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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
http://folk.uio.no/ravi/CMP2015
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
Lattice Energy-Madelung constants – Born Haber cycle
1
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Introduction
Cohesive energy
energy required to
break up crystal into
neutral free atoms.
Lattice energy (ionic
crystals) energy
required to break up
crystal into free ions.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycleKcal/mol = 0.0434 eV/molecule KJ/mol = 0.0104 eV/molecule
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Interactions between Ions6
Typically consider an ionic solid with many cations and many anions
All ions are interacting with each other: repulsion and attraction
Lattice energy of a solid – ΔE of ions in gas vs solid
High LE – strong interaction between ions, tightly bonded solid
Start with the CPE of 2 ions with charges z1 and z2:
Total PE of ionic solid is sum of CPE interactions between all ions
120
2
21
120
2112
r4
ezz
r4
ezezCPE
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Electronegativity and Chemical Bonds
1. Ionization enthalpy
The enthalpy change when one mole of electrons are removed from one mole of atoms or positive ions in gaseous state.
X(g) X+
(g) + e-
H 1st
I.E.
X+
(g) X2+
(g) + e-
H 2nd
I.E.
Ionization enthalpies are always positive.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
X(g) + e-
X-(g) H 1
stE.A.
X
(g) + e-
X2
(g) H 2nd
E.A.
Electronegativity and Chemical Bonds
2. Electron affinity
The enthalpy change when one mole of electrons are added to one mole of atoms or negative ions in gaseous state.
Electron affinities can be positive or negative.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
I.E. and E.A. only show e- releasing/attracting power of
free, isolated atoms
However, whether a bond is ionic or covalent depends on the ability of atoms to attract electrons in a chemical bond.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Electronegativity and Chemical Bonds
3. Electronegativity
The ability of an atom to attract electrons in a chemical bond.
Mulliken’s scale of electronegativity
Electronegativity(Mulliken) )(2
1EAIE
Nobel Laureate in Chemistry, 1966
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Pauling’s scale of electronegativity
Nobel Laureate in Chemistry, 1954
Nobel Laureate in Peace, 1962
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Pauling’s scale of electronegativity
calculated from bond enthalpies
cannot be measured directly
having no unit
always non-zero
the most electronegative element, F, is arbitrarily assigned a value of 4.00
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
IA IIA IIIA IVA VA VIA VIIA VIIIA
H
2.1
He
-
Li
1.0
Be
1.5
B
2.0
C
2.5
N
3.0
O
3.5
F
4.0
Ne
-
Na
0.9
Mg
1.2
Al
1.5
Si
1.8
P
2.1
S
2.5
Cl
3.0
Ar
-
K
0.8
Ca
1.0
Ga
1.6
Ge
1.8
As
2.0
Se
2.4
Br
2.8
Kr
-
Rb
0.8
Sr
1.0
In
1.7
Sn
1.8
Sb
2.0
Te
2.1
I
2.5
Xe
-
Cs
0.7
Ba
0.9
Tl
1.8
Pb
1.8
Bi
1.9
Po
2.0
At
2.2
Rn
-
Pauling’s scale of electronegativity
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
What trends do you notice about the EN values in the Periodic Table? Explain.
IA IIA IIIA IVA VA VIA VIIA VIIIA
H
2.1
He
-
Li
1.0
Be
1.5
B
2.0
C
2.5
N
3.0
O
3.5
F
4.0
Ne
-
Na
0.9
Mg
1.2
Al
1.5
Si
1.8
P
2.1
S
2.5
Cl
3.0
Ar
-
K
0.8
Ca
1.0
Ga
1.6
Ge
1.8
As
2.0
Se
2.4
Br
2.8
Kr
-
Rb
0.8
Sr
1.0
In
1.7
Sn
1.8
Sb
2.0
Te
2.1
I
2.5
Xe
-
Cs
0.7
Ba
0.9
Tl
1.8
Pb
1.8
Bi
1.9
Po
2.0
At
2.2
Rn
-
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
What trends do you notice about the EN values in the Periodic Table? Explain.
The EN values increase from left to right across a Period.
The atomic radius decreases from left to right across a Period.Thus,the nuclear attraction experienced by the bonding electrons increases accordingly.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
The EN values decrease down a Group.
The atomic radius increases down a Group, thus weakening the forces of attraction between the nucleus and the bonding electrons.
