Correlation between thermal expansion and heat capacity
Jozsef Garai
Florida International University, Department of Earth Sciences
University Park, PC 344, Miami, FL 33199, USA
Ph: 1-786-247-5414
Fax: 1-305-348-3877
E-mail address: [email protected]
Abstract
Theoretically predicted linear correlation between the volume coefficient of thermal expansion and the
thermal heat capacity was investigated for highly symmetrical atomic arrangements. Normalizing the data
of these thermodynamic parameters to the Debye temperature gives practically identical curves from zero
Kelvin to the Debye temperature. This result is consistent with the predicted linear correlation. At
temperatures higher than the Debye temperature the normalized values of the thermal expansion are always
higher than the normalized value of the heat capacity. The detected correlation has significant
computational advantage since it allows calculating the volume coefficient of thermal expansion from one
experimental data by using the Debye function.
Keywords: Thermal Properties; Thermal expansion; Heat capacity; Correlation; Highly symmetrical atomic
arrangements
1. Introduction
The volume coefficient of thermal expansion [α V ] is described as
αV =
1  dV 


V  dT  p
(1)
where V is the volume, T is the temperature, and P is the pressure. The volume
coefficient of thermal expansion is temperature dependent requiring numerous
experiments for its complete description. The experiments are time consuming and
technically difficult at extreme temperatures.
The theory of thermal expansion is well defined. The volume expansion can be
determined by calculating the anharmonic term of the lattice vibration [ex. 1, 2]. This
traditional approach has been challenged by a simple classical model [3, 4] which gave
good and excellent agreement with the experimental data of Ne, Ar, Kr, and Xe. In these
calculations it was assumed that the potential is Lennard-Jones and that the mean value of
the interatomic atomic distance [〈 R 〉 ] can be used to calculate the thermal expansion.
[α(T)] P=0 =
1  d〈 R (T)〉 
R 0  dT 
(2)
P =0
The mean interatomic distance with a good approximation is proportional to the mean
vibrational energy per atom in a solid [2]. The thermal energy of a system [Q thermal ] is the
product of the mean vibrational energy of an atom and the number of atoms in the
system. The zero-term energy of the atoms is not included in the thermal energy.
Introducing a constant [a ] allows to substitute the mean interatomic distance with the
thermal energy of a system. Equation (2) can be rewritten then
 dQ thermal (〈 R (T)〉 ) 

n 
dT

[α(T)] P=0 = a
(3)
P =0
where n is the number of moles.
The molar thermal heat capacity c(ϕ)
thermal
c(ϕ)
thermal
is defined as:
1 T + δT
1 dQ(ϕ)
thermal
δQ(ϕ)
=
∫
nδT T
n
dT
thermal
=
(4)
where ϕ = g (gas), s(solid), l(liquid ) . Combining equation (3) and (4) gives
[α V (T)] P=0 = a [c(s )thermal ] P=0 .
(5)
The molar thermal heat capacity is pressure independent. The pressure has effect on the
equilibrium separation [R 0 ] of the atoms, which is incorporated in the constant. The
pressure effect on R 0 can be accommodated by assuming that the introduced constant [a]
is pressure dependent.
[α V (T)] P = a Pthermal [c(s )thermal ]
(6)
Replacing the molar thermal heat capacity with the molar heat capacity at constant
pressure modifies the pressure dependent multiplier.
[α V (T)] P = a P [c(s )] P
(7)
Equation (7) predicts linear correlation between the volume coefficient of thermal
expansion and the molar heat capacity at constant pressure. This correlation will be
considered in detail.
2. Correlation between the volume coefficient of thermal expansion and molar heat
capacity
Experimental data of highly symmetrical mono-atomic arrangements [5] Ag, Al, Au,
Ba, Co, Cr, Cu, Fe, K, Ni, Pb, Pt, Ti, V have been used to investigate the predicted
correlation between the volume coefficient of thermal expansion and the molar heat
capacity at atmospheric pressure. The experimental values of these two thermodynamic
parameters were not necessarily determined at the same temperature. In order to make
these parameters comparable, the two data sets were normalized to the Debye
-2-
temperature. It has been found that the normalized values of the volume coefficient of
thermal expansion and the heat capacities are practically identical below the Debye
temperature. Above the Debye temperature the normalized values of the volume
coefficient of thermal expansion are higher than the normalized values of the heat
capacity. The characteristic behavior is shown on Fig. 1. These results are interpreted as
follows. The identical curves of the normalized values of the volume coefficient of
thermal expansion and molar heat capacity at atmospheric pressure confirm the predicted
linear correlation between these thermodynamic parameters from zero Kelvin to the
Debye temperature. The higher normalized values of the volume coefficient of thermal
expansion above the Debye temperature might be the result of the additional expansion
caused by lattice vacancies [6, 7].
3. Computational advantages
Employing the detected linear correlation between the volume coefficient of thermal
expansion and the molar thermal heat capacity allows calculating the volume coefficient
of thermal expansion from any experiment. Using equation (6) for two different
temperatures and dividing equation one by equation two gives
 α V (T1 ) 


