay-Lussac, 1802)

15. Step 1: Isobaric change Step 2: Isothermal change } = const. ! • Combination of 1 and 3 for: •  1 mol gas at •  p0 = 1.013 bar •  T0 = 273.15 K •  v0 = 22.42 l A region of the p,V,T surface of a fixed amount of perfect gas. The points forming the surface represent the only states of the gas that can exist. Sections through the surface shown in the figure at constant temperature give the isotherms shown for the Boyle-Mariotte law and the isobars shown for the Gay-Lussac law. p  v = R  T p  V = n  R  T ‘perfect gas equation’ R : ‘gas constant’ (= 8.31434 J K-1 mol-1)

16. Step 1: Isothermal change Step 2: Isobaric change } = const. ! • Swap the changes: • a combination of 3 and 1 for •  1 mol gas at •  p0 = 1.013 bar •  T0 = 273.15 K •  v0 = 22.42 l The change of a state variable is independent of the path, on which the change of the state has been made, as long as initial and final state are identical.  • Some mathematical consequences: • The change can be described as an ‘exact differential’, i.e. the variables can be varied independently; e.g. for z=f(x,y): • The mixed derivatives are identical (Schwarz’s theorem): • Upon variation of x, y for z=const (Euler’s theorem):  same result !!!

17. A more general approach to thermal expansion and compression:  V = f(T) for p=const.:V = V0 (1 + ) (Gay-Lussac)  : (thermal) expansion coefficient  p = f(T) for V=const.:p = p0 (1 + )  p = f(V) for T=const.:pV = const. and d(pV) = pdV + Vdp = 0 (Boyle-Mariotte)  : (isothermal) compressibility generally valid!  Due to generally valid! Exact differential of V=f(p,T):

18. The mole fraction, xJ, is the amount of J expressed as a fraction of the total amount of molecules, n, in the sample: When no J molecules are present, xJ=0; when only J molecules are present, xJ=1. Thus the partial pressure can be defined as: The partial pressure of a gas is the pressure that it would exert if it occupied the container alone. If the partial pressure of a gas A is pA, that of a perfect gas B is pB, and so on, then the partial pressure when all the gases occupy the same container at the same temperature is where, for each substance J, and The partial pressures pA and pB of a binary mixture of (real or perfect) gases of total pressure p as the composition changes from pure A to pure B. The sum of the partial pressures is equal to the total pressure. If the gases are perfect, then the partial pressure is also the pressure that each gas would exert if it were present alone in the container. Mixtures of gases: Partial pressure and mole fractions Dalton’s law: The pressure exerted by a mixture of perfect gases is the sum of the partial pressures of the gases.

19. Compression factor For a perfect gas, Z=1 under all conditions. Deviation of Z from 1 is a measure of departure from perfect behaviour. Molecular interactions Real gases show deviations from the perfect gas law because molecules (and atoms) interact with each other: Repulsive forces (short-range interactions) assist expansion, attractive forces (operative at intermediate distances) assist compression. At very low pressures, all the gases have Z1 and behave nearly perfect. At high pressure, all gases have Z>1, signifying that they are more difficult to compress than a perfect gas, and repulsion is dominant. At intermediate pressure, most gases have Z<1, indicating that the attractive forces are dominant and favor compression. The variation of the potential energy of two molecules on their separation. High positive potential energy (at very small separations) indicates that the interactions between them are strongly repulsive at these distances. At intermediate separations, where the potential energy is negative, the attractive interactions dominate. At large separations (on the right) the potential energy is zero and there is no interaction between the molecules. The variation of the compression factor Z = pv/RT with pressure for several gases at 0C. A perfect gas has Z = 1 at all pressures. Notice that, although the curves approach 1 as p  0, they do so with different slopes. Real gases: An introduction

20. Below, some experimental isotherms of carbon dioxide are shown. At large molar volumes v and high temperatures the real isotherms do not differ greatly from ideal isotherms. The small differences suggest an expansion in a series of powers either of p or v, the so-called virial equations of state: The virial equation can be used to demonstrate the point that, although the equation of state of a real gas may coincide with the perfect gas law as p  0, not all of its properties necessarily coincide. For example, for a perfect gas dZ/dp = 0 (because Z=1 for all pressures), but for a real gas as p  0, and as v  , corresponding to p  0. Experimental isotherms of carbon dioxide at several temperatures. The `critical isotherm', the isotherm at the critical temperature, is at 31.04 C. The critical point is marked with a star. Real gases: The virial equation of state The third virial coefficient, C, is usually less important than the second one, B, in the sense that at typical molar volumes C/v2<<B/v. In simple models, and for p  0, higher terms than B are therefore often neglected.

