4.4 Cohesive Energy
77
apply, estimation through vapor pressure is often utilized. The temperature dependence of the saturated vapor pressure, which is the pressure of vapor coexisting with
a condensed phase, is thermodynamically given as
dp v
dT
=
s v − s s
v v − v s
≈
Δh
T v v
,
(4.11)
where thermodynamic quantities written by lower case letters are those per mole.
Ignoring the temperature dependence of the difference in enthalpy between the vapor
and crystal (Δh) while assuming the ideal gas behavior for the vapor, the integration
gives
ln p v (T ) = −
Δh
RT
+ const.
(4.12)
The formula is known as the Antoine equation. Thus, the information on the vapor
pressure gives the enthalpy of sublimation. Since the molecular shape is not kept
upon the change in its environments, the experimental estimates of cohesive energy,
in principle, always contains the energy assignable to the change in the molecular
geometry. For example, benzene molecules sit on centers of inversion in crystal,
resulting in the loss of its ideal sixfold symmetry. However, the symmetry recovers
in a usual gaseous state resembling the isolated state.
When the compound is ionic, the sublimation is practically impossible. For such
ionic crystals, the cohesive energy is estimated through the so-called Born-Harber
cycle [42, 43], which is based upon the first law of thermodynamics. For example,
consider the cohesive energy of the rocksalt, NaCl. The difference in energy between
crystalline NaCl and (mixed) gas of Na
+ and Cl
− can be decomposed as:
Decomposition to elements : NaCl (c) → Na (c) +
1
2
Cl 2 (g)
Sublimation : Na (c) → Na (g)
Atomization :
1
2
Cl 2 (g) → Cl (g)
Ionization : Na (g) → Na
+ (g) + e
−
Electron attachment : Cl (g) + e
−
→ Cl
− (g),
where (c) and (g) distinguish states of substance, crystalline or gaseous. The important
aspects here are that each reaction is practically achievable for measuring energies.
Notably, the energies related to two last reactions are known as ionization energy (or
potential) and electron affinity.
For both ionic and nonionic cases, experimental estimates of cohesive energy are
smaller than the ideal ones by the zero-point energy of lattice vibration, which is
treated in Chap. 5. The so-called harmonic approximation becomes appropriate in
most cases. Since the zero-point energy of a harmonic oscillator with the angular
77
apply, estimation through vapor pressure is often utilized. The temperature dependence of the saturated vapor pressure, which is the pressure of vapor coexisting with
a condensed phase, is thermodynamically given as
dp v
dT
=
s v − s s
v v − v s
≈
Δh
T v v
,
(4.11)
where thermodynamic quantities written by lower case letters are those per mole.
Ignoring the temperature dependence of the difference in enthalpy between the vapor
and crystal (Δh) while assuming the ideal gas behavior for the vapor, the integration
gives
ln p v (T ) = −
Δh
RT
+ const.
(4.12)
The formula is known as the Antoine equation. Thus, the information on the vapor
pressure gives the enthalpy of sublimation. Since the molecular shape is not kept
upon the change in its environments, the experimental estimates of cohesive energy,
in principle, always contains the energy assignable to the change in the molecular
geometry. For example, benzene molecules sit on centers of inversion in crystal,
resulting in the loss of its ideal sixfold symmetry. However, the symmetry recovers
in a usual gaseous state resembling the isolated state.
When the compound is ionic, the sublimation is practically impossible. For such
ionic crystals, the cohesive energy is estimated through the so-called Born-Harber
cycle [42, 43], which is based upon the first law of thermodynamics. For example,
consider the cohesive energy of the rocksalt, NaCl. The difference in energy between
crystalline NaCl and (mixed) gas of Na
+ and Cl
− can be decomposed as:
Decomposition to elements : NaCl (c) → Na (c) +
1
2
Cl 2 (g)
Sublimation : Na (c) → Na (g)
Atomization :
1
2
Cl 2 (g) → Cl (g)
Ionization : Na (g) → Na
+ (g) + e
−
Electron attachment : Cl (g) + e
−
→ Cl
− (g),
where (c) and (g) distinguish states of substance, crystalline or gaseous. The important
aspects here are that each reaction is practically achievable for measuring energies.
Notably, the energies related to two last reactions are known as ionization energy (or
potential) and electron affinity.
For both ionic and nonionic cases, experimental estimates of cohesive energy are
smaller than the ideal ones by the zero-point energy of lattice vibration, which is
treated in Chap. 5. The so-called harmonic approximation becomes appropriate in
most cases. Since the zero-point energy of a harmonic oscillator with the angular