Chapter 14. The Amazing Chemical Potential Introduction

Transcription

1 Gibbs Equilibrium of Heterogeneous Substances has practically unlimited scope. It spells out the fundamental thermodynamic theory of gases, mixtures, surfaces, solids, phase changes, chemical reactions, electrochemical cells, sedimentation and osmosis. It has been called, without exaggeration, the Principia of Thermodynamics. Chapter 14 William H. Cropper, Great Physicists, Oxford University Press US, The Amazing Chemical Potential 14.1 Introduction The Chemical Potential µ, a creation of J. W. Gibbs 1, is the essential state variable for studying the thermodynamics of open systems, which includes chemical reactions, phase transitions, non-uniform systems, surfaces and other cases which benefit from modeling as open systems. Although sometimes regarded as vague, 2 interfacing Schrödinger s fixed particle number quantum mechanics with field theory s open system interaction H open = µn op, where µ the chemical potential an energy per particle conjugate to N op, gains clarity in meaning and application. Indeed, N op and µ are key to open system particle hamiltonians and corresponding variable particle processes. Least Bias Lagrangians are modeled as before and lead to intuitive and transparent rules governing diffusive thermodynamic processes. In order to develop an understanding of the thermodynamics of reactions, phase transformations and heterogeneous systems, this chapter consist, mainly, of examples and applications of the chemical potential. 1 J. W. Gibbs, A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Transactions of the Connecticut Academy (1873) 2 In the preface to Introduction to Solid State Physics, Wiley New York 1971, C. Kittel writes: A vague discomfort at the thought of the chemical potential is still characteristic of a physics education. This intellectual gap is is probably due to the obscurity of the writings of J. Willard Gibbs who discovered and understood the matter 100 years ago. 1

2 2 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL 14.2 Diffusive Equilibrium Consider a system composed of two distinct species, say A and B. Allow them, initially, to be thermally and diffusively isolated and chemically un-reactive. This condition of isolation can be expressed, succinctly, in terms of a composite, open system quantum hamiltonian Ĥ composite = [H A op µ A N A op] + [H B op µ B N B op]. (14.1) Its corresponding Least Bias Lagrangian is L = k B [ˆP A ln ˆP A + ˆP B ln ˆP B ] λ 0 (ˆP A + ˆP B ) λ 1A [ H A op µ A N A op ] λ 1B [ H B op µ B N B op ]. (14.2) where for total species isolation λ 1A λ 1B and µ A µ B. Now let the two systems exchange energy so that only the total average energy H A op + H B op = H op. (14.3) is distinct. This is reflected in the Least Bias Lagrangian, Eq.14.2, by setting λ 1A = λ 1B = λ 1 so that L E = k B [ˆP A ln ˆP A + ˆP B ln ˆP B ] λ 0 (ˆP A + ˆP B ) λ 1 [ H A op + H B op ] + λ 1 [µ A N A op + µ B N B op ]. (14.4) Having previously identified λ 1 = 1/T, this simply announces the zeroth law of thermodynamics, i.e. when systems freely exchange energy, at equilibrium the systems are at the same temperature. If the two systems A and B are also in diffusive contact then only average total energy and average total particle number are distinct and the corresponding Least Bias Lagrangian becomes L E,N = k B [ˆP A ln ˆP A + ˆP B ln ˆP B ] λ 0 (ˆP A + ˆP B ) λ 1 [ H A op + H B op ] + λ 1 µ [ N A op + N B op ], (14.5) In summary: if two systems A and B freely exchange energy and particles, at thermodynamic equilibrium they have the same value of temperature T and the same value of chemical potential µ, i.e. µ A = µ B = µ.