21. Because the virial coefficients depend on the temperature (see table above), there may be a temperature at which Z1 with zero slope at low pressure p or high molar volume v. At this temperature, which is called the Boyle temperature, TB, the properties of a real gas coinicide with those of a perfect gas as p 0, and B=0. It then follows that pvRTB over a more extended range of pressures than at other temperatures. Real gases: The Boyle temperature The compression factor approaches 1 at low pressures, but does so with different slopes. For a perfect gas, the slope is zero, but real gases may have either positive or negative slopes, and the slope may vary with temperature. At the Boyle temperature, the slope is zero and the gas behaves perfectly over a wider range of conditions than at other temperatures.

22. Reconsider the experimental isotherms of carbon dioxide. What happens, when gas initially in the state A is compressed at constant temperature (by pushing a piston)? Real gases: Condensation and critical point • Near A, the pressure rises in approximate agreement with Boyle’s law. • Serious deviations from the law begin to appear when the volume has been reduced to B. • At C (about 60 bar for CO2), the piston suddenly slides in without any further rise in pressure. Just to the left of C a liquid appears, and there are two phases separated by a sharply defined surface. • As the volume is decreased from C through D to E, the amount of liquid increases. There is no additional resistance to the piston because the gas can respond by condensation. The corresponding pressure is the vapour pressure of the liquid at this temperature. • At E, the sample is entirely liquid and the piston rests on its surface. Further reduction of volume requires the exertion of a considerable amount of pressure, as indicated by the sharply rising line from E to F. This is due to the low compressibility of condensed phases. • The isotherm at the temperature Tc plays a special role : • Isotherms below Tc behave as described above. • If the compression takes place at Tc itself, a surface separating two phases does not appear, and the volumes at each end of the horizontal part of the isotherm have merged to a single point, the critical point of the gas. The corresponding parameters are the critical temperature, Tc, critical pressure, pc, and critical molar volume, vc, of the substance. • The liquid phase of a substance does not form above Tc.

23. Real gases: Critical constants, compression factors, Boyle temperatures, and the supercritical phase A gas can not be liquefied if the temperature is above its critical temperature. To liquefy it - to obtain a fluid phase which does not occupy the entire volume - the temperature must first be lowered to below Tc, and then the gas compressed isothermally. The single phase that fills the entire volume at T> Tc may be much denser then is normally considered typical of gases. It is often called the supercritical phase, or a supercritical fluid.

24. van der Waals equation Comparison to the virial equation of state: The surface of possible states allowed by the van der Waals equation. The van der Waals equation of gases: A model Starting point: The perfect gas lawpv = RT Correction 1: Attractive forces lower the pressure  replace p by (p+), where  is the ‘internal pressure’. More detailed analysis shows that =a/v2. Correction 2: Repulsive forces are taken into account by supposing that the molecules (atoms) behave as small but impenetrable spheres  replace v by (v-b), where b is the ‘exclusion volume’. More detailed analysis shows that b is approximately the volume of one mole of the particles. a, b: van der Waals coefficients

25. (3) The critical constants are related to the van der Waals constants. At the critical point the isotherm has a flat inflexion. An inflexion of this type occurs if both the first and second derivative are zero: at the critical point. The solution is and the critical compression factor, Zc, is predicted to be equal to for all gases. Van der Waals isotherms at several values of T/Tc. The van der Waals loops are normally replaced by horizontal straight lines. The critical isotherm is the isotherm for T/Tc = 1. (2) Liquids and gases coexist when cohesive and dispersing effects are in balance. The ‘van der Waals loops’ are unrealistic because they suggest that under some conditions an increase in presure results in an increase of volume. Therefore they are replaced by horizontal lines drawn so the loops define equal areas above and below the lines (‘Maxwell construction’) Analysis of the van der Waals equation of gases (1) Perfect gas isotherms are obtained athigh enough temperatures and large molar volumes.

