Ch2102 - Bonding And Structure

CH2102 – Bonding and Structure The Breakdown of Classical Ideas 1898 – JJ Thompson introduced the ‘fruit cake’ model, a positive mass containing the negatively charged electrons within it 1909 – Rutherford proposed a model involving a positive nucleus with orbiting electrons. The problem with this model is that opposite forces attract and, since the proposed centrifugal force of the electrons would irradiate energy, it would simply collapse into itself. Atomic Spectra When an atom is given energy it is absorbed, allowing it to reach an excited state. This energy is then emitted again in order to reach the ground, stable, state and as it is released it is seen as discrete lines of colours. This shows that light is quantised into energy gaps equivalent to those between the ground and excited state of the atom, and as such can be used to determine the energy levels within atoms according to the formula; ΔE=hν (where h=Planck’s constant=6.626x10-34Js) Bohr Theory    

Electrons possess fixed energy orbits Each energy level is characterised by the quantum number, ‘n’ Lines in the atomic spectra show the transitions between orbits However; Each line in the atomic spectrum is, in fact, several lines close together. Therefore, each orbit must contain ‘sub-shells’.

Considering Wave Mechanics in Remaking Our Description of the Electron Light is both a wave and a particle, everything with a mass also has a wavelength and vice-verse, light is therefore defined as being composed of photons, or ‘packets’ of light energy. If E=hν defines the energy of light, and E=mc2 defines the energy of a particle then the two must be equal for a wave-particle duality, this gives rise to the de Broglie relationship; hν=mc2 but ν=c/λ So; hν h = mc 2 , = mc and mc = ρ (momentum) λ λ Therefore; h λ

= ρ, relating wave and particle behaviour, The de Broglie Relationship.

Electrons also act similarly to light, experimentally shown as they can be diffracted. Note however, most objects do not display wave-particle behaviour.

The Heisenberg Uncertainty Principle For a particle it is possible to measure position and momentum at the same time. For a wave-particle this is not possible, the more accurately we measure the position, the less accurately we can find momentum. This is because to view the electron we must use light, which in turn gives the electron energy, distorting the results. This is an inherent property of nature and cannot be circumvented. When position=x, y, z and error=Δx, Δy, Δz and momentum =ρx, ρy, ρz and error=Δρx, Δρy, Δρz; Δx + Δρx ≥

h 4π

This makes it meaningless to talk of electron position in orbits since both values simply can’t be evaluated effectively. We therefore exchange orbits for orbitals – a probability distribution of space in which the electron moves and not a precise position. Wave Mechanics 

An electron in a stable state must also be in a stationary state, meaning it returns to the same point repeatedly in order to give the diameter of the orbital an integer number of wavelengths since wave mechanics will only allow for these values.

The Schrodinger Equation δ2 ψ δ2 ψ δ2 ψ 8π2 + + + 2 E−v ψ=0 δx 2 δy 2 δz 2 h Where the blue describes the position of the electron, the red is a constant value and the green, v, is the potential energy (the sum of all attractive/repulsive forces). Ψ is the wave function, it is the equivalent of y in y=sinx ψ itself is simply numerical and provides no information, ψ2 however gives the probability of finding an electron at any given Cartesian point defined by (x, y, z). ψ has two parts;

𝜓 = 𝑅𝑛,ℓ . 𝑌𝑙,𝑚 ℓ ,𝑛 where R= the radial wave function, dependant on distance from nucleus and Y= the angular wave function, dependant on the Cartesian co-ordinates. The Quantum Numbers 



n – the principle quantum number -takes integral values, n≠0 -determines the energy level of the orbital ℓ – the azimuthal quantum number -takes integral values from 0 ->n-1 -determines orbital shape



i.e. ℓ=0=s-orbital ℓ=1=p=orbital etc. mℓ– the magnetic quantum number -takes integral values from -ℓ-> ℓ -determines the orbital’s orientation in 3d space

Radial Wave Function, R When n=1, mℓ and ℓ must equal 0; this gives rise to a 1s orbital. The radial wave function for a 1s orbital is, graphically, shown as;

R (Å-3/2)

N.B. This point is asymptotic

And for 2s (where n=2, mℓ=0 and ℓ=0);

r(Å)

Note that in this graph R is negative in places;  this is nothing to do with electron presence or electrostatic charges  the sign is purely mathematical and is part of R, the description of the electron. It describes the phase, an important factor in bonding.