3 14.2. DIFFUSIVE EQUILIBRIUM Examples The the ideal gas provides a basic model for several physical examples that illustrate the role of the chemical potential Ideal Gas in the Earth s Gravitational Field At a distance z above but close to the earth s surface, ideal gas molecules experience a quasi-classical gravitational potential energy per particle P E = mgz. Here g is the earth s gravitational constant and m the molecular mass. As introduced in Chapter 13, Z gr can be re-written to reflect this additional gravitational energy contribution Z gr = e β(µ mgz)n Z (N) (14.6) N=0,1,... where Z (N) is the fixed particle, i.e. canonical, partition function for the ideal gas [see Chapter 8] After summing N Eq.14.6 becomes Z (N) = 1 N! (n QV ) N. (14.7) Z gr = exp [e β(µ mgz) n Q V ]. (14.8) Then, as was also shown in Chapter 13 [see Eq.13.14] the average particle number at a distance z > 0 above the earth s surface is N op (z) = 1 β ( µ ln Z gr)t,v (14.9) = 1 β ( µ ) eβ(µ mgz) n Q V (14.10) = e βµ e βmgz n Q V. (14.11) The chemical potential of gas molecules at any height z is, therefore, µ (z) = 1 β ln N op (z) n Q V + mgz. (14.12)

4 4 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL Such gas particles (at height z) can freely diffuse to any adjacent level, say z ± z, so that in diffusive equilibrium µ must be uniform throughout the atmosphere, i.e. independent of z. At ground level, i.e. z = 0, the chemical potential for ideal gas molecules is µ (0) = 1 β ln N op (0), (14.13) n Q V so in diffusive equilibrium Eqs and are equated to give the well known barometric equation. N op (z) = N op (0) e βmgz (14.14) Charged Ideal Gas Between Capacitor Plates Ideal gas molecules, each with charge q, lie between a pair of large parallel capacitor plates placed a distance d apart and charged to a potential V = E x d, where E x is a uniform electric field between the plates. An electric potential energy per particle, P E = qxe x adds, quasi-classically, to the N-particle ideal gas eigen-energies so that, as in Eq.14.11, the number of charged molecules a distance x from the positive plate is Solving for the chemical potential, When x = 0 (the positive plate) N op (x) = e βµ e βqxex n Q V. (14.15) µ (x) = 1 β ln N op (x) n Q V µ (0) = 1 β ln N op (0) n Q V q x E x. (14.16). (14.17) For diffusive equilibrium, in the region between the plates µ is independent of x and At x = d, i.e. the negative plate, N op (x) = N op (0) e βqxex. (14.18) N op (d) = N op (0) e βqv (14.19) where V = E x d is the electric potential difference between the plates.

5 14.2. DIFFUSIVE EQUILIBRIUM Ideal Gas in a Rotating Cylinder Ideal gas particles, mass m, in a cylinder rotating with angular frequency ω, acquire a rotational potential energy per particle (in the rotating frame) which, quasiclassically, is P E rot = 1 2 mω2 r 2. Here r is a particle s radial distance from the cylinder axis. The grand partition function is Z gr = e β(µ+ 1 2 mω2 r 2 )N Z (N) (14.20) N=0,1,... where Z (N) is, as in Eq.14.7, the N particle (canonical) ideal gas partition function. After summing N, at a distance r from the axis of rotation the average particle number is therefore Hence, the chemical potential at r is N op (r) = 1 β ( µ ln Z gr)t,v (14.21) = 1 β ( µ ) eβ(µ 1 2 mω2 r 2) n Q V (14.22) = e βµ e β 2 mω2 r 2 n Q V. (14.23) µ (r) = 1 β ln N op (r) n Q V whereas the chemical potential at r = 0 is 1 2 mω2 r 2, (14.24) µ (0) = 1 β ln N op (0). (14.25) n Q V In diffusive equilibrium µ must be uniform throughout the cylinder, i.e. independent of r, giving the radial distribution of particles in a centrifuge N op (r) = N op (0) exp ( β 2 mω2 r 2 ). (14.26) A Star is Born Star formation takes place inside dense, cold (T 10K) regions of interstellar gas and dust (mostly molecular hydrogen (H 2 ) and carbon monoxide (CO)) known as