26. Van der Waals constants of selected gases

27. Idea If the critical constants are characteristic properties of gases, than characteristic points, like melting or boiling point, should be unitary defined states. We therefore introduce reduced variables and obtain the reduced van der Waals equation: b) Reduced melting temperature c) Reduced boiling temperature Examples a) Compression factors The compression factors of four gases, plotted for three reduced temperatures as a function of reduced pressure. The use of reduced variables organizes the data on to single curves. The principle of corresponding states d) Trouton’s rule (or: Pictet-Troutons’s rule) A wide range of liquids gives approximately the same standard entropy of vaporization of S  85 J K-1 mol-1 But: approximation!  works best for gases composed of spherical particles  fails, sometimes badly, when the particles are non-spherical or polar

28. The First Law of Thermodynamics In thermodynamics, the total energy of a system is called its internal energy, U. The internal energy is the total kinetic and potential energy of the molecules (atoms) composing the system. We denote by U the change in internal energy when a system changes from an initial state i with an internal energy Ui to a final state f of internal energy Uf: U = Uf – Ui. The internal energy is a state function in the sense that its value depends only on the current state of the system, and is independent of how this state has been prepared. The internal energy is an extensive property. Please note: In thermodynamics, only changes of state functions are of importance. Their absolute values are usually not known. The First Law of Thermodynamics: The internal energy of an isolated system is constant. or: No perpetual motion machine (perpetuum mobile) of the first kind, i.e. a machine that does work without consuming fuel or some other source of energy, has ever been build. or: The sum of work W and heat Q which is transferred between a system and its surroundings is equal to its resulting change in internal energy: U = Q + W

29. It is found that the same quantity of work must be done on an adiabatic system to achieve the same change of state even though different means of achieving that work may be used. This path independence implies the existence of a state function, the internal energy. The change in internal energy is like the change in altitude when climbing a mountain: its value is independent of path. As the volume and temperature of a system are changed, the internal energy changes. An adiabatic and a non-adiabatic path are shown as Path 1 and 2, respectively: they correspond to different values of q and w but to the same value of U. Rationale for the path independency of U, and the path dependencies of Q and W Please note: Although the internal energy U is a state function, and depends only on the current state and not on how this state has been prepared, the exchange work W and the exchange heat Q depend on the path! In the language of infinitesimal changes, this is often expressed by the notation dU = Q + W, where dU characterizes an exact, i.e. path-independent differential, while the changes Q and W are inexact, or path-dependent, differentials.

30. Since U is an extensive quantity, it can be described by two state variables and the amount of substance: U = f(T, V, n1, n2, … nk) This equation is often called the ‘caloric equation of state’. Since U is a state function, its change can be described via an ‘exact differential’: b) isobaric changes of state (dp = 0): dU = Q – pdV (UII – UI) = Q – p(VII – VI) (UII + pVII) - (UI + pVI) = Q  (U + pV): exchanged heat  introduction of the enthalpy H = U + pV As in case of U, H is an extensive quantity, which is usually described by the two state variables T and p and the amount of substance: H = f(T, p, n1, n2, … nk) and: Isochoric versus isobaric changes: ‘internal energy’ and ‘enthalpy’ Closed system (dni = 0): a) isochoric changes of state (dV = 0): dU = Q (UII – UI) = Q  U: exchanged heat

31. assume: 1 phase, 1 mol, pure substance cv: molar heat capacity at constant volume cp: molar heat capacity at constant pressure experimentally, cp is often easier to determine than cvcalculate cv from cp The temperature dependence of internal energy and enthalpy: Heat capacities Perfect gas: the internal energy is independent of the volume v

32. closed system: changes in the amounts of substance, ni, are linked to each other ! e.g. for a reaction: AA + BB  CC + DD we find  : ‘extent of reaction’ note: stoichiometric coefficients of reactants have negative, of products positive sign since all nistart = const.: Reaction Enthalpy and Internal Reaction Energy so far considered: pure, homogeneous systems U=f(V,T,n) and H=f(p,T,n) now considered: - closed system, pure substance, two (or more) phasesphysical changes or - closed system, one phase, several components chemical changes  U=f(V,T,n1,n2, … nk) and H=f(p,T,n1,n2, … nk) it is: dni = ni – nistart = id and I = const.  U=f(V,T,n1start,n2start, … nkstart,) and H=f(p,T,n1start,n2start, … nkstart,)

