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CHEM*2060 F2010
Atomic and Molecular Dimensions
Equilibrium Interatomic Distances
When two atoms approach each other, their positively
charged nuclei and negatively charged electronic clouds
interact. The total interaction is the sum of attraction (plus –
minus) and repulsion (plus – plus; minus – minus). If the curve
of the total energy has a minimum, the minimum occurs for a
certain interatomic distance r0.
Once two atoms approached each other at the distance r0 and
some energy was released, the two atoms will stay together
until they receive this energy back.
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If the amount of energy is very significant, the two atoms will
stay together for a long time. These atoms are defined as
bonded, and the attraction relation is called chemical bond.
If the amount of energy is not very significant, the two atoms
may stay together for a moment and fly apart, or may stay
longer due to some external restrictions. These atoms are
regarded as interacting weakly.
There is the whole spectrum of interatomic interactions: the
strongest is the chemical bond (covalent, ionic, etc), the
weakest is a so called van der Waals interaction.
Two hydrogen atoms can form a covalent bond at a distance
of r0 = 0.74 Å (bond energy is 432 kJ/mol) or van der Waals
contact at a distance of r'0 = 2.32 Å (energy is <0.1 kJ/mol)
The equilibrium interatomic distances can be predicted for
various combinations of two atoms through the concept of
atomic, ionic, and van der Waals radii.
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Atomic Radii
The equilibrium bond length can be partitioned into
contributions from each atom involved in the bond. The
contribution of an atom to a covalent bond is called
effective atomic radius (DeKock) or covalent radius of
the element (Shriver).
The covalent radius can be defined as one half the
internuclear distance between neighboring atoms of the
same element in a molecule.
Two atoms A with covalent radius rA and van der Waals radius
rvdW form the A2 molecule. The A–A bond distance is equal to
rAA = 2rA; the "length" of the molecule is (2rvdW + 2rA); the
"thickness" of the molecule is 2rvdW.
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For example, the separation of two protons in the H2
molecule is 0.74 Å. Therefore the covalent radius of H is
0.37 Å.
The covalent radius of another atom rB can be determined as
the A–B bond distance in the AB molecule rAB minus rA
As a rule, covalent radii:
• decrease from left to right along a given period;
• increase as we go down a given group
(not always for transition metals and f-elements)
Can you explain these trends?
(For explanations of these effects read DeKock 2-2 or Shriver 1,9a.)
Can you explain why the greatest changes are between H and
Li and between Li and Be (see next page)?
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(from DeKock)
Also, it is evident that, for the same element, covalent
radius will depend on the order of the bond, for example:
Bond
r C, Å
single
0.77
aromatic
0.70
double
0.67
triple
0.60
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CHEM*2060 F2010
(from Li)
Metallic Radii
The metallic radius of a metallic element is defined as
half the experimentally determined distance between the
centers of nearest-neighbor atoms in the solid
(crystalline) metal (Shriver, 1.9a).
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PART 3
Metal
Cov. rad., Å
Met. rad., Å
Li
1.34
1.57
Na
1.54
1.91
CHEM*2060 F2010
K
1.96
2.35
Rb
2.16
2.50
Cs
2.72
Can you explain why metallic radius is greater than the
covalent radius of the same element?
Ionic Radii
By analogy with atomic radius, the ionic radius of an ion
is the contribution of this ion to the ionic bond. The ionic
radius is related to the distance between the centers of
neighboring cations and anions in an ionic (crystalline)
solid.
The system of ionic radii depends on an arbitrary
decision on how to apportion the cation – anion distance
between the two ions. One common scheme is based on
assumption that the radius of O2– is 1.40 Å.
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Compare ionic and covalent radii (see p.5). What conclusions
can you make?
Van der Waals Radii
Van der Waals radius of an element is defined as a half
distance between two approaching identical atoms of the
element when their van der Waals interaction is minimal.
Van der Waals Radii vs Covalent Radii (Å)
H
1.16
0.37
Li
2.1
1.34
Na
2.3
1.54
K
2.7
1.96
He
1.4
Be
1.8
0.91
B
1.75
0.82
Al
2.0
1.26
C
1.71
0.77
Si
1.95
1.17
N
1.50
0.74
P
1.90
1.10
O
1.41
0.72
S
1.84
1.04
F
1.35
0.71
Cl
1.9
0.99
Br
1.9
1.14
I
2.15
1.33
Ne
1.5
Ar
1.8
Kr
2.0
Xe
2.2
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Explain the trends in the rows and groups.
Why the two sets of radii reveal similar trends?
What interatomic distance do you expect in liquid and solid Xe?
