Advances in Interfacial Adsorption Thermodynamics:
Metastable-Equilibrium Adsorption (MEA) Theory

529 Fig. 8. Comparison between calculated and measured isotherms under different
C
p

conditions in Cd–goethite system. Lines are calculated from the
C
p
effect isotherm equation
0.435
1.778
eq
C  . Points are adsorption data from Figure 1b.
According to MEA theory, for the ideal reversible adsorption reactions, changes in
C
p
have
no influence on the reversibility of MEA states, and it should have no
C
p
effect in such
systems when experimental artifacts are excluded.
11, 18
For partially irreversible adsorption
reactions, changes in
C

2

was highly reversible (there was no apparent hysteresis between the adsorption and
desorption isotherms, see Figure 10). This contrast adsorption behavior between the two
forms of manganese oxides could be explained from the different microscopic structures

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

530
between δ-MnO
2
and manganite, as well as the linkage modes of adsorbed Zn(II) on δ-
MnO
2
and manganite.
19
Fig. 9. Adsorption (closed symbols) and desorption (open symbols) isotherms of Zn(II) on
manganite. EXAFS samples were indicated by arrows. Fig. 10. Adsorption (■) and desorption (□) isotherms of Zn(II) on δ-MnO
2
. EXAFS samples
were symboled with blank triangles (Δ).
Manganite had a structure with rows of edge-sharing Mn(II)O
6
octahedra linked to adjacent

Advances in Interfacial Adsorption Thermodynamics:
Metastable-Equilibrium Adsorption (MEA) Theory

531

Fig. 11. Corner-sharing linkage (a) and interlayer structures of Zn(II) adsorbed on δ-MnO
2

(b). (a) R
Zn–O
= 2.07 Å, R
Mn–O
= 1.92 Å, R
Zn–Mn
= 3.52 Å. (b) Squares were vacant sites,
illustration diagram adapted from Wadsley,
27
Post and Appleman,
28
and Manceau et al
25
Fig. 12. Two types of linkage between adsorbed Zn(II) (octahedron and tetrahedron) and
MnO
6
octahedra on the γ-MnOOH surfaces. (a) Double-corner linkage mode; (b) edge-
linkage mode.
Extended X-ray absorption fine structure (EXAFS) analysis showed that Zn(II) was adsorbed

δ-MnO
2
and manganite may
therefore be used as a pair of model systems for comparative studies of metastable-
equilibrium adsorption.
5. Temperature dependence of metastable-equilibrium adsorption
Since temperature (T) is expected to affect both adsorption thermodynamics and kinetics,
the adsorption–desorption behavior may be
T-dependent. The adsorption irreversibility of
Zn(II) on anatase at various temperatures was studied using a combination of macroscopic
thermodynamic methods and microscopic spectral measurement.
Adsorption isotherm results
29
showed that, when the temperature increased from 5 to 40 °C,
the Zn(II) adsorption capacity increased by 130% (Figure 13). The desorption isotherms
significantly deviate from the corresponding adsorption isotherms, indicating that the
adsorption of zinc onto anatase was not fully reversible. The thermodynamic index of
irreversibility (TII) proposed by Sander et al.
30
was used to quantify the adsorption
irreversibility. The TII was defined as the ratio of the observed free energy loss to the
maximum possible free energy loss due to adsorption hysteresis, which was given by

eq eq
eq eq
ln ln
TII
ln ln
D
SD

C

is the solution concentration of hypothetical reversible desorption state γ (
eq
C

,
eq
q

).
eq
S
C and
eq
D
C are determined based on the experimental adsorption and desorption
isotherms, and are easily obtained from the adsorption branch where the solid-phase
concentration is equal to
eq
q
D
.
Based on the definition, the TII value lies in the range of 0 to 1, with 1 indicating the
maximum irreversibility. The TII value (0.63, 0.34, 0.20) decreased by a factor of >3 when the
temperature increased from 5 to 40 °C. This result indicated that the adsorption of Zn(II) on
the TiO
2
surfaces became more reversible with increasing temperature.
29

