1 5.6 Isobaric thermal expansion and isothermal compression (Hiroshi Matsuoka)
5.6.2 The coefficient of thermal expansion as a response function to a temperature change
The coefficient of thermal expansion
!
" is basically a measure for the response of a system’s
volume to an increase in the system’s temperature T: it tells us how a material changes its
volume as T is varied. It is therefore important to know
!
" when we design roads, railways,
buildings, airplanes, ships, cars, etc. The larger ! gets, the more sensitive the volume of a piece
of material is to a change of temperature. Typical values for
!
" in different phases are:
Gases:
!
" #10$3
K-1
Liquids:
!
" #10$4
~ 10$3
K-1
Solids:
!
" #10$6
~ 10$4
K-1
The values for
!
" for solids are smaller than those for liquids while the values for
!
" for
liquids are smaller than those for gases. The values of
!
" are basically determined by inter-
atomic or inter-molecular forces on the microscopic level. Inside a solid, atoms are close
together and tightly bound together so that by a small increase in atomic vibrational speed
induced by an increase in the temperature does not allow the atoms to increase their inter-atomic
distances very much resulting in a very small value for
!
" . Similarly, inside a liquid, the atoms
are still close together but they are a bit further away from each other compared to the atoms
inside a solid so that they are less strongly bound to each other and they can increase their inter-
atomic distances more readily in response to an increase in their speeds induced by an increase in
the temperature. As we found in the last chapter, inside a gas, the atoms are far apart and hardly
exert forces on each other so that they can increase its volume most readily in response to an
increase in their speeds.
!
" for low-density gases The ideal gas law or v = RT P leads to
! =1
v
"v
"T
#
$ %
&
' ( P
=1
T, (HW#5.6.4: show this)
2
from which we find that at T ! 300 K , ! " 3 #10$3
K-1 and that
!
" decreases as T is increased,
which means that it gets harder to expand a gas at higher T.
!
" also remains constant when the
pressure is kept constant.
!
" for solids and liquids
Typically,
!
" for solids and liquids are positive and increases as T is increased as we will find
in the next subsection.
Substance ! 300 K( ) K-1( )
Water 2.1! 10"4
Ethanol 1.1 !10"3
Diamond 3.0 !10"6
Copper 5.0 !10"5
NaCl 1.2 ! 10"4
How do we measure
!
" of a solid?
We can measure the molar volume of a solid directly by using the x-ray diffraction
technique, with which we can find a crystal lattice structure of atoms inside the solid and its
lattice constant, which is directly related to the average inter-atomic distance, from which we can
estimate the molar volume as a function of temperature and pressure and finally the coefficient
of thermal expansion
!
" . As mentioned above, it is quite time-consuming to directly obtain the
molar volume and
!
" this way.
We can also measure
!
" indirectly by measuring the linear coefficient of thermal expansion
defined by
!l"1
l
# l
#T
$
% &
'
( ) P
v = l3( ) .
!
" and !l are then related by
! = 3!l. (HW#5.6.5: show this. Hint: substitute v = l3 into the definition of ! )
3 To measure
!
"l, we must amplify a small length change by using
• Interference fringes of light
• Variation of capacitance
• Variation of light intensity
For more detail on these specific techniques, see, for example, “Heat and Thermodynamics (sixth
edition” by Zemansky and Dittman (McGraw-Hill).
Using
!
"l
!
"l is also a useful quantity in engineering. For example, we can answer the following
question by using
!
"l. How much does the length of a steel arch bridge, whose length at –20
!
°C
is 500 m, change when its temperature is increased to 40
!
°C? Assume that the value of
!
"l for
steel between –20
!
°C and 40
!
°C is roughly constant at
!
"l=1#10
$5 K
$1 . According to the
definition of
!
"l, the length increase
!
"l can be estimated by
!
"l =#l
#T
$
% &
'
( ) P
"T = l*l"T = 500 m( ) 1+10
,5 K
,1( ) 40°C( ) , ,20°C( ){ }
= 500 m( ) 1+10,5
K,1( ) 60 K( )
= 0.3 m
This length change is sizable enough that when constructing such a bridge a designer must take
into account this change.
5.6.3 Phenomenology of
!
" for solids at
!
P =1 atm
Under the atmospheric pressure, the coefficient of thermal expansion
!