What trends do you notice about the EN values in the Periodic
Table? Explain.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Born-Mayer Equation:
U0 = (e2 / 4 0) * (N zA zB / d0) * A * (1 – (d* / d0))
U0 = 1390 (zA zB / d0) * A * (1 – (d* / d0)) in kJ/mol
Kapustinskii equation :
U0 = (1210 kJ Å / mol) * (n zA zB / d0) * (1 – (d* / d0))
Where:
e is the charge of the electron, 0 is the permittivity of a vacuum
N is Avogadro’s number
zA is the charge on ion “A”, zB is the charge on ion “B”
d0 is the distance between the cations and anions (in Å) = r+ + r-
A is a Madelung constant
d* = exponential scaling factor for repulsive term = 0.345 Å
n = the number of ions in the formula unit
Ionic Bonding
The equations that we will use to predict lattice energies for crystalline solids are the
Born-Mayer equation and the Kapustinskii equation, which are very similar to one
another. These equations are simple models that calculate the attraction and
repulsion for a given arrangement of ions.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
In this case the energy of coulombic forces (electrostatic attraction and repulsion) are:
Ecoul = (e2 / 4 0) * (zA zB / d) * [+2(1/1) - 2(1/2) + 2(1/3) - 2(1/4) + ....]
because for any given ion, the two adjacent ions are each a distance of d away, the next two
ions are 2d, then 3d, then 4d etc. The series in the square brackets can be summarized to
give the expression:
Ecoul = (e2 / 4 0) * (zA zB / d) * (2 ln 2)
where (2 ln 2) is a geometric factor that is adeqate for describing the 1-D nature of the infinite
alternating chain of cations and anions.
For an Infinite Chain of Alternating Cations and Anions:
Ionic Bonding
The origin of the equations for lattice energies.
The lattice energy U0 is composed of both coulombic (electrostatic) energies and an additional
close-range repulsion term - there is some repulsion even between cations and anions because of
the electrons on these ions. Let us first consider the coulombic energy term:
U0 = Ecoul + Erep
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
This means that the general equation of Coulombic energy for any 3-D ionic solids is:
Ecoul = (e2 / 4 0) * (zA zB / d) * A
Note that the value of Ecoul must be negative for a stable crystal lattice.
Ionic Bonding
For a 3-dimensional arrangement, the geometric factor will be different for each different
arrangement of ions. For example, in a NaCl-type structure:
Ecoul = (e2 / 4 0) * (zA zB / d) * [6(1/1) - 12(1/2) + 8(1/3) - 6(1/4) + 24(1/5) ....]
The geometric factor in the square brackets only works for the NaCl-type structure, but
people have calculated these series for a large number of different types of structures
and the value of the series for a given structural type is given by the Madelung constant,
A.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
The numerical values of Madelung constants for a variety of different structures are
listed in the following table. CN is the coordination number (cation,anion) and n is the
total number of ions in the empirical formula e.g. in fluorite (CaF2) there is one cation
and two anions so n = 1 + 2 = 3.
Ionic Bonding
lattice A CN stoich A / n
CsCl 1.763 (8,8) AB 0.882
NaCl 1.748 (6,6) AB 0.874
Zinc blende 1.638 (4,4) AB 0.819
wurtzite 1.641 (4,4) AB 0.821
fluorite 2.519 (8,4) AB2 0.840
rutile 2.408 (6,3) AB2 0.803
CdI2 2.355 (6,3) AB2 0.785
Al2O3 4.172 (6,4) A2B3 0.834
Notice that the value of A is fairly constant for each given stoichiometry and that the
value of A/n is very similar regardless of the type of lattice.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Ionic Bonding
This is the Born-Mayer equation, when the constants are evaluated
we get the form of the equation that we will use:
U0 = 1390 (zA zB / d0) * A * (1 - (d* / d0)) in kJ/mol
Note: d* is the exponential scaling factor for the repulsive term and
a value that we will use for this is 0.345 Å.
If only the point charge model for Coulombic energy is used to estimate the lattice energy (i.e. if U0 = Ecoul) the
calculated values are much higher than the experimentally measured lattice energies.
E.g. for NaCl (rNa+ = 0.97Å, rCl- = 1.81Å):
U0 = 1390 (zA zB / d0) * A = 1390 ((1)(-1)/2.78) * (1.748) kJ/mol = - 874 kJ/mol
But the experimental energy is -788 kJ/mol. The difference in energy is caused by the repulsion between the
electron clouds on each ion as they are forced close together. A correction factor, Erep, was derived to
account for this.