energetically unstable MEA state compared with the BB (-8.58 kcal/mol)
and BM (edge-
sharing bidentate mononuclear, -15.15 kcal/mol) adsorption modes,
13
indicating that the
MM linkage mode would be a minor MEA state, compared to the BB and BM MEA state. In
the X-ray absorption near-edge structure analysis (XANES), the calculated XANES of BB
and BM complexes reproduced all absorption characteristics (absorption edge, post-edge
absorption oscillation and shape resonances) from the experimental XANES spectra (Figure
15).
13
Therefore, the overall spectral and computational evidence indicated that the corner-
sharing BB and edge-sharing BM complexation mode coexisted in the adsorption of Zn(II)
on anatase.
As the temperature increased from 5 to 40 °C, the number of strong adsorption sites (edge
linkage) remained relatively constant while the number of the weak adsorption sites (corner
linkage) increased by 31%.
29
These results indicate that the net gain in adsorption capacity
and the decreased adsorption irreversibility at elevated temperatures were due to the
increase in available weak adsorption sites or the decrease in the ratio of edge linkage to
corner linkage. Both the macroscopic adsorption/desorption equilibrium data and the
molecular level evidence indicated a strong temperature dependence for the metastable-
equilibrium adsorption of Zn(II) on anatase.Thermodynamics – Interaction Studies – Solids, Liquids and Gases

534


5-coord. BM
4-coord. BB
exp. pH=6.3
exp. pH=6.8
Photon Energy (eV)
Normalized Relative Absorption

Fig. 15. Calculated XANES spectra of 4-oxygen coordinated BB and 5-oxygen coordinated
BM complex and experimental XANES spectra.
6. pH dependence of metastable-equilibrium adsorption
According to MEA theory, both adsorbent/particle concentration (i.e., Cp) and adsorbate
concentration could fundamentally affect equilibrium adsorption constants or isotherms
when a change in the concentration of reactants (adsorbent or adsorbate) alters the reaction
irreversibility or the MEA states of the apparent equilibrium. On the other hand, a general
theory should be able to predict and interpret more phenomena. To test new phenomenon
predicted by MEA theory can not only cross-confirm the theory itself but also provide new
insights/applications in broadly related fields. The influence of adsorbate concentration on
adsorption isotherms and equilibrium constants at different pH conditions was therefore
studied in As(V)-anatase system using macroscopic thermodynamics and microscopic
spectral and computational methods.
14, 31, 32

The thermodynamic results
14
showed that, when the total mass of arsenate was added to the
TiO
2
suspension by multiple batches, the adsorption isotherms declined as the multi-batch
increased, and the extent of the decline decreased gradually as pH decreased from 7.0 to 5.5
(Figure 16). This result provided a direct evidence for the influence of adsorption kinetics (1-

2
suspension in 3 times every 4 hours. EXAFS samples
were marked by ellipse, in which the initial total As (V) concentration is 0.80 mmol/L.

Sample
As-O
As-Ti
Res. CN
1
/CN
2
BB MM
CN R(Å) σ
2
CN
1
R
1
(Å) σ
2
CN
2
R
2
(Å) σ
2

1-batch pH5.5 3.9 1.68 0.002 1.9 3.17 0.008 1.1 3.60 0.01 8.6 1.8
3-batch pH5.5 4.0 1.68 0.002 2.2 3.26 0.01 0.9 3.61 0.008 14.2 2.4
1-batch pH6.2 4.0 1.68 0.002 1.8 3.16 0.007 1.0 3.59 0.006 11.0 1.7

occupying the adjacent surface site (MM
1
); (b) monodentate mononuclear arsenate H-bonded
to a -OH surface functional group occupying the adjacent surface site (MM
2
); (c) bidentate
binuclear (BB) complex; (d) H-bonded complex. Red, big gray, small gray, purple circles
denote O, Ti, H, As atoms, respectively. Distances are shown in angstroms.
The EXAFS coordination number of
CN
1
and CN
2
represented statistically the average
number of nearest Ti atoms around the As atom corresponding to a specific interatomic
distance. We used the coordination number
ratio of CN
1
/CN
2
to describe the relative
proportion of BB mode to MM mode in adsorption samples. The
CN
1
/CN
2
was 1.6 and 2.2
for 1-batch and 3-batch adsorption samples at pH 7.0, respectively (Table 1),
14
indicating that