6 6 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL giant molecular clouds (GMC). The process of star formation is one of gravitational collapse, triggered by mutual gravitational attraction within the GMC, to form a region called a protostar. GMC s have typical masses of solar masses, densities of 100 particles per cm 3 and diameters of km. The process of collapse resembles a phase transition in that its onset occurs at critical GMC values of radius (r c ), mass (m c ) and pressure (p c ). Once begun the collapse process continues until the GMC is so hot and compact that nuclear fusion begins and a star is born ( years) Thermodynamic Model A thermodynamic model for a GMC is an ideal gas with mutual gravitational attraction between all the gas particles. Were it not for gravitational effects the GMC equation of state would simply be pv = Nk B T. But the GMC has an additional attractive potential energy from gravitational interactions proportional to 1/ r V grav = 1 2 G dr dr [ρ (r) 1 r r ρ (r )] (14.27) with ρ (r) the mass density of the cloud and G the universal gravitational constant. This many-particle gravitational interaction is manageable in a mean field approximation (MFA), 3 in which case each particle sees the average potential of all the other particles as an effective potential per particle, i.e. V eff = G dr ρ Φ (r), (14.29) where ρ = m H V 3 The essence of a mean field approximation is the identity: (14.30) = 3m H 4πR 3. (14.31) ρ (x ) ρ (x) = [ρ (x ) ρ ] [ρ (x) ρ ] + ρ (x ) ρ + ρ ρ (x) ρ ρ (14.28) where ρ is the average density. The MFA neglects the first term, i.e. fluctuations around the average density.

7 14.2. DIFFUSIVE EQUILIBRIUM 7 is an average density per particle, m H is the mass of a hydrogen molecule (assumed to be the dominant species in the GMC) and where V is the cloud s volume. The potential Φ (r) can be determined from Poisson s equation with the solution 2 Φ (r) = 4πρ (r) (14.32) Φ (r) = 1 r 0 r ρ (r ) dr. (14.33) To simplify the model, all GMC radial mass dependence is ignored the cloud being assumed spherical with uniform density 4,5 ρ (r) = N op m H /V, (14.34) where N op (r) is the average number of particles at r. 6 Therefore 4πG 0 R Φ (r)r 2 dr = 3G N op m H V 5R, (14.35) where N op m H = M, the cloud mass and R is the cloud radius. Thus we can write and a grand partition function Z gr = which from Eqs.14.7 and 14.8 is NV eff = Nm H Φ (14.36) e β [µ m H Φ ] N Z (N) (14.37) N=0,1,2... Z gr = exp {n Q V e β [µ m H Φ ] } (14.38) 4 This simplification has obvious shortcomings. Nevertheless, some features of gravitational collapse are preserved. 5 W.B. Bonnor, Boyle s Law and Gravitational Instability, Mon. Notices Roy. Astron. Soc. 116, 351 (1956). 6 Notice the self consistent use of N op!!

8 8 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL where 7 Using this we find: Φ = 3MG 5R. (14.39) 1. The mean number of hydrogen molecules in the GMC N op = 1 β ( µ ln Z gr)t,v = 1 β ( µ ) n QV e β [µ + m H ( (14.40) 3GM 5R )] (14.41) = n Q V e β [µ + m H ( 3GM 5R )], (14.42) 2. The chemical potential µ = m H ( 3GM 5R ) + 1 β ln N op n Q V (14.43) 3. The pressure in the GMC p = 1 β ( V ln Z gr)t,µ (14.44) which, with R = ( 3V 4π ) 1/3 and Eq.14.43, gives the GMC Equation of State p = N op k B T V N op m H 5 ( 4π 3 ) 1/3 GMV 4/3. (14.45) Restoring M = N op m H the equation of state becomes p = N op k B T V N op 2 m 2 H 5 ( 4π 3 ) 1/3 GV 4/3. (14.46) 7 In what follows the factor 3/5 is due to assumed spherical symmetry of the GMC.

