Download Atomic Structure - Physics of Biological Systems, Notes | ME 498 and more Study notes Mechanical Engineering in PDF only on Docsity!
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
ATOMIC STRUCTURE The purpose of these first 2 lectures is to revisit chemistry concepts that are important to Biophysics.^ •^ Electron Orbitals and the Periodic table^ •^ Molecular Bonding from the perspectives of valence bond theory and molecular orbital theory^ •^ Intermolecular Forces Electron orbitals and the periodic table
In the simplistic planetary model (Bohr) of the atom, electrons orbit thenucleus, but if this actually happened, they should lose energy in a dyingburst of electromagnetic radiation and crash into the nucleus. This is becausethe negatively charged electron is inherently attracted to the positivelycharged nucleus, and crashing into it represents a lower energy, and thereforemore favorable state. But they do not crash. The paradox is removed by the
quantum^
model of the atom.
In quantum theory, the energy of an electron is only allowed to be gained orlost in discrete chunks called
quanta. No 'burst' of energy is allowed, only
discrete jumps between energy states. This makes the allowed energy levelsin an atom like the discrete steps of a staircase, rather than a continuous
gentle slope. Furthermore, transitions are only allowed to
unoccupied
energy states (this is called the Pauli
exclusion principle
). Hence, energies higher than the lowest energy state are stable, because there is nowhere unoccupied further down the staircase of energy for the electrons to fall to.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
However, this still leaves the 'ground state' (the lowest energy state). Why is this stable? Equivalently, why is theground state not collapsing on the nucleus? To account for this, we invoke Heisenberg's
uncertainty principle
(important maxim of quantum theory), which
states that an electron's position (say x) and momentum (correspondingly p
) cannot be simultaneously known withx
certainty. One way we can tell where an electron is in space is by hitting it with light waves. However, the moreprecisely we need to know the position of the electron, the
smaller^ the wavelength of light we will need: a dinghy
will leave an obvious, observable shadow behind short-wavelength water waves, but a long-wavelength tsunamiwill not even notice it's there. Mathematically, the uncertainty principle can be expressed as Δx^ Δp≥x^
½ h/2π,^
where^ Δx = { ^ −^
1/2^ }and expresses the average of A. Similarly for p
.x
Also,^ λ^ = h
⁄^ p, or equivalently, E = m c
2 = h^ ν^ (de Broglie's and Planck's relationships), where
λ^ is wavelength,
ν
frequency, E energy, p momentum, h Planck's constant, m mass, c speed of light) So, short wavelengths carry more energy: X-rays (short wavelength) are more energetic than radio waves (longwavelength). These energetic light waves will kick the electron sufficiently to change its movement, so althoughwe know exactly where the electron is, we will have no idea in which direction it is now traveling. Mathematically, as
Δx→^ 0,^ Δ
px→^ ∞^ and
^ →^ ∞x
From the above, we can see that if an electron tries to get nearer the nucleus, its position will become
more^ certain.
Hence its velocity must become
less^ certain, and therefore higher on average. This means its energy of motion is
higher, and therefore there is no net energy advantage in crashing into the nucleus, because the increase in kineticenergy would exceed any loss in potential energy. Hence the ground state is also stable.
_ME 498 Physics of Biological Systems –Spring 2007_
Lectures 1-
The principal quantum number (n) corresponds to the appropriate
radial^ function
R(r)^ (Laguerre polynomials) and n
describes the order of the^ •^ Largest energy differences ('shells').
o^ For the hydrogen atom, there are n
2 wave functions with the same energy (degenerate)
E= - 2^ πn^
2 4 μq/(he^ e
2 2 n)
-^ Size of orbital. •^ Numbered 1, 2, 3, 4, 5,
etc^ (although the shells it describes are often lettered K, L, M,
etc. ).
-^ Low numbers are closest to the nucleus, and have the lowest energies. •^ Higher values of n indicate larger orbitals The angular momentum quantum number (
l ) corresponds to the appropriate
polar^ function
Φ^ (φ)^ (Legendre l
polynomials) and describes the:^ •^ Small energy differences ('subshells') due to angular momentum of the electron^ •^ Shape of orbital (number of lobes).^ •^ Given letters inspired from spectroscopy: s (sharp), p (principal), d (diffuse), f (fine), g, h, i, etc.^ •^ The blocks of the periodic table are named after the azimuthal quantum number of the orbitals being filled(alkali metals are the s-block, the nonmetals are in the p-block, the transition metals are the d-block, thefootnote is the f-block).^ •^ The values of
l^ run from 0 to n
−^ 1, where n is the principal quantum number of the shell in question.