The good agreement of EXAFS results of
CN
1
/CN
2
with XANES and FTIR analysis also
validated the reliability of the CN ratio used as an index to approximate the proportion
change of surface complexation modes. BB complex occupies two active sites on adsorbent
surface whereas MM occupies only one. For monolayer chemiadsorption, a unit surface area
of a given adsorbent can contain more arsenate molecules adsorbed in MM mode than that
in BB mode. Therefore, the increase of the proportion of BB complex from 1-batch to 3-batch
addition mode was shown as the decrease of adsorption density in 3-batch isotherm
(Figure 16).
Table 1 showed that the relative proportion of BB and MM complex was rarely affected by
pH change from 5.5 to 7.0, indicating that the pH dependence for the influence of adsorption
kinetics (1-batch/multi-batch) on adsorption isotherm was not due to inner-sphere
chemiadsorption.
14
The influence of pH on adsorption was simulated by DFT theory
through changing the number of H
+
in model clusters. Calculation of adsorption energy
showed that the thermodynamic favorability of inner-sphere and outer-sphere adsorption
was directly related to pH (Table 2).
14
As pH decreased, the thermodynamic favorability of
inner-sphere and outer-sphere arsenate adsorption on Ti-(hydr)oxides increased. This DFT
result explained why the adsorption densities of arsenate (Figure 16) and equilibrium
adsorption constant (Table 2) increased with the decrease of pH.
Advances in Interfacial Adsorption Thermodynamics:

TiO
2

Fig. 19. ATR-FTIR spectra of adsorbed As(V) of 1-batch and 3-batch adsorption samples,
dissolved arsenate, and TiO
2
at pH 7.0.
Theoretical equilibrium adsorption constant (
K) of calculated surface complexes (BB, MM
and H-bonded complexes in this adsorption system) that constructed real equilibrium
adsorption constant were significantly different in the order of magnitude under the same
thermodynamic conditions (Table 2). The theoretical
K were in the order of BB (6.80×10
42
)
>MM (3.13×10
39
) >H-bonded complex (3.91×10
35
) under low pH condition, and in the order
of MM (1.54×10
-5
) > BB (8.72×10
-38
) >H-bonded complex (5.01×10
-45
) under high pH
condition. Therefore, even under the same thermodynamic conditions, the real equilibrium
adsorption constant would vary with the change of the proportion of different surface
complexes in real equilibrium adsorption.

) is common or inevitable.

HO/AsO
4
Adsorption reaction equations ΔG K
Bidentate binuclear complexes
0
H
2
AsO
4
-
( H
2
O)
12
+ [Ti
2
(OH)
4
(H
2
O)
6
]
4+
→
[Ti
2
(OH)

12
+ [Ti
2
(OH)
5
(H
2
O)
5
]
3+
→
[Ti
2
(OH)
4
(H
2
O)
4
AsO
2
(OH)
2
]
3+
(H
2
O)
2

2+
→
[Ti
2
(OH)
4
(H
2
O)
4
AsO
2
(OH)
2
]
3+
(H
2
O)
2

+ 2OH
-
(H
2
O)
10

211.5 8.72×10
-38

5
AsO
2
(OH)
2
]
3+
H
2
O + 12H
2
O
-225.4 3.13×10
39

1-1
H
2
AsO
4
-
( H
2
O)
12
+ [Ti
2
(OH)
5
(H

-6

1-2
H
2
AsO
4
-
( H
2
O)
12
+ [Ti
2
(OH)
5
(H
2
O)
5
]
3+
→
[Ti
2
(OH)
5
(H
2
O)