9 14.2. DIFFUSIVE EQUILIBRIUM A Collapse Criterion A criterion for collapse is that the cloud s adiabatic compressibility κ S = 1 V ( V p ) S = 1 V γ ( V p ) T (14.47), (14.48) becomes infinite. 8 At that critical point the cloud spontaneously shrinks until the onset of some heat source (nuclear fusion) overcomes gravitational effects. 9 From the equation of state, Eq.14.46, the adiabatic compressibility is k S = 1 γv [ N op k B T V 2 4 1/ (4π 3 ) N op 2 m 2 H GV 7/3 ] (14.52) so that with ρ = N op /V the critical radius, r c, for gravitational collapse is and the critical pressure is r c = ( 45 1/2 16π ) kb T m 2 H Gρ (14.53) p c = ρk BT 4. (14.54) 8 As previously defined, γ = C p /C V. 9 This is equivalent to the Jeans criterion, [J. H. Jeans, The Stability of a Spherical Nebula, Phil. Trans. Royal Soc.(London). Series A,199, 1 (1902).] that the velocity of sound c s in the GMC becomes imaginary!!. Since the velocity of sound in a gas is c 2 s = K S ρ (14.49) where K S is the gas adiabatic bulk modulus K S = V ( p V ) S = γv ( p V ) T (14.50) (14.51) and ρ is its density, the Jeans criterion is that the adiabatic bulk modulus (the reciprocal of compressibility) becomes zero.

10 10 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL The trigger for star formation is GMC compression to p p c by collisions with other GMCs or compression by a nearby supernova event. Alternatively, galactic collisions can trigger bursts of star formation as the gas clouds in each galaxy are compressed Van der Waals Equation An ideal gas (classical or quantum) is defined by neglecting interactions between particles. More realistic models (theoretical or computational simulations) are generally based on pairwise forces with a long range attractive and short range repulsive component, the latter restricting gas molecules to finite volume. In 1873 Van der Waals pursued these ideas in his Ph. D. Thesis, replacing the ideal gas law with an improved equation of state that bears his name. 10 For this he was awarded the 1910 Nobel Prize in Physics. A derivation of the van der Waals equation is similar to the theory of GMC collapse, except that the interaction potential between gas (liquid) molecules can be more complicated than a gravitational interaction. In this case we choose a simple interaction U (r r) that displays hard core repulsion as well as a long range inter-molecular attraction, as pictured in Figure Once again we write an interaction energy V int = 1 2 dr dr [ρ (r) U (r r) ρ (r )] (14.55) with the density in units of particles per unit volume. The many-particle inter-molecular interaction is also manageable in a mean field approximation (MFA) [see Eq.14.28], in which case the interaction energy per particle is V int = dr ρ Φ (r), (14.56) where ρ = 1 V. (14.57) 10 In this era the reality of microscopic models was still being challenged by influential skeptics.

11 14.2. DIFFUSIVE EQUILIBRIUM 11 Figure 14.1: Intermolecular potential. Assuming gas uniformity Φ (r) =4π ρ (r)u (r)r 2 dr (14.58) r 0 =4π r 0 ( N op V )U (r)r2 dr. (14.59) Now, with the abbreviation a = 4π U (r)r 2 dr (14.60) r 0 we have the potential per particle V int = a N op V (14.61)

12 12 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL and the grand partition function Z gr = N=0,1,2... β [µ + a N op e V ] N Z (N) (14.62) β [µ + a N op = exp n Q V e V ]. (14.63) From this we find N op = 1 β ( µ ln Z gr)t,v (14.64) = 1 β ( β [µ + a N op µ ) n QV e V ] (14.65) β [µ + a N op = n Q V e V ], (14.66) and the chemical potential µ = a N op V + 1 β ln N op n Q V. (14.67) Finally, the pressure is p = 1 β ( V ln Z gr)t,µ = N op βv An expression for the equation of state is, so far (14.68) a ( N op V ) 2. (14.69) p + a ( N op V ) 2 = N op βv. (14.70) The final step in the van der Waals argument assigns the total space occupied by the gas molecules as a minimum gas volume V min = b N op, where b = π 6 r3 0 is the