" of a solid becomes a
function of its temperature T only:
!
" =" T,P =1 atm( ). For solids,
!
" T,P =1 atm( ) as a function
of T has the following two common features (see the figures on the next page):
•
!
" T,P =1 atm( ) approaches zero when T is decreased toward absolute zero.
•
!
" T,P =1 atm( ) increases as T is increased.
4 With further examination of data for
!
" T,P =1 atm( ) of various solids, we can identify two
universality classes according to low-temperature behaviors of
!
" T,P =1 atm( ) .
Insulators as a universality class
Solid insulators such as sodium chloride (NaCl) share the same temperature dependence of
!
" T,P =1 atm( ) at low temperatures (see the top figure on the next page). More specifically, at
low temperatures, we find
!
" #T 3 low T( ),
where the proportionality constant between
!
" and
!
T3 varies from one solid to another. As you
can see in the figure below, because of this
!
T3 dependence, the slope of
!
" as a function of T is
zero at
!
T = 0 and
!
" increases very slowly as T is increased near
!
T = 0.
0 100
5 10-5
1 10-4
1.5 10-4
0 50 100 150 200 250 300
NaCl
! (K
-1)
T (K)
“Simple” metals as a universality class
!
" T,P =1 atm( ) for “simple” metals such as alkali metals (e.g., sodium, etc.) and noble metals
(e.g., copper) behaves, at low temperatures, as
!
" = AT + BT 3 low T( ),
where the constants A and B vary from one metal to another.
5
0 100
2 10-5
4 10-5
6 10-5
8 10-5
1 10-4
0 200 400 600 800 100012001400
Cu
! (K
-1)
T (K)
As you can see in the figure above, because of the linear temperature term, the slope of
!
" as a
function of T is not zero at
!
T = 0 and
!
" increases relatively sharply as T is increased near
!
T = 0.
Expansion of a crystal lattice and of a free electron gas
For insulators, thermal expansion comes from an expansion of a crystal lattice of atoms or
ions (e.g., inside a NaCl crystal, positive Na ions and negative Cl ions are placed at corners of
cubes and each positive Na ion is surrounded by 6 negative Cl ions). The crystal lattice expands
because the amplitudes of thermal vibrations of atoms or ions become larger at higher
temperatures so that each atom or ion will claim a larger volume to itself.
At low temperatures, inter-atomic or inter-ionic forces strongly bind atoms or ions together
and generate collective vibration of atoms or ions or lattice waves. The cubic term in T at low
temperatures reflect how these lattice waves affect thermal expansion. At high temperatures,
individual atoms or ions vibrate almost independently so that the temperature dependence of !
becomes different from that at low temperatures.
For metals, besides a crystal lattice of positive ions, we must also take into account the
presence of “conduction” electrons that can almost freely move through the crystal lattice. It
turns out that for a class of “simple” metals such as sodium and copper, we can regard these
electrons as a gas of non-interacting particles (i.e., a free electron gas). For temperatures much
lower than roughly 10,000 K, this free electron gas contributes a linear term in T to ! . In other
6 words, temperatures much lower than 10,000 K (for example, the room temperature) is still low
temperatures for the free electrons inside a simple metal (more on this below).
The Debye temperature ! separates “low T” from “high T”.
The characteristic temperature that separates “high temperatures” from “low temperatures”
for lattice waves in a solid is called the “Debye temperature” ! . Temperatures that are much
lower than ! are low temperatures, while those that are much higher than ! are high
temperatures. The Debye temperatures for various solids range roughly between 100 K and
1000 K. For example, ! = 321 K for NaCl while ! = 343 K for copper.
Where does ! come from?