Erep = - (e2 / 4 0) * (zA zB d*/ d2) * A and since
Ecoul = (e2 / 4 0) * (zA zB / d) * A the total is given by
U0 = (e2 / 4 0) * (zA zB / d0) * A * (1-(d*/d0))
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Ionic Bonding
Using the Born-Mayer equation, for our example with NaCl.
U0 = 1390 (zA zB / d0) * A * (1 - (d* / d0))
= 1390 ((1)(-1)/2.78) * (1.748) * (1-(0.345/2.78) kJ/mol = - 765 kJ/mol
Which is much closer to the experimental energy of -788 kJ/mol.
Kapustinskii observed that A/n is relatively constant but increases slightly with
coordination number. Because coordination number also increases with d, the value of
A/nd should also be relatively constant. From these (and a few other) assumptions he
derived an equation that does not involve the Madelung constant:
Kapustinskii equation :
U0 = (1210 kJ Å / mol) * (n zA zB / d0) * (1 – (d* / d0))
One advantage of the Kapustinskii equation is that the type of crystal lattice is not
important. This means that the equation can be used to determine ionic radii for non-
spherical ions (e.g. BF4-, NO3-, OH-, SnCl6-2 etc.) from experimental lattice energies. The
self-consistent set of radii obtained in this way are called thermochemical radii.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Interactions between Ions24
Consider a line of alternating cationsand anions:
CPE of an ion in center:
A = 2 ln 2
CPE is negative, net attraction between the ions
Now expand the model to 3D:
Coefficient A – Madelung constant –related to arrangement of ions
E A z2NAe
2
40d
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Interactions between Ions25
As ions are separated, the attraction decreases
If ions are too close, past the point of contact, they repel each other
There is an ideal separation between ions:
Born-Meyer equation
d* = 34.5 x 10-12 m
PEmin NA z1z2 e
2
40d1d*
d
A
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Ionic Crystals
ions: closed outermost shells
~ spherical charge
distribution
Cohesive/Binding energy
= 7.9+3.615.14 = 6.4 eV
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Electrostatic (Madelung) EnergyInteractions involving ith
ion:i i j
j i
U U
2/
2
. .R
i j
i j
qn ne
RU
qotherwise
p R
tot iU NUFor N pairs of
ions:
2/R q
N z eR
z﹦number of n.n.
ρ ~ .1 R0
j i i jp
﹦Madelung constant
At equilibrium: 0totdU
dR
2/
2
Rz qN e
R
→ 0
2/2
0
R qR e
z
2
0
0 0
1tot
N qU
R R
2
0
N qMadelung Energy
R
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Evaluation of Madelung Constant
App. B: Ewald’s method
j i i jp
1 1 12 1
2 3 4
2ln2
KCl
i fixed
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Kcal/mol = 0.0434 eV/molecule
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Neighbors Signs Numbers Distance
1st - 6 r
2nd + 12
3rd - 8
4th + 6 2r
31
31
Madelung Constant
r2
r3
Arrangement of ions in sodium chloride structures.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle3232
Total Coulombic potential:
r
e
r
e
r
e
r
e
r
e
r
eU
oA
0
2
0
2
2
0
2
0
2
0
2
4
748.1
4
...)2(4
634
824
124
6
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
33
33
Crystal Potential
The crystal potential is the summation of repulsive
energy and the Coulombic attraction
r
e
r
B
rUrUrV
on
AR
4
)()()(
2
Attractive energy EA
Net energy EN
Repulsive energy ER
Interatomic separation r
Ed
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Cry
sta
l P
ote
ntial V
(r)
distance r
Coulombic attraction UA
Exclusion repulsive force UR
V(r)
equilibrium position
r = ro
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Equilibrium position
The position at which the energy is minimum.
Balance between the repulsive force and attractive
force.
To evaluate, the first order derivative of V with
respect to r is zero, i.e.
02
2
1
o
norr r
e
r
nB
dr
dV
o
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
12
no
e
nBr
The equilibrium position is then
given by
And the minimum energy is
)1
1(4
)(2
minnr
erVV
oo
o
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
37
Example
Parameters for Sodium Chloride:
– B = 1.209x10-96 eV m9
– n = 9
– =1.7576
– Ionization energy for sodium = 5.14eV, and electron
affinity of chlorine = 3.61 eV
Find the cohesive energy Ecoh
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle3838
Definitions
Ionization Energy
A loss in energy for the ionization
+
Sodium ion
Na+
Sodium atom
Eion
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Electron affinity
Capture
electron
-
Chorine atom ClChorine ion Cl-
Energy released by capturing electron
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
NaCl molecule
dissociation
Sodium
atom
Chlorine
atom
Cohesive energy is the energy required to
dissociate the molecule into atoms..
