Unit Cells and Crystal Structures: Bravais Lattices and Adsorption, Study notes of Physical Chemistry

Download Unit Cells and Crystal Structures: Bravais Lattices and Adsorption and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3520 Spring 2004 Introduction to Solid-State and Surface Chemistry The solid materials can be: – crystalline (= substances that have a periodic structure) – amorphous (= substances that don’t have a periodic structure) The structure of crystals – The smallest collection of atoms (or molecules) in the crystals such that its replication in the three directions will generate the crystal is called the unit cell. The unit cell cannot have any arbitrary shape; it should be a geometric structure that will fill all space when replicated. The unit cell is not unique for a crystal. – The most general unit cell is three- dimensional parallelepipedic. Consider the origin of the coordinate system in the lower left corner and the positive a, b, and c axes pointing along the sides of the unit cell. a b c α β c a bγ – The geometry of the unit cell can be described by specifying the length along the a, b, and c axes (that is a, b, and c, respectively) and the angles between pairs of axes (that is α, β, and γ, respectively). – The atoms (or molecules) will always occupy the corners of the unit cell and, in addition, the center of the unit cell, and centers of faces. (In the case of a crystal containing more than one type of particles, like ionic crystals, other positions can be occupied.) – The number of atoms in a unit cell is given by the formula: )cellunit theinside atoms #()faces on the atoms #( 2 1 )edgeson atoms #( 4 1)corners in the atoms #( 8 1cell atoms/unit # ++ += – August Bravais showed that only 14 distinct unit cells are necessary to generate all possible crystal lattices, and they are called Bravais lattices. The lattices are organized in seven classes: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, trigonal, and cubic based on the general geometric features of the unit cell. CHEM 3520 Spring 2004 P I C F R Telelinle ambeac ae fey 5 Monoclinic = oebeo os ypetee ba So wt ra phe Orthorhombic erbzc asfsys 30° ee ot _ Terra gonsl Gg@edeac a= fay = SP Trigonal Héxagonal asbec gebeo anbey SOF a cee fim SOF a ae yedao ch fie Ae ye 20" Cube a=abec an faye SOF CHEM 3520 Spring 2004 The crystal surface – The simplest model of a crystal surface is perfectly flat with the distance between atoms being the same as in the bulk. Even in this case, the structure of the crystal surface is not unique and it depends upon how the crystal is cut. One specifies the structure of the surface by specifying the three Miller indices for the plane of the surface that corresponds to the crystal plane in the bulk. hkl – Example: Platinum crystallizes in a face-centered cubic (or cubic closed-packed) subsurface surface 111 surface 100 surface 110 surface – Usually, the surface atoms occupy sites that are shifted from the atomic positions in the bulk. Most of atomic metals exhibit a significant contraction (< 40%) of interlayer distance between the first and second layers of atoms. Often there are compensating expansion between the second and third layer (~ 1%) and smaller expansion between the third and the fourth layer. – In general, without special treatment, the atomic structure of a crystal surface may show numerous irregularities that give roughness to the surface. Understanding the atomic structure of the surface would help understanding chemical reactions on the surfaces (like heterogeneous catalysis). – Some of the imperfections of the surfaces are terraces, steps, and adatoms. – The reactivity of these imperfections is usually different (higher) than that of a perfect surface. CHEM 3520 Spring 2004 Adsorption on crystal surfaces – Adsorption is the process of trapping molecules or atoms on a surface. (It was confirmed to be the first step in a surface-catalyzed reaction.) The adsorbed molecule (atom) is called adsorbate, and the surface is called the substrate. – Adsorption is always an exothermic process ⇔ 0ads <∆ H . – Physisorption or physical adsorption (full line) is the process in which the attractive forces between substrate and adsorbate are weak (< ) and are mainly van der Waals in nature. kJ/mol20 – Chemisorption or chemical adsorption (dotted line) is the process in which the adsorbate is bound to the substrate by covalent or ionic forces. The interaction is strong (~ 250-500 kJ/mol). Chemisorption involves a bond broken in the molecule and new bond made between the molecular fragments and the surface (dissociative chemisorption). Only a single layer of molecules (monolayer) can chemisorb on the surface. z V(z) A+B A–B – Lennard-Jones modeled the adsorption in terms of one-dimensional potential energy curves. The potential energy depends on the distance between the substrate and adsorbate (z). V at infinite separation between the substrate and non-dissociated adsorbate. 0)( =z – The equilibrium substrate-adsorbate bond length for the physisorbed molecule ( ) is longer than the substrate-adsorbate bond length for the chemisorbed molecule ( ). phz chz – Molecules that chemisorb can be initially trapped in a physisorbed state. The physisorbed molecule is a precursor to the chemisorbed molecule. The physisorption and chemisorption potential energy curves cross at a distance . The transformation from the physisorbed state to the chemisorbed state can be seen as a chemical reaction characterized by activation energy . In some cases the energy at is bigger than 0. cz aE cz Monitoring the adsorption – An adsorption isotherm is a plot of surface coverage as a function of the gas pressure at constant temperature. The adsorption isotherms can be used to determine the equilibrium constant for the adsorption-desorption reaction, the concentration of surface sites (active sites) for the adsorption, and the enthalpy of adsorption. – A simple model was proposed by Langmuir in 1918. Assumptions: adsorbed molecules do not interact to each other; the enthalpy of adsorption is independent of surface coverage; there are a finite number of surface sites where the molecule can adsorb. CHEM 3520 Spring 2004 – The processes of adsorption and desorption are reversible elementary processes: ; S(s)AS(s)A(g) a d −⇒⇐+ k k [A][S] S][A d a −== k k Kc where k and k are the rate constants for adsorption and desorption a dbP θ A – Consider 0σ to be the concentration of surface sites (in 2m− ), θ to be the fraction of active sites occupied by adsorbates ⇒ concentration of empty surface sites is 000 )1( σθθσσ −=− – Rate of adsorption: ]A[)1( 0aa σθ−= kv ; Rate of desorption: 0dd θσk=v – At equilibrium: ]A[ 111 cK += θ or A 111 bP += θ ([ TkP BA /]A = ; ) TkKb c B/= 1/θ 1/PA – Denote by V the volume adsorbed onto surface at m 1=θ ⇒ m/VV=θ ⇒ mm 111 VPbVV += ; A linear representation of V 1 versus P 1 has a slope of m 1 bV and an intercept of m 1 V . – V can be used to determine the concentration of surface sites m 0σ by knowing the are of the surface and transforming V to the number of molecules (=number of active sites). m – The Langmuir adsorption isotherm for a diatomic molecule: ; S(s)A2S(s)2(g)A a d 2 − ⇒ ⇐+ k k 2 2 2 d a ][S][A S][A − == k k Kc – Rate of adsorption: v ; Rate of desorption: v 20 2 2aa )1](A[ σθ−= k 2 0 2 dd σθk= – At equilibrium: v da v= ⇒ 1/2 A 1/2 A 22 111 Pb += θ – A representation of θ 1 versus 1/2 A2 1 P will be a linear representation. – The rate constant of desorption from a surface obey Arrhenius-like expression: where RTEek /10d ads −−= τ HE adsads ∆−≅ and is a constant with units of time. The reciprocal of is called residence time of a molecule on the surface: s10 120 −≈τ dk RTEe /0 adsττ = – There are more advanced models that Langmuir model that can account also multilayer adsorption.