The Debye temperature ! comes from the maximum energy that lattice waves can have
inside a solid. According to quantum mechanics, the lattice waves behave also as a type of
quantum particles called phonons with their energy given by ! = hf , where f is the frequency of
atomic vibrations accompanying the lattice waves. The maximum energy (called the Debye
energy !D) that the lattice waves can have in a solid is then controlled by the maximum
frequency (called the Debye frequency) for these waves. For any wave, we can use the
following general formula: f! = w , where ! is in our case the wavelength of a lattice wave and
w should be the speed of sound. The maximum frequency for the lattice waves then corresponds
to their minimum wavelength (called the Debye wavelength), which is on the order of the
average distance among the atoms or ions inside the solid or !D
~O 1 A( ) =O 10"1 0
m( ) so that
f D =w
!D
~O10
3 m s
10"1 0
m
#
$ %
&
' ( ~O 10
1 3 Hz( )
High temperatures for the lattice waves are the temperatures that satisfy
!
kBT >> "D = hfD or T >> ! "#D
kB=hf D
kB,
where kB is the Boltzmann constant, which is related with the universal gas constant R by
7
kB =R
NAvogadro
= 1.38 !10"23
J K ,
and kBT gives the order of magnitude for the average energy per atom.
!
kBT >> "
D then means
that at high temperatures the typical energy per atom is much higher than the maximum energy
for the lattice waves so that all the lattice waves are excited in the solid while at low
temperatures only the lattice waves with low energy are excited.
We can estimate the typical order of magnitude for the Debye temperature as
! ~Ohf D
kB
"
# $ $
%
& ' ' ~O
10(3 4
J )s( ) 101 3
1 s( )10
(23 J K
*
+ , ,
-
. / / ~O 100 K( )
The Debye model as a minimal model for quantum lattice waves or phonons
The minimal model for quantum lattice waves or phonons is called “the Debye model” and
treats phonons as a type of quantum particles called “bosons” and neglects forces or interactions
among them. Although the Debye model as it is cannot explain thermal expansion of the crystal
lattice, we can modify it to show the
!
T3-dependence of
!
" at low temperatures. This modified
version of the Debye model is sometimes called “the Gruneisen model.”
Free electrons in a simple metal are always in the low temperature regime
The characteristic temperature that separates high temperatures from low temperatures for
free electrons in a simple metal is called the “Fermi temperature” TF
, which is directly related
with the maximum energy (called the “Fermi energy”) for the free electrons in a simple metal at
T = 0. According to quantum mechanics, an electron behaves as a wave whose wavelength !
controls its energy ! through
! =h2
2m
2"
#
$ % & '
( ) 2
,
where m is the electron mass and h is Planck’s constant h divided by 2! . Inside a simple metal,
the minimum wavelength (called the “Fermi wavelength”) that corresponds to the maximum
energy is on the order of the average distance among the free electrons, which is comparable to
the average distance among positive ions inside the solid so that
8
!F
~O 1 A( ) = O 10"10
m( )
so that the Fermi energy is on the order of
!F
=h
2
2m
2"
#F
$
% &
'
( )
2
~O10-34 J *s( )
2
10+30 kg( ) 10+10 m( )2
$
% & &
'
( ) ) ~O 10
-18 J( ) ~O 10 eV( ) ,
where we have used
!
1 eV ~ 10"19
J . The Fermi temperature for the free electrons is then on the
order of
TF!"F
kB
~O10
-18 J
10-23
J K
#
$ % %
&
' ( ( ~O 10
5 K( )
so that the Fermi temperature is much higher than the melting temperature of the metal:
TF>>T
m~O 100 K !1000 K( ) . Therefore, for the free electrons in a simple metal, the
temperature range for the solid phase is always in their low temperature regime.
The free electron gas model as a minimal model for conduction electrons in simple metals
The minimal model for conduction electrons in simple metals is called “the free electron gas
model” and treats these electrons as a type of quantum particles called “fermions” and neglects
forces or interactions among them. We will show later that the free electron gas model
combined with the modified Debye model or Gruneisen model can explain the linear temperature
term in
!
" found for simple metals.
5.6.4 The molar volume v for solids at
!
P =1 atm
As mentioned above, if we know
!
" T,P( ) and
!
"TT,P( ) , we can calculate
!
v T,P( ). When the
pressure is kept constant at 1 atm, we then find
!
v T,P =1 atm( ) = v T0,P =1 atm( )exp " # T ,P( )d # T
T0
T
$%
& ' '
(
) * * .
9 As
!
" # T ,P( )d # T
T0
T
$ ~ O " T0,P( ) T %T
0( )[ ] <<1,
we can use
!
ex"1+ x for
!
x <<1 to get
!
v T,P =1 atm( ) = v T0,P =1 atm( ) 1+ " # T ,P( )d # T
T0
T
$%
& ' '
(
) * * .