Cohesive energy
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle4141
Equilibrium position of Sodium Chloride Crystal
12
no
e
nBr
819
96
10609.17476.1
10209.19
xx
xx
Did I make mistake? It must
be e2
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
There is no mistake. Sine the unit of B is eV m9. It must be
converted into Joule by multiplying the electron charge e. The
electron charge cancels out one of the e factor in the
denominator.
Effectively, it can express B in eV m9, while eliminating one e
in the denominator.
The calculation result for ro becomes
r0 = 2.887x10-10 m
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle4343
The energy required to dissociate a sodium
chloride molecule into separate ions:Ed
ClNaENaCl d
)1
1(4
)(2
nr
erVE
oo
od
9
8
1085.81416.44
10602.1747565.112
19
xxxx
xx
Why e instead of e2
?
= 7.96 eV
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
44
Ionization energy of Na
eNaENa ion
Electron affinity of Cl
affECleCl
affion EEClNaClNa
Eion = 5.14 eV
Eaff = 3.61 eV
recall
dENaClClNa
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle45
45
affiond EEENaClClNa
cohENaClClNaClNa
where
Ecoh = Ed -Eion + Eaff
In our present case Ecoh = 7.96 -5.14 +3.61 =
6.43 eV
Cohesive energy per ion = 6.43/2 = 3.21eV
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Born-Haber Cycles
Calculating lattice enthalpies.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Lattice enthalpy terms
∆Hө L.E = Lattice enthalpy. The standard enthalpy change when one mole of
crystalline substance is formed from its constituent gaseous ions.
Na+(g) + Cl-(g) → NaCl(S)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Born-Haber cycles
Define and apply the terms enthalpy of formation,
ionisation enthalpy, enthalpy of atomisation of an
element and of a compound, bond dissociation
enthalpy, electron affinity, lattice enthalpy (defined
as either lattice dissociation or lattice formation),
enthalpy of hydration and enthalpy of solution.
Construct Born–Haber cycles to calculate lattice
enthalpies from experimental data.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
For an ionic compound the lattice
enthalpy is the enthalpy change
when one mole of solid in its
standard state is formed from its
ions in the gaseous state.
The lattice enthalpy cannot be measured
directly and so we make use of other
known enthalpies and link them together
with an enthalpy cycle.
This enthalpy cycle is the Born-Haber cycle.
What do we mean by lattice enthalpy?
- -
-
--
-
-
-
For an ionic compound the lattice
enthalpy is the enthalpy change
when one mole of solid in its
standard state is formed from its
ions in the gaseous state.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
1 Sublimation of Sodium
Na(s) + 1/2 Cl2(g)
Na(g) + 1/2 Cl2(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
kJmol-1
-400
-300
-200
-100
H = +107kJmol-1θS
Born-Haber Cycle for Sodium Chloride
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
2 Bond Dissociation of Chlorine
Na(s) + 1/2 Cl2(g)
Na(g) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
H = +121kJmol-1θD
½
Born-Haber Cycle for Sodium ChloridekJmol-1
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
e-
e-
e-
e-
e-
Na(s) + 1/2 Cl2(g)
Na(g) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na+(g) + Cl(g)
H = +502kJmol-1θI
Born-Haber Cycle for Sodium Chloride
3 First Ionisation of Sodium
kJmol-1
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
4 Electron Affinity of Chlorine
e
-
Na(g) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(s) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na+(g) + Cl(g)
Na+(g) + Cl-(g)
H = -355kJmol-1θE
Born-Haber Cycle for Sodium ChloridekJmol-1
-
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Na(s) + 1/2 Cl2(g)
Na(g) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
NaCl(s)
Born-Haber Cycle for Sodium Chloride
H = -411kJmol-1θF
Na+(g) + Cl(g)
kJmol-1
- -
-
--
5 Formation of Sodium Chloride
Na+(g) + Cl-(g)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Na(s) + 1/2 Cl2(g)
Na(g) + 1/2 Cl2(g)
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
Na(g) + Cl(g)
0
+100
+200
+300
+400
+500
+600
+700
+800
-400
-300
-200
-100
NaCl(s)
Lattice Enthalpy for
Sodium Chloride
Na+(g) + Cl-(g)
H = -786 kJmol-1θL
Na+(g) + Cl(g)
Born-Haber Cycle for Sodium Chloride
- -
-
--
kJmol-1
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Born-Haber cycles
Explain and use the term: lattice enthalpy.