33. What is the meaning of and ? a) isothermal and isochoric processes b) isothermal and isobaric processes and correspond to the heat q produced or absorbed by a chemical reaction or a phase transition for one formula conversion. The expression / is usually abbreviated by the Greek   U, H • 1)U and H are state functions, i.e. path independent. • 2) U and H are composed of the internal energies and enthalpies of reactants and products: • U = iui H = ihi • for e.g. ammonia synthesis: 1/2 N2 + 3/2 H2  NH3 • H =h(NH3) – 1/2 h(N2) – 3/2 h(H2) Reaction Enthalpy and Internal Reaction Energy (cont’d)

34. The different types of enthalpies encountered in ‘thermochemistry’ – the study of the heat produced or required by physical changes or chemical reactions - is listed in the table below. Standard Enthalpy Changes and Transition Enthalpies Changes in enthalpy are normally reported for processes under a set of standard conditions: The standard state of a substance at a specified temperature is its pure form at 1 bar. Thestandard enthalpy changeis denoted byH°. Since H is a state function, it is e.g. subH = fusH + vapH

35. for dT = dp = 0 holds: A constant-volume bomb calorimeter. The `bomb' is the central vessel, which is massive enough to withstand high pressures. The calorimeter (for which the heat capacity must be known) is the entire assembly shown here. To ensure adiabaticity, the calorimeter is immersed in a water bath with a temperature continuously readjusted to that of the calorimeter at each stage of the combustion. The correlation between H and U  U is markedly different from H only in case of noticeable changes V of the volume. for the perfect gas we find because of (U/V)T=0 : H = U + pV = U + pivi = U + iRT e.g. ammonia synthesis (1/2 N2 + 3/2 H2  NH3): I = 1 – 1/2 – 3/2 = -1 H = U - RT The most common device for measuring U is the adiabatic bomb calorimeter, where the change in temperature, T, of the calorimeter is proportional to the heat that a reaction releases or absorbs. The above correlation allows to determine also H.

36. Standard reaction enthalpies at different temperatures may be estimated from heat capacities and the reaction enthalpy at some other temperature. It is: and: Integration yields Kirchhoff’s law for the standard reaction enthalpy change from rH°(T1) to where rCp° is the difference in the molar heat capacities of products and reactants weighted by the stoichiometric coefficients that appear in the chemical equation. e.g. ammonia synthesis (1/2 N2 + 3/2 H2  NH3): rCp° = cp(NH3) – 1/2 cp(N2) – 3/2 cp(H2) An illustration of the content of Kirchhoff's law. When the temperature is increased, the enthalpies of the products and the reactants both increase, but may do so to different extents. In each case, the change in enthalpy depends on the heat capacities of the substances. The change in reaction enthalpy reflects the difference in the changes of the enthalpies. A similar expression is found for rU°: The temperature dependence of reaction enthalpies H: Kirchhoff’s law

37.  for perfect gases H is independent of pressure, since T(V/T)p = V  for condensed phases V is very small, and H almost independent of pressure It is: Hess’s law Standard enthalpies of individual reactions can be combined to obtain the enthalpy of another reaction. This application of the First Law of Thermodynamics is called Hess’s law (1840): The thermodynamic basis is the path-independence of rH. The individual steps need not be realizable in practice: they may be hypothetical reactions, the only requirement being that their chemical equations should balance. The importance of Hess’s law is that information about a reaction of interest, which may be difficult to determine directly, can be assembled from information on other reactions. Example: formation of carbon monoxide 1. C + ½ O2 CO H1, immeasurable 2. CO + ½ O2 CO2 H2 = -283.1 kJ mol-1 3. C + O2  CO2 H3 = -393.7 kJ mol-1  H1 = H3 - H2 = -110.6 kJ mol-1 3 III CO2 I C+O2 1? II CO+ ½O2 2 The pressure dependence of reaction enthalpies H The standard enthalpy of an overall reaction is the sum of the standard enthalpies of the individual reactions into which a reaction may be divided.