2
O)
4
]
2+
→
[Ti
2
(OH)
5
(H
2
O)
4
AsO
2
(OH)
2
]
2+
H
2
O + OH
-
( H
2
O)
11

27.5 1.54×10

2
O)
6
AsO
2
(OH)
2
]
3+
+ 12H
2
O
-203.1 3.91×10
35

1
H
2
AsO
4
-
( H
2
O)
12
+ [Ti
2
(OH)
5
(H

2
H
2
AsO
4
-
( H
2
O)
12
+ [Ti
2
(OH)
6
(H
2
O)
4
]
2+
→
[Ti
2
(OH)
4
(H
2
O)
6
AsO

7. Acknowledgment
The study was supported by NNSF of China (20073060, 20777090, 20921063) and the
Hundred Talent Program of the Chinese Academy of Science. We thank BSRF (Beijing),
SSRF (Shanghai), and KEK (Japan) for supplying synchrotron beam time.
8. References
[1] Atkins , P. W.; Paula, J. d., Physical Chemistry, 8th edition. Oxford University Press:
Oxford, 2006.
[2]
Sverjensky, D. A., Nature 1993, 364 (6440), 776-780.
[3]
O'Connor, D. J.; Connolly, J. P., Water Res. 1980, 14 (10), 1517-1523.
[4]
Voice, T. C.; Weber, W. J., Environ. Sci. Technol. 1985, 19 (9), 789-796.
[5]
Honeyman, B. D.; Santschi, P. H., Environ. Sci. Technol. 1988, 22 (8), 862-871.
[6]
Benoit, G., Geochim. Cosmochim. Acta 1995, 59 (13), 2677-2687.
[7]
Benoit, G.; Rozan, T. F., Geochim. Cosmochim. Acta 1999, 63 (1), 113-127.
[8]
Cheng, T.; Barnett, M. O.; Roden, E. E.; Zhuang, J. L., Environ. Sci. Technol. 2006, 40, 3243-
3247.
[9]
McKinley, J. P.; Jenne, E. A., Environ. Sci. Technol. 1991, 25 (12), 2082-2087.
[10]
Higgo, J. J. W.; Rees, L. V. C., Environ. Sci. Technol. 1986, 20 (5), 483-490.
[11]
Pan, G.; Liss, P. S., J. Colloid Interface Sci. 1998, 201 (1), 77-85.
[12]
Pan, G.; Liss, P. S., J. Colloid Interface Sci. 1998, 201 (1), 71-76.

Manceau, A.; Lanson, B.; Drits, V. A., Geochim. Cosmochim. Acta 2002, 66 (15), 2639-2663.
[26]
Silvester, E.; Manceau, A.; Drits, V. A., Am. Mineral. 1997, 82 (9-10), 962-978.
[27]
Wadsley, A. D., Acta Crystallographica 1955, 8 (3), 165-172.
[28]
Post, J. E.; Appleman, D. E., Am. Mineral. 1988, 73 (11-12), 1401-1404.
[29]
Li, W.; Pan, G.; Zhang, M. Y.; Zhao, D. Y.; Yang, Y. H.; Chen, H.; He, G. Z., J. Colloid
Interface Sci.
2008, 319 (2), 385-391.
[30]
Sander, M.; Lu, Y.; Pignatello, J. J. A thermodynamically based method to quantify true
sorption hysteresis
; Am Soc Agronom: 2005; pp 1063-1072.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

542
[31] He, G. Z.; Pan, G.; Zhang, M. Y.; Wu, Z. Y., J. Phys. Chem. C 2009, 113 (39), 17076-17081.
[32]
Zhang, M. Y.; He, G. Z.; Pan, G., J. Colloid Interface Sci. 2009, 338 (1), 284-286.
0
Towards the Authentic Ab Intio Thermodynamics
In Gee Kim
Graduate Institute of Ferrous Technology,
Pohang University of Science and Technology, Pohang
Republic of Korea
1. Introduction
A phase diagram is considered as a starting point to design new materials. Let us quote the

equilibrium statistical mechanics for grand canonical ensemble by introducing the grand
21
2 Will-be-set-by-IN-TECH
partition function
Ξ
(
T, V,
{
μ
i
})
=
∑
N
i
∑
ζ
exp

−β

E
ζ
(
V
)
−
∑
i
μ

= −
β
−1
ln Ξ.(3)
The Legendre transformation relates the grand potential Ω and the Helmholtz free energy F as
Ω
(
T, V,
{
μ
i
})
=
F −
∑
i
μ
i
N
i
= E − TS −
∑
i
μ
i
N
i
.(4)
It is noticeable to find that the Helmholtz free energy F is able to be obtained by the relation
F