13 14.2. DIFFUSIVE EQUILIBRIUM 13 hard core restricted volume per molecule. Thus the van der Waals equation of state becomes p + a ( Nop 2 V ) [V N op b] = N op β [V N op b] is sometimes called V the effective volume.. (14.71) Chemical Reactions Consider a generic chemical reaction involving three species A, B and AB, with A + B AB. (14.72) Initially allow all three species to be non-interacting so that a Least Bias Lagrangian is L = k B [P A ln P A + P B ln P B + P AB ln P AB ] λ 0 (P A + P B + P AB ) λ 1A [ H A op µ A N A op ] λ 1B [ H B op µ B N B op ] λ 1AB [ H AB op µ AB N AB op ] (14.73) where µ A, µ B and µ AB are the chemical potentials for each of the isolated species. Introducing a uniform reaction temperature T the Least Bias Lagrangian becomes λ 1A = λ 1B = λ 1AB = λ 1 = 1/T (14.74) L = k B [P A ln P A + P B ln P B + P AB ln P AB ] λ 0 (P A + P B + P AB ) λ 1 [ H A op + H B op + H AB op ] + λ 1 [µ A N A op + µ B N B op + µ AB N AB op ]. (14.75) Eq defines atom number conservation in the reaction, i.e. A for atom type N A op + N AB op = ν A (14.76)

14 14 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL while for atom type B N B op + N AB op = ν B (14.77) where ν A and ν B are distinct constants. These reaction constraints are described in the Least Bias Lagrangian L R = k B [P A ln P A + P B ln P B + P AB ln P AB ] λ 0 (P A + P B + P AB ) λ 1 [ H A op + H B op + H AB op ] + λ 1 [µ A ( Nop A + Nop AB ) + µ B ( Nop B + Nop AB )]. (14.78) which is obtained by requiring as the condition for chemical (diffusive) equiibrium µ A + µ B = µ AB. (14.79) As a second example consider a hypothetical reaction among the diatoms A 2, B 2 and the molecule A B A 2 + B 2 2 AB (14.80) Assuming uniform reaction temperature λ 1 = 1/T, consider the chemical species initially non-reactive. In that case the Least Bias Lagrangian is L = k B [P A2 ln P A2 + P B2 ln P B2 + P AB ln P AB ] λ 0 (P A2 + P B2 + P AB ) λ 1 [ H A 2 op + H B 2 op + H AB op ] However, this time the atom conservation constraints are: for atom A for atom B + λ 1 [µ A 2 N A 2 op + µ B 2 N B 2 op + µ AB N AB op ]. (14.81) 2 N A 2 op + N AB op = ν A (14.82) 2 N B 2 op + N AB op = ν B (14.83) where ν A and ν B are distinct constants required by Eq These reaction constraints are embodied in the Least Bias Lagrangian L R = k B [P A2 ln P A2 + P B2 ln P B2 + P AB ln P AB ] λ 0 (P A2 + P B2 + P AB ) λ 1 [ H A 2 op + H B 2 op + H AB op ] + λ 1 [µ A 2 ( N A 2 op + 1 AB Nop ) + µ B 2 ( N B 2 op + 1 AB Nop )]. (14.84) 2 2

15 14.2. DIFFUSIVE EQUILIBRIUM 15 which is obtained by requiring the condition for chemical (diffusive) equiibrium µ A 2 + µ B 2 = 2µ AB. (14.85) These results can be generalized to any chemical reaction, e.g. n A A + n B B + n C C n X X + n Y Y + n Z Z, (14.86) in which case the condition for chemical (diffusive) equilibrium among reactants A, B, C and products X, Y, Z is n A µ A + n B µ B + n C µ C = n X µ X + n Y µ Y + n Z µ Z (14.87) where n A, n B,... n Z are the coefficients in Eq A Law of Mass Action Chemical reactions such as (hypothetically) defined in Eqs.14.72, 14.80, do not usually go to completion, with the sealed reaction vessel usually containing an equilibrium mixture of reactants and products. For example, the gaseous reaction for which at thermal equilibrium 3H 2 + N 2 2 N H 3 (14.88) 3µ H 2 + µ N 2 2µ NH 3 = 0, (14.89) yields only a small amount of NH 3 (ammonia), with reactants H 2 and N 2 having a continued presence. The equilibrium concentrations of reactants and products in the closed reaction vessel is, however, well described by a Law of Mass Action which is a consequence of Eq This law is arrived at by first exponentiating or exp (3µ H 2 ) exp (µ N 2) exp (2µ NH 3 ) = 1 (14.90) [exp (µ H 2 )] 3 [exp (µ N 2 )] [exp (µ NH 3 )] 2 = 1. (14.91)