38. Illustration: The standard reaction enthalpy of 2HN3(l) + 2NO(g)  H2O2(l) + 4N2(g) is calculated as follows: rH° = {fH°(H2O2,l) + 4fH°(N2,g)} - {2fH°(HN3,l) + 2fH°(NO,g)} = {-187.78 + 4(0)} – {2(264.0) + 2(90.25)} = -892.3 kJ mol-1 Standard enthalpies of formation The standard enthalpy of formation, fH°, of a substance is the standard reaction enthalpy for the formation of the compound from its elements in their reference state. The reference state of an element is its most stable state at the specified temperature and 1 bar. Examples: at 298 K the reference state of nitrogen is a gas of N2 molecules, that of mercury liquid Hg, that of carbon is graphite, and that of tin the white (metallic) form. Only exception: The reference state of phosphorous is the white form since this allotrope is the most reproducible form of this element. The standard enthalpies of formation of elements in their reference states are zero at all temperatures. The reaction enthalpy in terms of enthalpies of formation: Conceptually, a reaction can be regarded as proceeding by decomposing the reactants into their elements, and then forming those elements into the products. The value for fH° is the sum of these ‘unforming’ and ‘forming’ enthalpies.

39. Expansion work: Assume a system with massless, frictionless, rigid, perfectly fitting piston of area A und an external pressure pex. i: initial state f: final state • All processes considered here: •  perfect gas in cylinder with perfect piston • work done by the gas is stored in a (virtual) ‘work storage’ • modus operandi either isothermal (in a thermostat; left) or adiabatically (right) When a piston of area A moves out through a distance dz, it sweeps out a volume dV = A dz. The external pressure, pex, is equivalent to a weight pressing on the piston, and the force opposing expansion is F = pex A. Starting point: Conversion of heat and expansion work General expression for work: dw = -F dz where dw is the work required to move an object a distance dz against an opposing force F. =0 for a perfect gas

40. - Expansion against constant pressure Piston e.g. pressed on by the atmosphere, which exerts the same pressure throughout the expansion  constant external pressure pex, which can be taken outside the integral: This type of expansion is irreversible. The work done by a gas when it expands against a constant external pressure, pex, is equal to the shaded area in this example of an indicator diagram. - Isothermal reversible expansion A reversible change is a change that can be reversed by an infinitesimal modification of a variable. The system is in equilibrium with its surroundings, and the pressure p=pex at each stage p=nRT/V: The maximum work available from a system operating between specified initial and final states and passing along a specified path is obtained when the change takes place reversibly. The work done by a perfect gas when it expands reversibly and isothermally is equal to the area under the isotherm p = nRT/V. The work done during the irreversible expansion against the same final pressure is equal to the rectangular area shown slightly darker. Note that the reversible work is greater than the irreversible work. - Free expansion i.e. expansion against zero opposing force, or pex = 0  w = 0

41. Starting point: no exchange with surroundings, i.e. q = 0 Since R = cp – cv: With T = pV/nR and the heat capacity ratio  = cp/cv we find: i.e.:  expansion work at the expense of U  temperature change of the perfect gas compression  heating expansion  cooling it is: ‘Poisson’s law’ The variation of temperature as a perfect gas is expanded reversibly and adiabatically. The curves are labelled with different values of c = cV/R. Note that the temperature falls most steeply for gases with low molar heat capacity. for a change of state from Vi, pi, Ti Vf, pf, Tf: Adiabatic changes • When a perfect gas expands adiabatically, a change in temperature is to be expected: • Because work is done, the internal energy falls, and therefore the temperature of the working gas also falls. • In molecular terms, the kinetic energy of the molecules falls, so their average speed decreases, and hence the temperature falls.