An s orbital is just shorthand for
l^ = 0, a p orbital is shorthand for
l^ = 1, a d orbital
l^ = 2, an f orbital
l^ = 3,
etc. • This also means that (for example) the n = 3 shell can only contain subshells with l = 0, 1 or 2,
i.e.^ only s, p
and d orbitals.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
http://www.orbitals.com/orb/index.html NOTATION: n=1^ Æ^ 1s
n=2, l=
Æ2p^
n=3, l=^
Æ^ 3d^
n=4, l=^
Æ^ 4f
The orbitals in the same row differ in their angular momentum quantum numbers: s orbitals are spherical and havejust one lobe. p orbitals have two lobes, and a 'node' between the lobes where the electrons do not spend muchtime. Consequently, 2p orbitals have higher energy than 2s orbitals, because the electron spends a little longerdistant from the nucleus. d and f orbitals have increasing numbers of lobes. In fact, d and f orbitals can have evenmore bizarre shapes than this…
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
The magnetic quantum number (m
) corresponds to the appropriate l
azimuthal
function^ Θ
(θ)^ (sines and cosines) m
and describes:
√^6 √^6 √^6 √^6 √ 6
- 2 + 1^0 - 1^ -2 Example of
-^ No energy differences at all (the orbitals are
degenerate
-^ Orientation of the orbitals in space. •^ Named after the directions they point in (sort of) x, y, z,
etc.
-^ Can also be given numbers ranging from 0, ±1, ±2, ±3 … ±
l****. Hence
the three sorts of p orbitals, p
, pand pxy^
have mz l^ of -1, 0 and +
respectively. • This restriction is called Space Quantization • Consequently, there's only one sort of s orbital (because l = 0), but:^ o^ 3 sorts of p^ o^ 5 sorts of d^ o^ 7 sorts of f^ o^ 9 sorts of g …
Space Quantization
: The 5 (that is 2
l^ +1) allowed orientations of the angularmomentum with
l =2. The length of each vector is1/2{ l (^ l^ +1)}
=^ √^6 m l
http://winter.group.shef.ac.uk/orbitron/ The 2p, 2px
and 2porbitals all have the same energy, but different orientations in space.y z^
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
Summary of Orbitals:^ •^ n=1 shell
o^ One s-orbital: 1s • n=2 shell o^ One s-orbital: 2s o^ Three p-orbitals: 2p
2p2px y^ z
-^ n=3 shell^ o^
One s-orbital: 3s o Three p-orbitals: 3p
, 3p, 3pxyz o^ Five d-orbitals: 3d
, 3d, 3dxyxz
(^22) , 3d, 3dyzx−y (^2) z
-^ n=4 shell^ o^
One s-orbital: 4s o Three p-orbitals: 4p
, 4p, 4pxyz o^ Five d-orbitals: 4d
, 4d, 4dxyxz
(^22) , 4d, 4dyzx−y (^2) z
o^ Seven f-orbitals: 4f
32 , 4f4fzxz
22 , 4fz(x−y)xy 2 , 4f, 4fxyz
22 , 4fx(x−3y)
(^22) y(3x−y)
The final quantum number is the spin quantum number (m
). Each orbital (r
e.g.^ a 2p) can hold two electrons.x
The Pauli
exclusion principle
states that no two electrons in a single atom can have the same quantum numbers,
hence they must differ in their spin quantum number, which takes the values +½ or -½. Two electrons in the sameorbital have paired (opposite) spins. The number of electrons a set of degenerate orbitals can contain is thereforejust 2 for an s orbital, 6 for a set of p orbitals (3p
3p3pcan hold 2 electrons each), 10 for a set of five d orbitals,x y^ z^
14 for a set of f,
etc.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
Properties of elements exhibit periodic behavior as we traverse the table: elements in a vertical group, like thehalogens (F, Cl, Br, I) and alkali metals (Li, Na, K, Rb) have similar properties. The reason for this is that theiroutermost orbitals have a similar electronic configuration: the alkali metals all have s
1 and the halogens all have
(^25) spin their outermost shell. Elements with similar electronic configurations have similar properties. As we move across the horizontal periods, we are filling up electron orbitals from the lowest energy (nearest tonucleus) to the highest (furthest out). The electrons actually fill in the orderbelow: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p 8s … So, the only rules for filling up energy levels are that they
fill from the
lowest energy upwards
, and that^
electrons only pair if there's no orbital of
the same energy available
(Hund's rule). Looking at the periodic table, and the filling order, we can easily see that the following orbitals are filled bythe elements listed:^ •^ 1s : H He.^ •^ 2s : Li Be.^ •^ 2p : B C N O F Ne.^ •^ 3s : Na Mg.^ •^ 3p : Al Si P S Cl Ar.^ •^ 4s : K Ca.^ •^ 3d : Sc - Zn. And so on: f-orbitals are filled in the f-block lanthanides/actinides (the footnote at the bottom of most periodictables).