16 16 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL Continuing in the spirit of the example, assume all participating gases g are ideal and at sufficiently low temperature that no internal modes of the reactive molecules (rotational, vibrational or electronic) are excited. In which case βµ g = ln [ C g ] (14.92) n g Q where m g is the molecular mass of the gas g and = ln {[ mg 2π h 2 β ] 3/2 C g } (14.93) C g = N g op V is the gas concentration. From this follows a typical mass action result (14.94) [C NH 2 3 ] [C H 2 ] 3 = {n H 2 Q }3 {n N 2 Q } (14.95) [C N 2 ] {n NH 3 Q }2 = K (T ) (14.96) where K depends only on T. Specific mass action constants K are associated with physically different molecular models. If, for example, in the previous model temperature is raised sufficiently to access a molecule s rotational and vibrational modes (internal modes) the N-particle canonical partition function becomes Z (N) = 1 N! [Z int] N [n Q V ] N (14.97) where Z int is the partition function for the internal (rotational and vibrational) modes. In that case we redefine n Q for the molecular gas g so that the mass action expression becomes where n g Q Zg int ng Q = ng int (14.98) [C NH 3 ] 2 [C H 2 ] 3 [C N 2 ] = K int (T ), (14.99) K int (T ) = {nh 2 int }3 {n N 2 int } {n NH 3 int } 2. (14.100)

17 14.2. DIFFUSIVE EQUILIBRIUM Surface Adsorption Langmuir s Model In the process of film growth or surface doping, atoms or molecules in gas phase or in dilute solution, bind to the film surface. A theory of solid surface coverage by these molecules was formulated by Irving Langmuir (1916), the acknowledged pioneer of surface chemistry. Assumptions of the model are: 1. The solid surface is in contact with ideal monatomic gas atoms at temperature T and pressure p There are a fixed number of sites N on the surface available for bonding. 3. Each surface site can be only singly occupied (monolayer coverage). 4. Adsorption at a given site is independent of occupation of neighboring sites (no interactions). 5. A gas molecule bound to the surface has the non-degenerate eigen-energy with respect to the energy of a free gas atom. E bound = ɛ (14.101) 6. The energy difference between adsorbed gas atoms and free gas atoms is independent of surface coverage. 7. Atoms bound to the surface (bd) are in thermal and diffusive equilibrium with gas phase atoms as described by the reaction With Langmuir s model the fractional surface coverage θ = A gas A bd (14.102) N bd op N max (14.103) can be determined, where N bd op is the average number of gas atoms bound to N max accessible solid surface sites. Diffusive equilibrium in the surface reaction of Eq requires that µ gas = µ bd (14.104)

18 18 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL gas bound Figure 14.2: Gas molecules interacting with solid surface. where µ bd is the chemical potential of atoms bound to surface sites and µ gas is the chemical potential for ideal gas atoms. Finding the average number of atoms bound to surface sites N bd op follows discussions in Chapter 13 where it was shown that N bd op = N max Ne β{[es(n)] µ bdn} N=0 s N max e β{[es(n)] µ bdn} s N=0. (14.105) where the average number of bound atoms Nop bd is limited to N max, the number of available surface sites. The denominator in Eq is the Grand Partition Function Z gr, Z gr = N max e βµ bdn Z (N) (14.106) N=0 where Z (N) is the (canonical) partition function for N-bound atoms Z (N) = g (N, N max ) e Nβ ɛ. (14.107) Since the bound atom s only eigen-energy [see Eq ] has the configurational degeneracy g (N, N max ) = N max! (N max N)! N! (14.108)