What interatomic distances do you expect to see between two I atoms
in I2 molecule and between two contacting I atoms of neighboring I2
molecules?
Summary on Atomic Size Measures
Type of
interaction
Contribution of the atom
to the interatomic distance
(radius type)
Main factor
responsible for
variations
covalent bond
(molecule)
effective atomic radius
covalent radius
the order of the
bond
ionic bond
(solid)
ionic radius
coordination
number
metallic bond
(solid)
metallic radius
coordination
number
van der Waals
interaction
van der Waals radius
various
It is expected that the length of the A–B interaction is
approximately the sum of the corresponding radii for
elements A and B
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Molecular Dimensions
For the purpose of modeling, the molecule is considered as a
shape formed by spherical atoms with the corresponding van
der Waals radii. Some molecules are rigid and so their shape
and dimensions are constant. Other molecules are flexible
and can adopt two or more conformations.
Molecular diameter is defined as the diameter of the
molecule assuming it to be spherical:
h = 1/3(hx + hy + hz)
where h is the mean molecular diameter and the hi are
the dimensions of the molecule in three perpendicular
directions.
The molecular (maximal) size hmax is the maximal linear
dimension of the molecular shape (max of hi).
Ellipsoid approximation is the most straightforward way to
roughly describe the size and shape of a molecule. Find the
maximal linear dimension of the molecule (hx = hmax). Than
find a minimal dimension in the plane ⊥ to the x axis (hz). The
last dimension is in the direction ⊥ to both x and z (hy).
Ellipsoid approximation may not be adequate for irregular
geometries such as a concave shape.
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Working example: molecular dimensions of para-dichlorobenzene
hmax = hx = 2rvdW(Cl) + 2rC + 2rCl +2rC-C(ar) = 3.8 + 1.54 + 1.98 + 2.78 ≈ 10 Å
hy ≈ 6.6 Å
hz = 3.8 Å
h ≈ 6.8 Å
!!! Useful notes:
The methyl group is usually in fast rotation and can be approximated
by a ball. Since it is known experimentally that the attached Cl creates
similar volume as CH3, the methyl group has approximate radius of 2Å.
In molecular size calculations, it is helpful to use the cosine formula:
c2 = a2 + b2 – 2ab cos γ
The translation length in a stretched normal hydrocarbon is about 2.5Å
(two 1.53Å bonds at the angle of 109.5o). For example, the translational
period in crystalline polyethylene is 2.54Å.
The distance between two C atoms in the peptide backbone is ~3.8Å.
The contribution of one nucleotide unit to the length of DNA is 3.3Å.
For big molecules the contribution of van der Waals radii to the overall
size becomes small.
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Relative size of some molecules: water, ethanol, glycine (the simplest
amino acid) and fullerene (C60). The size of the bar is 1 nm
How Can We See the Molecule?
There are various methods to measure the bond lengths
and overall size of the molecules. There are techniques
making it possible to literally "see" a molecule.
Most experimental data on bond lengths and molecular
geometries come from spectroscopy measurements
(especially rotational/vibrational spectroscopy) and
crystal diffraction experiments (especially single-crystal
X-ray diffraction). Spectroscopy techniques work best for
gaseous samples, while the diffraction analysis requires
crystals or at least crystalline materials.
Discuss the strengths and weaknesses of the two methods and how
they can complement each other.
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Relative size of the molecule of oxygen (2 atoms) and oxy-hemoglobin
(crystal structure data: one half of oxy-hemoglobin in vivo that
contains >9000 atoms; H-atoms omitted). The size of the bar is 1 nm
Relative size of the molecule of ethanol and alcohol dehydrogenase
(crystal structure data). The size of the bar is 10 nm
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CHEM*2060 F2010
Dynamic Light Scattering and Nanoparticle Tracking
Analysis help to determine the overall size of particles
down to 1 nm (DLS) or 10 nm (NTA). Both methods
analyze the Brownian motion of nanoparticles
suspended in a fluid and give size distribution of the
particles in a sample.
Scanning Tunneling Microscopy and Atomic Force
Microscopy are the two most common techniques used
to study individual molecular-size objects at atomic
resolution. In both techniques the signal comes from a tip
that moves over and in very close proximity to a surface
(2D scan). STM records the differences in electric
current between the tip and the surface as the tip moves.
AFM records the changes in the mechanical force
("contact force") as the tip moves across the surface.