If
!
" > 0 , the molar volume then increases as T is increased.
Insulators (e.g., NaCl)
At low temperatures, we have found ! " AT3 , where A is a positive constant specific to a
particular insulator, so that
!
v T,P =1 atm( ) " v T = 0 K,P =1 atm( ) 1+A
4T
4#
$ %
&
' ( . (HW#5.6.6: show this)
As shown on the figure on the next page, because of the
!
T4 term, the slope of the molar volume
v as a function of T is zero at
!
T = 0 and v increases very slowly as T is increased near
!
T = 0.
Simple metals (e.g., copper)
At low temperatures, we have found
!
" # AT + BT3, where A and B are positive constants
specific to a particular simple metal, so that
!
v T,P =1 atm( ) " v T = 0 K,P =1 atm( ) 1+A
2T
2 +B
4T
4#
$ %
&
' ( . (HW#5.6.7: show this)
As shown on the figure on the next page, because of the
!
T2 term, the slope of the molar volume
v as a function of T is zero at
!
T = 0 and v increases relatively sharply compared with v for NaCl
as T is increased near
!
T = 0.
10
2.6 10-5
2.65 10-5
2.7 10-5
2.75 10-5
0 50 100 150 200 250 300
NaCl
v (
m3/m
ol)
T (K)
6.5 10-6
7 10-6
7.5 10-6
0 200 400 600 800 1000 1200 1400
Cu
T (K)
v (
m3/m
ol)
11 SUMMARY OF SEC.5.6.2 THROUGH SEC.5.6.4
1. Typical orders of magnitude for
!
" are:
Gases:
!
" #10$3
K-1
Liquids:
!
" #10$4
~ 10$3
K-1
Solids:
!
" #10$6
~ 10$4
K-1
2. For low-density gases:
!
" =1
v
#v
#T
$
% &
'
( ) P
=1
T.
3. We measure the linear coefficient of thermal expansion defined by
!
"l#1
l
$l
$T
%
& '
(
) * P
!
v = l3( )
and calculate
!
" using
!
" = 3"l.
4. The Debye temperature of a solid separates the low temperature regime from the high
temperature regime for lattice waves in the solid and ranges between 100 K and 1000 K.
5. The Fermi temperature of a simple metal separates the low temperature regime from the high
temperature regime for free electrons in the metal and is on the order of
!
105 K .
6. For solid insulators such as NaCl,
!
" # AT3 at low temperatures so that their molar volumes
at low temperatures behave as
!
v T,P =1 atm( ) = v T = 0 K,P =1 atm( ) 1+ " A T4( ).
7. For simple metals such as Cu,
!
" # AT + BT3 at low temperatures so that their molar volumes
at low temperatures behave as
!
v T,P =1 atm( ) " v T = 0 K,P =1 atm( ) 1+ # A T2 + # B T
4( ) .
12 Answers for the homework questions in Sec.5.6.2 and Sec.5.6.4
HW#5.6.4
!
" =1
v
#v
#T
$
% &
'
( ) P
=1
v
#
#T
RT
P
$
% &
'
( )
*
+ ,
-
. / P
=R
vP=R
RT=1
T
HW#5.6.5
!
" =1
v
#v
#T
$
% &
'
( ) P
=1
l3
#l3
#T
$
% &
'
( ) P
=3l2
l3
#l
#T
$
% &
'
( ) P
= 31
l
#l
#T
$
% &
'
( ) P
= 3"l
HW#5.6.6
!
v T,P =1 atm( ) = v T = 0 K,P =1 atm( ) 1+ " # T ,P( )d # T
0
T
$%
& '
(
) *
+ v T = 0 K,P =1 atm( ) 1+ A # T 3d # T
0
T
$%
& '
(
) *
= v T = 0 K,P =1 atm( ) 1+A
4T
4,
- .
/
0 1
HW#5.6.7
!
v T,P =1 atm( ) = v T = 0 K,P =1 atm( ) 1+ " # T ,P( )d # T
0
T
$%
& '
(
) *
+ v T = 0 K,P =1 atm( ) 1+ AT + BT3( )d # T
0
T
$%
& '
(
) *
= v T = 0 K,P =1 atm( ) 1+A
2T
2 +B
4T
4,
- .
/
0 1