Since R is not measureable we look at the radial distribution function for which we consider the electron density (related to ψ2) At the nucleus, r=0, there is a high probability of finding an electron but only a tiny amount of space available to possibly occupy. For small values of r there are more points to occupy but less probability, until you reach large values of r where there are

many possible spaces to occupy but a much lower probability again. This leads to; Probability Distribution= probability of (x) x the number of ways of getting (x) =ψ2 x x4πr2 (the surface area of a sphere)

For 1s this gives;

 



For 2s this gives;

the area of maximum electron density increases with larger values of n, therefore n determines the size of the orbital since there is technically a small but non-zero probability of finding an electron metres away from the nucleus it is difficult to determine the actual size of the orbital, a ‘boundary surface’ is therefore used, the area containing 90% of the probability is deemed the orbital size There are distances from the nucleus with a zero probability of containing an electron; these are called nodes or terminals. The number of nodes is equal to; No. of nodes=n- ℓ -1

Once the fixed boundary surface is determined we can consider the angular wave function, Y, and establish the shape of orbitals within 3d space, determined by mℓ and ℓ. This shape is independent of n, but the size of the shape is reliant upon it.

Solving the Schrodinger Equation for Multiple Electrons To solve the Schrodinger equation in multiple electron systems is problematic as the term v (potential energy) includes information on both nuclear attraction and, in multiple electron systems, electron repulsion. This means, for example, that to work out the Schrodinger equation for a 1s electron in a multiple electron system, we must first know the electron distribution, in essence this means me must solve the Schrodinger equation before we can solve the Schrodinger equation.

We therefore approximate using the hydrogen atom orbital wave function with one large difference; the orbital energies now rely on both the n and ℓ quantum numbers. Electrons fill orbitals starting with the lowest energy;

i.e. 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f etc Why s -> p -> d -> f?  Electrons in s-orbitals, and to a lesser extent p-orbitals, spend a small amount of time at, or very close to, the nucleus, thus making them very stable and in turn low in energy  Outer electrons are screened from nuclear attraction by the shielding effect of inner electrons, hence for example 3s electrons are more tightly bound than 3p electrons

The Aufbau Principle “To move from one element to the next (in the Periodic Table) add 1 proton, x neutrons and 1 electron into the lowest energy orbital available.” A fourth quantum number, ms (magnetic spin), goes on to define how many electrons may fit into each orbital. 1

ms=± 2 Pauli’s Exclusion Principle “No two electrons in a given system may have identical values for all 4 quantum numbers.” Since ms may only have two values, this therefore means there may only be 2 electrons in any given orbital. Hund’s Rule “The most stable state is one with identical (parallel) spins.” Therefore in a system with 2 electrons occupying a 2p sub-shell, each electron will take up a position in a separate orbital and with the same value of ms i.e. having parallel spin values

Bonding in Molecules The octet rule – atoms bond together in order to achieve a stable electronic configuration (a full outer shell of electrons) Ionic Bonding     

Applies to salts, e.g. NaCl Regular array of cations and anions Bonding is non-directional and based on electrostatic interaction One atom loses one or more electrons to obtain an octet, the other gains one or more electrons Common for one or two electrons to be lost or gained, more than that is hindered by increasing successive ionisation energies

Covalent Bonding Valence bond theory;   

Not correct but useful to visualise simple molecules Directional bonding caused by sharing of electrons Simple for molecules like H2, for some molecules hybridisation must first occur

Hybridisation;   

i.e. methane, it can be experimentally shown that all C-H bonds are equivalent but C contains an 1s22s22p2 arrangement of electrons and promotion wouldn’t be a valid route as the 2s and 2p are far from equivalent sp3 hybridisation therefore occurs creating 4 equivalent orbitals, each containing a lone electron in the case of multiple bonds, i.e. C=C, different hybridisations occur, in this case 3sp2 hybrid orbitals form with one p orbital remaining, the sp2 orbitals form ς bonds, the p orbital then overlaps to form a π bond between the carbons