42. Starting point: Provided the heat capacity is independent of temperature, the adiabatic expansion work is To achieve a change of state from one temperature and volume to another temperature and volume, we may consider the overall change as composed of two steps. In the first step, the system expands at constant temperature; there is no change in internal energy if the system consists of a perfect gas. In the second step, the temperature of the system is increased at constant volume. The overall change in internal energy is the sum of the changes for the two steps. An adiabat depicts the variation of pressure with volume when a gas expands reversibly and adiabatically. (a) An adiabat for a perfect gas. (b) Note that the pressure declines more steeply for an adiabat than it does for an isotherm because the temperature decreases in the former. The work of (reversible) adiabatic changes

43. The Carnot Cycle Starting point:- ideal gas - alternating isothermal and adiabatical expansion and compression 1) Isothermal expansion from V1 to V2 dT = 0, dU = 0 2) Adiabatic expansion from V2 to V3  Q = 0 3) Isothermal compression from V3 to V4 dT = 0, dU = 0 4) Adiabatic compression from V4 to V1  Q = 0 Determination of T2 (Tc):

44. The Carnot cycle transports heat from the reservoir 1 into the colder reservoir 2, and delivers work The Carnot Cycle (cont’d) + Suppose an energy qh (for example, 20 kJ) is supplied to the engine and qc is lost from the engine (for example, qc = -15 kJ) and discarded into the cold reservoir. The work done by the engine is equal to qh + qc (for example, 20 kJ + (-15 kJ) = 5 kJ). The efficiency is the work done divided by the heat supplied from the hot source. Reversal: heat pump

45. work output supply of heat from the hot reservoir Efficiency of the Carnot-Cycle: A B A B + TATB with TA< TB TC with TA< TC < TB The Carnot Cycle (cont’d) •  becomes larger with increasing T1 and decreasing T2 • T2/T1 is the fraction of Q(T1) transferred as heat to the cold reservoir The direction of spontaneous change Experience 1 All spontaneous processes in nature proceed only in one direction. Experience 2 All spontaneous processes in nature are irreversible. They cause a loss of useable work and lead to a gain in heat.  not required by the First Law of Thermodynamics!

46. The Second Law of Thermodynamics Kelvin statement (Second Law of Thermodynamics): No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work. This means: • A gain of work is possible only with simultaneous transport of heat (efficiency ) • It is impossible to build an engine with a higher efficiency than the Carnot engine. This implies that a perpetual motion machine (perpetuum mobile) of the second kind is not feasible. Question: Which state function lets us assess the direction of spontaneous change? Answer: The Entropy S ! (a) The demonstration of the equivalence of the efficiencies of all reversible engines working between the same thermal reservoirs is based on the flow of energy represented in this diagram. (b) The net effect of the processes is the conversion of heat into work without there being a need for a cold sink: this is contrary to the Kelvin statement of the Second Law.

47. Each cyclic process can be regarded as a sequence of two lines of the Carnot cycle: The Entropy S Derivation via the Carnot process: Q = 0 for adiabatic processes: Extension to general cycles: “reduced heat”; state function! A general cycle can be divided into small Carnot cycles. The match is exact in the limit of infinitesimally small cycles. Paths cancel in the interior of the collection, and only the perimeter, an increasingly good approximation to the true cycle as the number of cycles increases, survives. Because the entropy change around every individual cycle is zero, the integral of the entropy around the perimeter is zero too.

48. The Entropy S (cont’d) Introduction of the Entropy S as new state function, with (Clausius) Consider e.g. an isothermal cyclic process, carried out irreversible from 1 to 2, and reversible from 2 to 1: Isolated system: dQ = 0  S2 – S1  0 for irreversible processes S2 – S1 = 0 for reversible processes or, more general: Clausius inequality Other formulation of the Second Law of Thermodynamics: The entropy of an isolated system increases in the course of a spontaneous change: Stot, irrev> 0 It remains constant in the course of a reversible change: Stot, rev= 0

49. Entropy changes accompanying specific processes General expression: Perfect gases: Reactions: Phase transitions: Caution: Chemical reactions are (in general) irreversible, and can not be described via for cv, cp = const. for p, T = const.

50. The variation of entropy with temperature: a more general approach usually cp is not independent of temperature This leads to the The determination of entropy from heat capacity data. (a) The variation of Cp/T with the temperature for a sample. (b) The entropy, which is equal to the area beneath the upper curve up to the corresponding temperature, plus the entropy of each phase transition passed. Nernst heat theorem: The entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero: S  0 as T  0 provided all the substances involved are perfectly ordered. Third Law of Thermodynamics: The entropy of all perfect crystalline substances is zero at T = 0.