= −SdT −PdV −
∑
i
N
i
dμ
i
,(8)
with the coefficients
S
= −

∂Ω
∂T

Vμ
, P = −

∂Ω
∂V

Tμ
, N
i
= −

∂Ω
∂μ
i


derivatives thereof, one of the tasks will be to develop methods to calculate the grand potential
Ω.
544
Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Towards the Authentic Ab Intio Thermodynamics 3
In principle, we can calculate any macroscopic thermodynamic states if we have the complete
knowledge of the (grand) partition function, which is abled to be constructed from first
principles. However, it is impractical to calculates the partition function of a given system
because the number of all accessible microstates, indexed by ζ,isenormouslylarge.
Struggles have been devoted to calculate the summation of all accessible states. The number of
all accessible states is evaluated by the constitutents of the system and the types of interaction
among the constituents. The general procedure in statistical mechanics is nothing more than
the calculation of the probability of a specific number of dice with the enormous number
of repititions of the dice tosses. The fundamental principles of statistical mechanics of a
mechanical system of the degrees of freedom s is well summarized by Landau & Lifshitz
(1980). The state of a mechanical system is described a point of the phase space represented
by the generalized coordinates q
i
and the corresponding generalized momenta p
i
,wherethe
index i runs from 1 to s. The time evolution of the system is represented by the trajectory in
the phase space. Let us consider a closed large mechanical system and a part of the entire
system, called subsystem, which is also large enough, and is interacting with the rest part of
the closed system. An exact solution for the behavior of the subsystem can be obtained only
by solving the mechanical problem for the entire closed system.
Let us assume that the subsystem is in the small phase volume ΔpΔq for short intervals. The
probability w for the subsystem stays in the Δ pΔq during the short interval Δt is
w
= lim

, q
1
, q
2
, ,q
s
)
dpdq, (13)
where ρ is a function of all coordinates and momenta in writing for brevity ρ
(
p, q
)
.This
function ρ represents the density of the probability distribution in phase space, called
(statistical) distribution function. Obviously, the distribution function is normalized as

ρ
(
p, q
)
dpdq = 1. (14)
One should note that the statistical distribution of a given subsystem does not depend on
the initial state of any other subsystems of the entire system, due to the entirely outweighed
effects of the initial state over a sufficiently long time.
A physical quantity f = f (p, q) depending on the states of the subsystem of the solved
entire system is able to be evaluated, in the sense of the statistical average, by the distribution
function as
¯
f
=

∂q
i
˙
q
i
+
∂ρ
∂p
i
˙
p
i

= 0 (17)
tells us that the distribution function is constant along the phase trajectories of the subsystem.
Our interesting systems are (quantum) mechanical objects, so that the counting the number of
accessible states is equivalent to the estimation of the relevant phase space volume.
2. Phenomenological Landau theory
A ferromagnet in which the magnetization is the order parameter is served for illustrative
purpose. Landau & Lifshitz (1980) suggested a phenomenological description of phase
transitions by introducing a concept of order parameter. Suppose that the interaction
Hamiltonian of the magnetic system to be
∑
i,j
J
ij
S
i
·S
j

S
i

, (20)
where N is the number of atomic sites. The physical order is the alignment of the microscopic
spins.
Let us consider a situation that an external magnetic field H is applied to the system. Landau’s
idea
1
is to introduce a function, L
(
m, H, T
)
, known as the Landau function, which describes
the “thermodynamics” of the system as function of m, H,andT. The minimum of
L indicates
the system phase at the given variable values. To see more details, let us expand the Ladau
function with respect to the order parameter m:
L
(
m, H, T
)
=
4
∑
n
a
n
(
H, T

= L
(
m
B
)
, (22)
1
The description in this section is following Negele & Orland (1988) and Goldenfeld (1992).
546
Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Towards the Authentic Ab Intio Thermodynamics 5
for the minima points A and B. For the case of the second-order phase transition,itisrequired
that
∂
L
∂m
=
∂
2
L
∂m
2
=
∂
3
L
∂m
3
= 0,
∂

=
b
n
+ c
n
h + d
n
t, (24)
and then the Ladau function near the critical point is
L
(
m, h, t
)
=
c
1
hm + d
2
tm
2
+ c
3
hm
3
+ b
4
m
4
, d
2

2
to a and b
4
to
1
2
b:
L = atm
2
+
1
2
bm
4
− Hm. (26)
Let us consider the second-order phase transition with H
= 0. For T > T
C
, the minimum of
L is at m = 0. For T = T
C
, the Landau function has zero curvature at m = 0, where the point
is still the global minimum. For T
< T
C
, the Landau function Eq. (26) has two degenerate
minima at m
s
= m
s