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
Molecular bonding The previous section described how electrons fill up atomic orbitals, and how this relates to periodicity. Whathappens when there is more than one nucleus present? In short, how do electrons arrange themselves in molecules,and how do they form chemical bonds? The short answer is that it's extremely difficult to work out! However, many models have been proposed, each with their advantages and disadvantages. The simplest is thevalence bond theory of the chemical bond. In this theory, we only consider the outermost s and p
valency^ electrons
as being responsible for most of the chemistry of an atom (or at least the ones biochemists are interested in).Having a completely full or empty
octet^ of these outermost electrons is a favorable low-energy state. Noble gases
already have this configuration, and are therefore unreactive, but other elements 'try' to completely fill or emptytheir outermost octet of electrons by losing, gaining or sharing them. In iodine chloride, we can see that both Cl and I are one electron short of an octet. Hence they can
share^ electrons
and form a
covalent^
bond between the two molecules, allowing both to have 8 electrons in their outermost orbitals, I-Cl. Note that both I and Cl have three pairs each of electrons
not^ involved in the bond itself. These are termed
lone pairs
, and contribute to the shape of the molecule as we will see. The two electrons that
are^ shared form a
single^ covalent bond. Other molecules may share more than one pair of electrons between two atoms: in oxygenmolecules (O
), a double bond is formed (O=O), sharing four electrons in total, and in nitrogen (N 2
) six electrons 2
are shared to form a triple bond N
≡N).
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
The bond between the F
−^ +^ and Cs
ions is called an
ionic^ bond, and is typical in compounds formed between a metal
and a nonmetal. Metallic^ bonds are formed between two metals, and indeed, between atoms of a single metallic element. They aresomewhat like covalent bonds (as in iodine fluoride), in that the electrons are shared between the metal atoms, butrather than a 1:1 sharing,
all^ the valency electrons are shared between
all^ the atoms, forming a sort of 'electron gas'
between the metal atoms, and accounting for the electrical conductivity of metals. These sorts of bond are not veryimportant in biophysics… Covalent^
bonds are formed when electrons are shared between two atoms, as in the iodine fluoride above. If two nonmetals react to form a compound, they usually form a covalent bond, because nonmetals have similarelectronegativities. However, a 'perfect' covalent bond is only formed between the atoms of the same element,
e.g.
in Cl. All other covalent bonds have a slight ionic character, called polarity, and such bonds are called^2
polar
covalent. In IF, the chlorine is rather more electronegative than the iodine, and hence the electrons will spendslightly longer around the chlorine than the iodine. This gives the chlorine a slight negative charge (
δ-), and the
iodine a slight positive charge (
δ+). Various weak bonds can be formed between such polar molecules. The most
important in biochemistry is the hydrogen bond (covered in detail later). An important consequence of polarity is the concept of
oxidation number
. The oxidation number of an atom in a
molecule is basically the charge that the atom would have if the molecule were broken up into the ions that theatoms might 'like' to be. For example, the O-H bond in water is very polar: H is slightly positive (and valency 1),and O slightly negative (and valency 2). Hence, the oxidation numbers are -2 for the oxygen, and +1 for thehydrogens, because if we could break H
O into the ions it might 'like' to be, we'd get H 2
+^2 −^ +^ OH. In fact, in almost
all biochemical compounds, H has an oxidation number of +1, and O an oxidation number of -2. Other elementshave more variable oxidation numbers, but a few simple rules will get you by in most cases:
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
-^ The sum of the oxidation numbers in the molecule must add to the charge on the molecule or ion (zero if it isuncharged). •^ The electronegativity series for the common biochemical elements goes (most negative) F > O > Cl > N > S> C > H > P > Fe > Mg > Ca > Na > K > Fr (least negative). •^ An element in combination with itself has oxidation number 0, hence Cl in Cl
has oxidation number 0, as 2
there's no polarity in the Cl-Cl bond. • Oxygen almost always has oxidation number
−2, except in
e.g.^ OF(where it's +2, because F is more^2
electronegative), and in peroxides H