Problems with valence bond theory; 1) Excited states are impossible to represent 2) Experimental observations are left unexplained in certain cases, e.g. O2 being blue and having two unpaired electrons Also some elements can have an extended octet of more than 8 electrons, i.e. PCl5 This, however, is explained by hybridisation of s, p and d orbitals however to form 5 dsp3 orbitals. Although this may be true of phosphorous, which can form both PCl3 and PCl5, the same is not true of nitrogen as the energy gap between the orbitals is too large, hybridisation may only occur between orbitals of the same principle quantum number. Electronegativity Electrons within a covalent bond are not always equally shared. Elements have a property known as electronegativity, a measure of the attractive force of an atom towards electrons. Elements with the highest electronegativity are found in the top right of the periodic table (N O F particularly). Electrons in a covalent bond will be closer to the more electronegative atom which leads to bond polarity; this is shown by δ+ and δ-.

Lewis Structures of Simple Molecules Among the easiest methods of representing a molecule, and commonly used, the Lewis structure method shows covalent bonds as solid lines, lone pairs as two dots .. (with only the lone pairs of interest shown, usually those involved in reactions or on the central atom). This method has absolutely no representation of 3d structure and requires a prior understanding of the connectivity within a molecule In some cases it may seem a mistake has been made in that the final result is charged, in this case formal charges are allocated providing a neutral molecule with the correct number of bond-pairs between all of the atoms involved. For example nitric acid, HNO3; O H O N+ O-

Molecular Orbital Theory – Diatomic Molecules Again wave mechanics are used, this time involving solving the Schrodinger equation for both a multiple electron and a multiple nuclei system, also relying upon an approximation as with multielectron atom systems. The solution in this case involves molecular orbitals, similar to the idea of atomic orbitals but centred around the entire molecule, they are again defined by ψ and ψ2. The approximate method;   

at any moment an electron near one nucleus will behave approximately as if in an atomic orbital of that atom over time the electron becomes associated with all of the nuclei involved in the molecule the molecular orbital is therefore constructed using the linear combination of atomic orbitals (LCAO)

The number of molecular orbitals will always equal the initial number of atomic orbitals. There are two types of molecular orbital; 1) A bonding molecular orbital -these molecular orbitals are formed by the sum of two atomic orbitals overlapping favourably as a result of in-phase combination of wave functions This molecular orbital strengthens the interaction between the two nuclei leading to large amount of electron density between the nuclei.

2) An anti-bonding molecular orbital -these molecular orbitals arise from the difference of two atomic orbitals, an unfavourable overlap arising from the out-of-phase combination of wave functions This molecular orbital reduces the bonding between the two atoms leaving little or no electron density between the two nuclei and creating a node.

In terms of stability (high to low) and energy (low to high); bonding molecular orbitals > non-interacting atomic orbitals > anti-bonding molecular orbitals

This is due to stabilising effect of several nuclei on an electron.

The bond order, essentially the number of bonds between atoms is defined as; Bond order= (no. of bonding electrons)-(no. of antibonding electrons)/2

Magnetic Properties -if all of the electrons in a molecule are paired then it will be repelled by a magnetic field, the molecule is then known as diamagnetic -if there are unpaired electrons present it will be attracted by a magnetic field and is known as paramagnetic Molecular Orbital Theory for First Row p-Block Elements This is an even more complicated process as there are p-orbitals to consider as well as s-orbitals. In order to do this; 1) Consider only the valence electrons, core electrons are too tightly bound to the nucleus to contribute to bonding 2) The most efficient overlap is between orbitals of the same energy, e.g. 2s/2s and 2p/2p (these are called homonuclear diatomic interactions, heteronuclear interactions (i.e. 1s/2p overlaps as in Hydrogen Fluoride) are different) 3) The nature of bonding from the 2 atomic orbitals is determined by how they overlap Note that in ς bonds there is no exchange of phase when rotated about the x-axis whereas for π orbitals the phases interchange upon rotation.

Heteronuclear Diatomics The energies of orbitals in more electronegative elements are lower than others, for example in HF only the 2p of Fluorine can interact with the 1s of Hydrogen as the 2s is too low in energy. There are therefore defined frontier orbitals to define the highest energy orbital containing electrons (the highest occupied molecular orbital or HOMO) and the lowest energy orbital that is empty (the lowest unoccupied molecular orbital or LUMO). The molecular orbital diagram for HF is therefore thus;