19 14.3. DISSOCIATIVE ADSORPTION 19 the grand partition function is Z gr = N max N=0 and the fractional surface coverage is [see Eq ] N max! (N max N)! N! enβ(µ bd+ ɛ ) (14.109) = [1 + e β(µ bd+ ɛ ) ] Nmax (14.110) θ = = = N bd op N max (14.111) 1 1 ln Z gr N max β µ bd (14.112) e βµ bd e β ɛ 1 + e. βµ bd e β ɛ (14.113) The ideal gas in contact with the surface has the chemical potential µ gas = 1 β ln N gas op n Q V. (14.114) Equating the chemical potentials [see Eq ] (diffusive equilibrium) and using the ideal gas equation of state p 0 V = coverage θ = gas Nop where Π 0 is the temperature dependent factor known as the Langmuir isotherm. β gives the Langmuir fractional surface p 0 Π 0 + p 0 (14.115) Π 0 = n Q e β ɛ β (14.116) 14.3 Dissociative Adsorption A diatomic gas molecule, say A 2, may not simply bind to a surface site, but may dissociate, with each component atom binding to a single site. This is referred to as dissociative adsorption and is described by the chemical reaction A 2 2A bound. (14.117)

20 20 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL Therefore, the thermal equilibrium chemical potentials satisfy µ A2 2µ A = 0, (14.118) where µ A2 is the chemical potential of the ideal diatomic gas and µ A is the chemical potential of surface bound atoms. The ideal diatomic gas in contact with the surface has the chemical potential µ A2 = 1 β ln N A 2 op n A 2 int V. (14.119) which includes the effect of internal (rotational and vibrational) modes [see Eq.14.97]. Once again taking E bound = ɛ the fractional surface occupation is, according to Eq , θ = Now applying Eq the surface coverage is Using the ideal gas result, Eq , we finally have e βµ A e β ɛ 1 + e βµ A e β ɛ. (14.120) e (1/2)βµ A 2 e θ = β ɛ. (14.121) 1 + e (1/2)βµ A 2 eβ ɛ θ = p Π D + p (14.122) where p is the gas pressure and the Langmuir isotherm is Π D = n A 2 int β e β ε. (14.123) 14.4 Crystalline Bistability In an ordered crystal it is common for atoms to be bistable, i.e. atoms at normal crystalline sites can migrate to abnormal sites (usually accompanied by a lattice distortion), where they have a different binding energy. If a crystal at a temperature T has normal sites and displaced sites which, for definiteness, lie on an interstitial sub-lattice [see Figure 14.3], after a long time a certain fraction of the atoms will

21 14.4. CRYSTALLINE BISTABILITY 21 occupy displaced sites. Since the total number of atoms normal plus defect is conserved, the atom constraint is Nop norm + Nop def = N (a constant). (14.124) The migration process can be interpreted as the chemical reaction A norm A inter. (14.125) Figure 14.3: Lattice with atoms migrating from normal to interstitial sites. Following details of Langmuir s model and assuming: 1. An equal number of normal and interstitial sites, N s. 2. Interstitial atoms have energy ɛ 0 > 0 with respect to normal sites. 3. Only single occupancy of normal and interstitial sites is permitted. The grand partition function for single occupancy of a normal site is Z norm gr = N s N=0 N e Nβµn s! { (N s N)!N! } (14.126) = (1 + e βµn ) Ns (14.127) while the grand partition function for single occupancy of an interstitial site is Z inter gr = N n N=0 e Nβµ N def s! { (N s N)!N! enβɛ 0 } (14.128) = (1 + e βɛ 0 e βµ inter ) Ns (14.129) The grand partition functions, Eqs and , yield the results Nop norm = 1 ln Zgr norm (14.130) β µ norm = N s λ norm 1 + λ norm (14.131)

22 22 CHAPTER 14. THE AMAZING CHEMICAL POTENTIAL and where Nop def = 1 ln Zgr inter (14.132) β µ inter = N s λ inter e βɛ λ inter e βɛ 0 (14.133) λ norm = e βµnorm (14.134) and λ inter = e βµ inter. (14.135) At thermal equilibrium µ norm = µ inter (14.136) or λ norm = λ inter = λ (14.137) and, of course, Eq applies, From Eqs , , and and we find Nop inter + Nop norm = N s. (14.138) λ 1 + λ + λe βɛ0 1 + λe βɛ 0 = 1 (14.139) λ = e βɛ 0/2. (14.140) Therefore, according to Eq the average fractional occupation of interstitial sites is as shown in Figure N inter op N s = exp (βɛ 0 /2) (14.141)

23 14.4. CRYSTALLINE BISTABILITY 23 def Figure 14.4: Fractional occupation number of interstitial sites.