H
)
≡
∂m
(
H
)
∂H




T
=
1
2

at + 3b
(
m
(
H
))
2

, (29)
where m
(
H
)

1
= 0 and changing the
coefficient symbols to yield
L = atm
2
+
1
2
m
4
+ Cm
3
− Hm. (30)
547
Towards the Authentic Ab Intio Thermodynamics
6 Will-be-set-by-IN-TECH
For H = 0, the equilibrium value of m is obtained as
m
= 0, m = −c ±

c
2
− at/b, (31)
where c
= 3C/4b. The nonzero solution is valid only for t < t
∗
, by defining t
∗
≡ bc
2

other words, the Landau theory is not valid. Hence, the first-order phase transition is arosen
by introducing the cubic term in m.
Since the Landau theory is fully phenomenological, there is no strong limit in selecting order
parameter and the corresponding conjugate field. For example, the magnetization is the order
parameter of a ferromagnet with the external magnetic field as the conjugate coupling field,
the polarization is the order parameter of a ferroelectric with the external electric field as
the conjugate coupling field, and the electron pair amplitude is the order parameter of a
superconductor with the electron pair source as the conjugate coupling field. When a system
undergoes a phase transition, the Landau theory is usually utilized to understand the phase
transition.
The Landau theory is motivated by the observation that we could replace the interaction
Hamiltonian Eq. (18)
∑
i,j
J
ij
S
i
S
j
=
∑
i
S
i ∑
j
J
ij

S

j
 by S
i
S
j

on average if we assume the translational invariance. The fractional error implicit in this
replacement can be evaluated by
ε
ij
=



S
i
S
j
−S
i
S
j




S
i
S
j

s
, (34)
where we assume the correlation function being written as
C
(
R
)
=
gf

R
ξ

, (35)
where f is a function of the correlation length ξ.ForT
 T
C
, the correlation length ξ ∼ R,
and the order parameter m is saturated at the low temperature value. The error is roughly
548
Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Towards the Authentic Ab Intio Thermodynamics 7
estimated as
ε
R


T
T
C

∼
1
|
t
|
2β

a
R

d
, (37)
which tends to infinity as t
→ 0. Hence, the Landau theory based on the mean-field
approximation has error which diverges as the system approaches to the critical point.
Mathematically, the Landau theory expands the Landau function in terms of the order
parameter. The landau expansion itself is mathematically non-sense near the critical point
for dimensions less than four. Therefore, the Landau theory is not a good tool to investigate
significantly the phase transitions of the system.
3. Matters as noninteracting gases
Materials are basically made of atoms; an atom is composed of a nucleus and the surrounding
electrons. However, it is convenient to distinguish two types of electrons; the valence electrons
are responsible for chemical reactions and the core electrons are tightly bound around the
nucleus to form an ion for screening the strongly divergent Coulomb potential from the
nucleus. It is customary to call valence electrons as electrons.
The decomposition into electrons and ions provides us at least two advantages in treating
materials with first-principles. First of all, the motions of electrons can be decoupled
adiabatically from the those of ions, since electrons reach their equilibrium almost
immediately by their light mass compared to those factors of ions. The decoupling of
the motions of electrons from those of ions is accomplished by the Born-Oppenheimer

β
(

i
−μ
)
+ 1
, (38)
where 
i
is the energy of the electronic microstate i and μ is the chemical potential of the
electron gas. At zero-temperature, the Fermi-Dirac distribution function becomes
1
e
β
(
−μ
)
+ 1
= θ
(
μ −
)
(39)
and the chemical potential becomes the Fermi energy 
F
. In the high-temperature limit, the
Fermi-Dirac distribution function recudes to
n
0

3
gV
3π
2

2m
¯h
2

3
2

∞
0
d

3
2
e
β
(
−μ
)
+ 1
(42)
and the number density is written as
N
V
=
g

3
gV
4π
2

2m
¯h

3
2

2
5
μ
5
2
+ β
−2
π
2
4
μ
1
2
+ ···

(44)
2
It is convenient to convert a summation over single-particle spectra to an integral over wavenumbers
according to

g

∞
−∞
dD
(

)
F
(

)
,whereD
(

)
is
the density of states.
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Thermodynamics – Interaction Studies – Solids, Liquids and Gases
Towards the Authentic Ab Intio Thermodynamics 9
and the chemical potential from the relation N =
(
∂
(
PV
)
/∂μ
)
TV