Probing surface with an Au and CO-modified tips (Gross)
"2ML" means "two monolayers"
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STM and AFM images of pentacene. (A) Ball-and-stick model. (B) STM
image. (C),(D) AFM images obtained with a CO-modified tip (Gross)
STM images: "Atom", iron atoms on Cu(111); "CO Man", CO molecules
on Pt(111). From http://www.almaden.ibm.com/vis/stm/gallery.html
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Typical Bond Lengths
(Data from the International Tables for Crystallography and DeKock, Å)
Bond
Range
Average
Bond
Range
Average
Csp3 – Csp3
Csp3 – Csp2
Csp3 – Caryl
Csp3 – Csp1
Csp2 – Csp2
Csp2 – Caryl
Csp3 – Csp1
Caryl – Caryl
Caryl – Csp1
Csp1 – Csp1
Caryl ÷ Caryl
Csp2 = Csp2
Csp1 ≡ Csp1
1.47-1.59
1.48-1.53
1.49-1.53
1.44-1.47
1.41-1.48
1.45-1.49
1.43
1.49
1.43-1.44
1.38
1.36-1.44
1.29-1.39
1.17-1.19
1.53(2)
1.51(2)
1.51(1)
1.47(1)
1.46(1)
1.48(2)
1.43(1)
1.49(1)
1.44(1)
1.38(1)
1.38(1)
1.32(2)
1.18(1)
C–F
C – Cl
C – Br
C–I
C–P
C–S
C=S
1.32-1.43
1.71-1.85
1.88-1.97
2.10-2.16
1.79-1.86
1.63-1.86
1.67-1.72
C–H
N–H
O–H
1.06–1.10
1.01-1.03
0.97-1.02
Csp3 – N
Csp2 – N
Caryl – N
Caryl ÷ N
Csp2 = N
Csp1 ≡ N
1.45-1.55
1.32-1.42
1.35-1.47
1.33-1.36
1.28-1.33
1.14-1.16
N–N
N÷N
N=N
N ≡ N*
1.35-1.45
1.30-1.37
1.12-1.26
1.10
in N3
N–O
N=O
N–S
1.23-1.46
1.22-1.24
1.60-1.71
Csp3 – O
Csp2 – Oa
Caryl – O
Csp2 = O
1.41-1.45
1.39-1.49
1.36-1.40
1.19-1.26
O–O
O=O
O–S
O=S
O – Pb
1.46-1.50
1.21
in O2
1.57-1.58
1.42-1.50
1.56-1.62
1.08
1.02
0.97
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PART 3
Bond
Range
H–F
H – Cl
H – Br
H–I
0.92
1.27
1.41
1.61
F–F
Cl – Cl
Br – Br
I–I
I–I
S – Te
Cl – Te
I – Te
Si – Si
1.42
1.99
2.28
2.67
in HF
in HCl
in HBr
in HI
in F2
in Cl2
in Br2
in I2
2.92
in I3–
2.41-2.68
2.52
2.93
2.36
CHEM*2060 F2010
Bond
Range
Li – Li
Na – Na
K–K
2.67
3.08
3.92
M – Hc
M – CO
M – CH3
M – NCR
M – NO
1.58-1.78
1.77-2.19
1.97-2.35
1.87-2.49
1.63-1.82
M – Nd
M – Ne
M – Nf
M – Ng
M – NH3
1.94-2.23
1.89-2.47
1.91-2.24
1.95-2.66
1.97-2.25
Only one oxygen in the molecule.
b Phosphorus with four or three bonds.
c In terminal metal hydrides. M is a transition metal.
d In porphinates.
e In phthalocyanines.
f In pyrazole complexes.
g In pyridine complexes.
a
in Li2
in Na2
in K2
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!!! Some numbers to remember:
H-H
C-H, N-H, O-H
C-C
C-C-C
C=C, C=N
C≡C, C≡N
C÷C, C÷N, N÷N
M-N, M-O
0.74 Å
~1 Å
~1.5 Å
~2.5 Å
~1.3 Å
~1.2 Å
~1.4 Å
~2 Å
the shortest bond
(in hydrocarbons)
(aromatics)
(metal complexes)
Additional literature/data used:
L Gross et al, Science 2009, 325, 1110.
International Tables for Crystallography, 5th Ed (T Hahn, ed.), IUCr,
Springer, 2005, Volume C.
Protein Data Bank; http://www.rcsb.org/pdb/
W-K Li et al, Advanced Structural Inorganic Chemistry, IUCr, Oxford
University Press, 2008.
Reading:
DeKock: 2-2, 2-5, 7-3 (pp 430-4); Tables 4-5, 4-9, 4-10
Shriver: 1.9a, 2.13, 3.7, 3.10a
Atkins: 17.6; Tables 10.2, 19.3.