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