 α V (T2 ) 
c(s )T1
thermal
=
P
and
c(s )T2
thermal
[α V (T2 )] P =
[α V (T1 )] P thermal
c(s )T
.
thermal
c(s )T
2
(8)
1
The pressure effect is canceled out in equation (8) by the division. The molar thermal
heat capacity can be calculated by using the Debye function [8].
c(s )
thermal
≈ c(s )
Debye
= 3Rf
 T
f = 3
 TD



3 x
D
x 4e x
∫0 (e x − 1) 2 dx
(9)
and
x=
hω
and
2πk BT
xD =
hωD
T
=
2πk BT TD
(10)
where h is the Planck's constant, ω is the frequency, ωD is the Debye frequency and TD
is the Debye temperature. This equation has to be evaluated numerically [9].
Using the experimental value of the volume coefficient of thermal expansion at the
Debye temperature [α(TD )] the volume coefficient of thermal expansion was calculated
between zero Kelvin and the Debye temperature. The calculated and the experimental
values are plotted in Fig. 2. Based on visual inspection the correlations are excellent for
elements (Ag; Al; Au; Cu; Ni; Pb; Pt) with face centered cubic structure, very good for
Co and Ti with hexagonal close packed structure and good for element (Ba; Cr; Fe; K; V)
with body centered cubic structure.
-3-
4. Temperature independent volume coefficient of thermal expansion
The detected linear correlation between the volume coefficient of thermal expansion and
the molar heat capacity at constant pressure allows describing the thermal expansion with
a constant coefficient. The volume coefficient of thermal expansion becomes constant if
the differential of the temperature is replaced with the differential of the enthalpy (H).
[α
V ( T < TD )
]
P
=
n  dV 
V  dH(s ) 
(11)
P
Above the Debye temperature the volume coefficient of thermal expansion is not
constant. For the investigated solids linear temperature related increase has been detected
for the volume coefficient of thermal expansion. This effect can be taken into
consideration by the introduction of a new constant [Cα ] .
[α
V ( T > TD )
]
P
= [1 + Cα (T − TD )]
n  dV 
V  dH(s ) 
(12)
P
The temperature independent nature of the new volume coefficient of thermal
expansion makes the thermodynamic calculations more convenient. Describing the
volume coefficient of thermal expansion by equation (11) instead of equation (1)
expresses the physics of the thermal expansion properly since the anharmonic term or the
mean atomic distance is proportional to the mean energy per atom and not to the
temperature.
5. Conclusions
The theoretically predicted linear correlation between the volume coefficient of
thermal expansion and the heat capacity has been confirmed for highly symmetrical
mono-atomic arrangements.
The detected correlation allows calculating the volume coefficient of thermal
expansion from an experiment conducted at a preferably chosen temperature.
Replacing the differential of the temperature in the derivative of the thermal
expansion with the differential of the enthalpy results in a temperature independent
volume coefficient of thermal expansion up to the Debye temperature. This new