Use the lattice enthalpy of a simple ionic solid and
relevant energy terms to construct Born–Haber cycles.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Cl¯ Br¯ F¯ O2-
Na+ -780 -742 -918 -2478
K+ -711 -679 -817 -2232
Rb+ -685 -656 -783
Mg2+ -2256 -3791
Ca2+ -2259
Lattice Enthalpy Values (kJ mol-1)
Smaller ions will have a greater attraction for each other
because of their higher charge density. They will have larger
Lattice Enthalpies and larger melting points because of the
extra energy which must be put in to separate the oppositely
charged ions.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Cl¯ Br¯ F¯ O2-
Na+ -780 -742 -918 -2478
K+ -711 -679 -817 -2232
Rb+ -685 -656 -783
Mg2+ -2256 -3791
Ca2+ -2259
Lattice Enthalpy Values
Smaller ions will have a greater attraction for each other because of their
higher charge density. They will have larger Lattice Enthalpies and larger
melting points because of the extra energy which must be put in to separate
the oppositely charged ions.
Cl¯Na+ Cl¯
The sodium ion has the same charge as a potassium ion but is smaller. It has a higher
charge density so will have a more effective attraction for the chloride ion. More energy
will be released when they come together.
K+
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
Calculating Lattice Enthalpy
SPECIAL POINTS
One CANNOT MEASURE LATTICE ENTHALPY DIRECTLY
it is CALCULATED USING A BORN-HABER CYCLE
greater charge
densities of ions = greater attraction
= larger lattice enthalpy
Effects
Melting point the higher the lattice enthalpy,
the higher the melting point of an ionic compound
Solubility solubility of ionic compounds is affected by the
relative
values of Lattice and Hydration Enthalpies
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
NaCl2(s)
Na(s) Na(g) Na+2(g)
Cl2(g) 2 Cl(g) 2Cl-(g)
H°eaH°d
(H°ie1 + H°ie2)
H°sub
H°f
Lattice Energy, Uo
Ionic BondingIf we can predict the lattice energy, a Born-Haber cycle analysis can tell us why
certain compounds do not form. E.g. NaCl2
H°f = H°sub + H°ie1 + H°ie2 + H°d + H°ea + Uo
H°f = 109 + 496 + 4562 + 242 + 2*(-349) + -2180
H°f = +2531 kJ/mol
This shows us that the formation of NaCl2 would be highly endothermic and very
unfavourable. Being able to predict lattice energies can help us to solve many problems so we
must learn some simple ways to do this.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
1
6
5
4
3
2
Mg(s) + Cl2(g)
MgCl2(s)
Mg(g) + Cl2(g)
Mg(g) + 2Cl(g) Mg2+(g) + 2Cl–
(g)
7
Mg+(g) + 2Cl(g)
Mg2+(g) + 2Cl(g)
Enthalpy of formation of MgCl2
Mg(s) + Cl2(g) ——> MgCl2(s)
Enthalpy of sublimation of magnesium
Mg(s) ——> Mg(g)
Enthalpy of atomisation of chlorine
½Cl2(g) ——> Cl(g) x2
Ist Ionisation Energy of magnesium
Mg(g) ——> Mg+(g) + e¯
2nd Ionisation Energy of magnesium
Mg+(g) ——> Mg2+(g) + e¯
Electron Affinity of chlorine
Cl(g) + e¯ ——> Cl¯(g) x2
Lattice Enthalpy of MgCl2
Mg2+(g) + 2Cl¯(g) ——> MgCl2(s)
1
2
3
4
5
7
6
Born-Haber Cycle - MgCl2
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
H
CaO (s)
Ca2+ (g) + O2- (g)
H lattice energy
of formation
Ca (s) + ½ O2 (g)
H formation
H atomisation(s)
Ca (g) + O (g)
Ca2+ (g) + 2 e- + O (g)
H ionisation energy/ies
H electron affinity/ies
CaO
193
248
590
1150
–142
+844
?
– 635 = 193 + 248 + 590 + 1150 – 142 + 844 + Hlattice
Hlattice = – 635 – 193 – 248 – 590 – 1150 + 142 – 844
= – 3518 kJ mol-1
–635
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2016 July: Lattice Energy-Madelung constants – Born Haber cycle
63
(1,0,0)
(0,1,0)