∂T

Vμ
=
gV
4π
2

2m
¯h

3
2
2
3

2π
2
4
k
B
β
+ ···

. (46)
It is thus the heat capacity of the noninteracting homogeneous electron gas to be
C
V
= T


g
4π
2

2m
¯h
2

3
2
2
5
μ
5
2
. (48)
All the necessary thermodynamic information of the homogeneous noninteracting electron
subsystem is acquired.
3.2 Elementary excitation as massive boson gas
For the case of ions, the treatment is rather complex. One can immediately raise the same
treatment of the homogeneous noninteracting ionic gas model as we did for the electronic
subsystem. Ignoring the nuclear spins, any kinds of ions are composed of fully occupied
electronic shells to yield the effective zero spin; ions are massive bosons. It seems, if the system
has single elemental atoms, that the ionic subsystem can be treated as an indistinguishable
homogeneous noninteracting bosonic gas, following the Bose-Einstein statistics (Bose, 1926;
Einstein, 1924; 1925). However, the ionic subsystem is hardly treated as a boson gas.
Real materials are not elemental ones, but they are composed of many different kinds of
elements; it is possible to distinguish the atoms. They are partially distinguishable each
other, so that a combinatorial analysis is required for calculating thermodynamic properties
(Ruban & Abrikosov, 2008; Turchi et al., 2007). It is obvious that the ions in a material

(

i
−μ
)
−1
. (49)
Since the chemical potential of a bosonic system vanishes at a certain temperature T
0
,aspecial
care is necessary during the thermodynamic property calculations (Cornell & Wieman, 2002;
Einstein, 1925; Fetter & Walecka, 2003). The grand potential of an ideal massive boson gas,
where the energy spectrum is also calculated as in Eq. (41), is
− βΩ
0
= βPV = −
gV
4π
2

2m
¯h
2

3
2

∞
0
d


3
2
e
β
(
−μ
)
−1
. (51)
The internal energy is calculated to be
E
=
∑
i
n
0
i

i
=
3
2
PV
=
gV
4π
2

2m


3
2

∞
0
d

1
2
e
β
(
μ−
)
−1
. (53)
A care is necessary in treating Eq. (53), because it is meaningful only if 
−μ ≥ 0, or
μ
≤ 0 (54)
with the consideration of the fact 
≥ 0.
In the classical limit T
→ ∞,orβ → 0, for fixed N,wehave
βμ
→−∞. (55)
Recall that the classical limit yields the Maxwell-Boltzmann distribution
n
0

βμ
c
= ln
⎡
⎣
N
gV

2π¯h
2
m

3
2
β
3
2
⎤
⎦
. (58)
As β increases at fixed density, βμ
c
passes through zero and becomes positive, diverging to
infinity at β
→ ∞. This contradicts to the requirement Eq. (54). The critical temperature β
0
,
where the chemical potential of an ideal boson gas vanishes, is calculated by using Eq. (53)
with μ
= 0tobe

3
, (59)
where Γ and ζ are Gamma function and zeta function, respectively. For μ
= 0andβ > β
0
,the
integral in Eq. (53) is less than N/V because these conditions increase the denominator of the
integrand relative to its value at β
0
.
The breakdown of the theory was noticed by Einstein (1925) and was traced origin of
the breakdown was the converting the conversion of the summation to the integral of the
occupation number counting in Eq. (53). The total number of the ideal massive Bose gas is
counted, using the Bose-Einstein distribution function Eq. (49), by
N
=
∑
i
1
e
β
(

i
−μ
)
−1
. (60)
It is obvious that the bosons tends to occupy the ground state for the low temperature
range β

−
3
2
, (61)
while the number density at the ground state is evaluated to be
N
=0
V
=
N
V

1
−

β
β
0

−
3
2

, (62)
with the chemical potential μ
= 0
−
for β > β
0
. The internal energy density of the degenerate

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Towards the Authentic Ab Intio Thermodynamics