O(where it's 22
−1, because the oxygen is half in combination with
itself). • Hydrogen almost always has oxidation number +1, except in
e.g.^ H(0) and NaH (^2
-^ Transition element compounds are usually named by their oxidation numbers in Roman numerals. Iron (III)chloride is FeCl
, iron (II) oxide is FeO. 3 From these rules, you can work out that the oxidation number of N in NH
is^ −3, that of S in SO 3
−^2 is +6, N in 4
−^ NOis +5, Fe in Fe 3
+2^ is +2,^ etc. The valence bond theory of covalent bonds is somewhat simplistic, and more quantum mechanical treatments ofmolecular bonding try to apply the same sort of reasoning to molecular bonds as to atomic orbitals. In the mostrefined treatment, molecular orbital theory, the valence electrons (in fact, all the electrons) are shared in
molecular
orbitals. However, somewhere between valence bond theory and full blown molecular orbital theory, we comeacross the concept of
hybridisation
, which is used to account for the shapes of covalent molecules. The valence
orbitals (outermost s and p orbitals) are hybridised (mathematically mixed) before bonding, converting some of thedissimilar s and p orbitals into identical hybrid sp
n^ orbitals.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
The overlap and pairing of the s orbitals of four hydrogen atoms with the sp
3 hybrids on a carbon forms four
covalent bonds in the methane molecule CH
Overlap four s orbitals from four hydrogens (blue) with four sp3 hybrids on carbon leads to formation of bonds,each containing one electron from the carbon and one from the hydrogen: these are represented by up and downpointing arrows, showing the pairing of the electron spins. The hybridisation also accounts for the shape of molecules like methane (tetrahedral), ammonia (trigonal pyramid),water (V-shaped), and hydrogen fluoride (linear). Note that the orbitals not involved in bonding to hydrogen arestill hybridised, but end up as lone pairs of electrons (symbolised by the two dots in the diagram to the right).
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
(^2) sp hybrids
are formed when only one s and two p orbitals are involved. This leaves one remaining p orbital, which may beinvolved in forming a double bond. The furthest these orbitalscan get from one another is a trigonal bipyramid, with the sp
2
hybrids arranged at 120° to each other in a plane. This ischaracteristic of molecules with double bonds.
(^2) sp hybrids. The top three images show the three sp
2 hybrids.
These particular
(^2) sp hybrids are combinations of 2
s^ and two
2 p^ functions. The bottom shows the relative positions of thesethree hybrids superimposed.
It is appropriate to invoke
(^2) sp
hybrid^ orbitals
in^ molecules
such^ as^ BF
or^ H 3 2C=CH
2
(ethene) where the F-B-F and H-C-C angles are ~ 120°
(^2) Three sphybrids in lilac, with two lobes of the remaining p orbital in orange.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
Ethane (top) has only sigma bonds, ethene (middle) andethyne (bottom) have pi bonds too.
(There are also
δ^ and^ φ^ bonds that describe the various In class, we will present discuss examples (Pt-catalysis) where theseways d and f orbitals can overlap.)
π^ bonds are important. In the diagrams to the left,
σ^ bonds are shown as simple lines. However, p orbitals can overlap
sideways
too: when this happens (as in ethene and ethyne), twolobes of electron density are formed between theatoms. This is termed a
pi^ bond (
π). From the diagram,
you can see that the double bond in ethene is composedof one^ σ^ plus one
π^ bond, and the triple bond in ethyne is one^ σ^ plus two
π. The formation of bonds involves the overlap of hybridorbitals with the orbitals of other atoms, as we sawwith methane. However, two sorts of bond can resultfrom different sorts of overlap. When s and/or hybridorbitals overlap 'end-on',
sigma^ bonds (
σ) are formed:
these have a single area of electron density between thenuclei of the two atoms whose orbitals are overlapping.
ME 498 Physics of Biological Systems –Spring 2007
Lectures 1-
Rotation is possible about a
σ^ bond, but not (at room temperatures) about a
π^ bond, which causes geometric
isomery. 2s^ σ^ and^ σ
*** bonds**. Bonding and anti-bonding interaction of nitrogen 2s orbitals on two nitrogen atoms as they approach. The two dots represent the N nuclei. The rather strange looking inner structures are the radial nodes ofthe two 2 s
orbitals. (http://winter.group.shef.ac.uk/orbitron/MOs/N2/2s2s-sigma /index.html)
2p^ π^ bond
Bonding interaction of nitrogen 2
p orbitals on twox^
itrogen atoms as they approach.n (http://winter.group.shef.ac.uk/orbitron/MOs/N2/2px2px- pi/index.html