thermodynamic parameter expresses the physical process of the thermal expansion more
appropriately and makes the calculations simpler.
References
[1] C. Kittel, Introduction to Solid State Physics 6th edition, New York, Wiley, 1986, p. 114.
[2] R.A. Levy, Principles of Solid State Physics, New York, Academic, 1968, p. 141.
-4-
[3] P. Mohazzabi and F. Behroozi, Phys. Rev. B 36 (1987) 9820-9823.
[4] P. Mohazzabi and F. Behroozi, Eur. J. Phys. 18 (1997) 237-240.
[5] Handbook of Physical Quantities, edited by I. S. Grigoriev and E. Z. Meilikhov, CRC Press, Inc. Boca
Raton, FL, USA, 1997.
[6] R.O. Simmons and R.W. Balluffi, Phys. Rev. 117 (1960) 52-61.
[7] Th. Hehenkamp, W. Berger, J.–E. Kluin, Ch. Lüdecke, and J. Wolff, Phys. Rev. B 45 (1992) 19982003.
[8] P. Debye, Ann. Physik 39, (1912) 789.
[9] Landolt-Bornstein, Zahlenwerte und Funktionen aus Physic, Chemie, Astronomie, Geophysic, und
Technik, II. Band, Eigenschaften der Materie in Ihren Aggregatzustanden, 4 Teil, Kalorische
Zustandsgrossen, Springer-Verlag, 1961.
-5-
Fig. 1. Normalized values of the volume coefficient of thermal expansion and the molar
volume heat capacity at constant pressure as function of temperature for K, Cu, and Ti.
α
α [10 K ]
-6
-1
20
-6
-1
[10 K ]
30
25
15
20
10
TD = 227 K
α = 18.1 10 K
-6
fcc
5
15
TD = 433K
-6
-1
αD = 25.06 10 K
10
fcc
-1
D
Ag
Al
5
0
0
0
100
50
150
250
200
300
Temperature [K]
α [10 K ]
-6
0
350
-1
14
100
200
300
400
500
600
Temperature [K]
α [10 −6 K −1 ]
18
16
12
14
10
12
8
6
TD = 162K
-6
-1
α D = 12.6 10 K
4
fcc
10
TD = 111K
-6
-1
α D = =17.1 10 K
8
6
Au
2
bcc
4
Ba
2
0
0
0
50
100
150
200
250
0
Temperature [K]
20
40
60
80
100
120
Temperature [K]
-6-
α
-6
α [10 −6 K −1 ]
-1
[10 K ]
16
10
14
8
12
10
6
TD = 460 K
α D = 13.9 10- 6 K- 1
8
6
TD = 606K
-6
-1
α D = 9.15 10 K
4
bcc
hcp
4
2
Co
2
Cr
0
0
100
0
200
300
400
500
0
600
100
200
300
400
Temperature [K]
500
700
α [10 −6 K −1 ]
α [10 −6 K −1 ]
20
16
18
14
16
12
14
10
12
10
8
TD = 347K
-6
-1
α D = 17.0 10 K
6
fcc
4
8
TD = 477K
-6
-1
α D = 14.1 10 K
6
bcc
4
Cu
2
Fe
2
0
0
0
100
200
300
400
0
500
100
200
300
400
Temperature [K]
α
600
Temperature [K]
-6
-1
[10 K ]
500
600
Temperature [K]
α [10 −6 K −1 ]
70
16
60
14
12
50
10
40
8
30
TD = 91 K
α D = 57.6 10- 6 K- 1
6
20
bcc
4
K
10
TD = 477K
-6
-1
α D = 14.6 10 K
fcc
Ni
2
0
0
0
20
40
60
80
100
0
120
100
200
400
300
Temperature [K]
α [10 −6 K −1 ]
500
600
Temperature [K]
α
30
-6
-1
[10 K ]
10
25
8
20
6
15
TD = 105K
-6
-1
α D = 25.5 10 K
10
TD = 237 K
4
α = 8.7 10 K
-6
fcc
5
Pb
0
20
40
60
80
100
fcc
2
0
-1
D
Pt
0
120
0
Temperature [K]
50
100
150
200
250
300
350
Temperature [K]
-7-
α
-6
α
-1
[10 K ]
-6
10
10
8
8
6
-1
[10 K ]
6
TD = 420 K
α D = 8.9 10 -6 K- 1
4
TD = 382 K
α D = 8.4 10 -6 K- 1
4
hcp
bcc
2
2
Ti
0
V
0
0
100
200
300
400
500
0
600
100
200
300
400
500
Te mperature [K]
Te mperature [K]
Fig. 2. The solid lines are the calculated volume coefficients of thermal expansion while
the dots represent the experimental values [5].
-8-