Thermal 1

Thermal Properties of Matter

Heat and internal energy
Consider rubbing two surfaces against one another in the presence of friction. We know we have done
work by translating a force through a distance and we know that some of this work has been done
against friction. But what happens to this work (energy)? We may be able to hear the surfaces rubbing
against each other, but more importantly, the surfaces will get hot (as anyone whose bicycle brakes
have ever seized will know). If we apply conservation of energy, we can conclude that the work has gone
into the internal energy of the objects…

We can now define the internal energy, U, of an object as the sum of all kinetic and potential energies of
the constituent particles of the object.

In practice, however, we are more interested in changes in internal energy, ΔU, rather than an absolute
value for the total internal energy. This allows us to write a simple conservation of energy statement as
follows:

∆𝑈 = 𝑄 + ∆𝑊 (17.1)

where 𝑄 is the heat flow into the object (in J) and ∆𝑊 is the work done on the object (again, in J). The
sum of the two will result in a change in internal energy of ∆𝑈 (in J). It additionally allows us to identify
the heat flow 𝑄 simply as the (thermal) flow of internal energy from one object to another.

The simple conservation of energy equation shown above is known as the “First law of
Thermodynamics” sometimes summarised by the common mnemonic “you can’t win”.

Temperature
If two objects at different temperatures are brought into thermal contact (where heat can flow between
them) there will be a spontaneous net heat flow from the hotter object to the colder object until the
two objects find themselves at the same temperature (or thermal equilibrium). Thus the temperature
has to be a quantitative measure of “hotness” which naturally leads us to the necessity for temperature
scales. In order to set up a temperature scale we have certain requirements, these are:

Fixed points: Conditions which can be independently set up to define fixed points for a scale, e.g. the
melting point of ice and the boiling point of pure water at one atmosphere.

Thermometric Property: This is a property of the equipment that will change with temperature, for
example, the length of a mercury column, the resistance of a platinum wire, the pressure exerted by a
gas contained din affixed volume or the voltage difference between the two metals of a thermocouple
junction.

Clearly, as scientists, we can define as many temperature scales as we can imagine given any set of fixed
points and intervals between them, here, however, we shall consider two sensible temperature scales at
the exclusion of all others, these are:

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Mechanics and Materials School of Physics and Astronomy
 Thermal 2

The Practical Centigrade Scale (Celsius)

This is a non SI scale of temperature using degrees Celsius ( oC) that has a very simple conversion to the
SI units (K) and has two fixed points. The ice point of water (where solid ice and liquid water are in
equilibrium), which is defined as 0 oC and the steam point of pure water (where the liquid water and
gaseous steam are in equilibrium) at one atmosphere defined as 100 oC. Any thermometric property
can be used and its value at a given temperature, 𝜃, is simply 𝑋𝜃 . It allows us to measure a temperature
in oC very simply by simply measuring the value of the thermometric property as long as we have carried
out a calibration at the two fixed points. The temperature 𝜃 in oC is defined as:
𝑋𝜃 −𝑋0
𝜃= × 100℃ (17.2)
𝑋100 −𝑋0

Where 𝑋100 and 𝑋0 are the values of the thermometric property at the steam point and ice point
respectively. One consequence we should be aware of is that any temperatures measured using
different thermometric properties will not necessarily agree except at the fixed points.

The Thermodynamic Scale

This measures in K (the SI unit of temperature) and uses two fixed points: absolute zero, defined as 0 K
and the triple point of water (where solid, liquid and gas water are at equilibrium, which happens at
0.01 oC and 6×102 Pa) defined as 273.16 K. In practical terms this means that a 1 K temperature interval
is the same as a 1 oC temperature interval. The absolute temperature 𝑇 measured in K can be defined
using an ideal gas thermometer as:
𝑃𝑉
𝑇 = (𝑃𝑉) × 273.16 𝐾 (17.3)
𝑡𝑟𝑖𝑝𝑙𝑒 𝑝𝑜𝑖𝑛𝑡

Where 𝑃 and 𝑉 are the pressure and volume of an ideal gas (which we will encounter later in the
course). The definitions of the Celsius and absolute temperature scales allow for simple conversion
between the two using:

𝜃 = 𝑇 − 273.15 (17.4)

For a Celsius temperature 𝜃 and an absolute temperature 𝑇.

Specific Heat Capacity

The change in internal energy required (and therefore heat required) to affect a given temperature
change will depend on both the amount of material considered and the material itself. We can define
the specific heat capacity, 𝑐, for a substance as the energy required to raise the temperature of one kg
of material by one oC (or indeed one K). Thus the heat 𝑄 required for a given process is:

𝑄 = 𝑚𝑐∆𝜃 = 𝑚𝑐∆𝑇 (17.5)

where 𝑚 is the mass of material, ∆𝜃 or ∆𝑇 the temperature change and 𝑐 is the specific heat capacity (in
J kg-1 K-1 or J kg-1 oC-1).

For a given object the heat capacity (in J K-1) is the energy required to change its temperature by one
degree Celsius (or one K). It could be calculated by the sum of all constituent mass- specific heat capacity
products, i.e. ∑𝑖 𝑚𝑖 𝑐𝑖

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Mechanics and Materials School of Physics and Astronomy
 Thermal 3

Latent Heat
When materials undergo changes of state (e.g. liquid to gas or solid to liquid) they will do so at a
constant temperature (the boiling and melting temperatures respectively) yet the change of state
requires energy. We call this energy the latent heat (in this context, latent has the meaning of potential).
For a given material we can define the specific latent heat, 𝑙 , for a phase change (e.g. latent heat of
fusion or latent heat of vaporisation) as the energy required to change the state of one kg of material.
The heat required, 𝑄, for a phase change thus becomes:

𝑄 = 𝑚𝑙 (17.6)

where 𝑚 is the mass of material and 𝑙 the specific latent heat (in J kg-1). The latent heats of fusion and of
vaporisation for a given material are in general numerically different, with the latent heat of
vaporisation generally (numerically) much larger than the heat of fusion.

We can relate the latent heat of vaporisation for a liquid (or latent heat of sublimation if a solid
undergoes this process) to its microscopic properties. If we recall the potential energy-separation graph
for a pair of molecules we see that the pair has a binding energy 𝜀0 (the minimum of the graph at the
𝜀
equilibrium separation, 𝑟0 ). If a molecule has 𝑛 nearest neighbours and it takes 20 to remove it from
𝑛𝜀0
each pair, then the energy required to fully remove the molecule from the bulk is simply . If there are
2
𝑁 molecules per kg of material then we can write the latent heat of fusion 𝑙 (or sublimation) as:
1
𝑙 = 2 𝑁𝑛𝜀0 (17.7)

Thermal Expansion
For solids we know that the linear thermal expansion is proportional to the linear dimension itself, i.e.

∆𝐿 ∝ 𝐿0 (17.8)

where ∆𝐿 is the change in length and 𝐿0 is the original length.

We also know that the linear thermal expansion is proportional to the temperature rise itself, i.e.:

∆𝐿 ∝ ∆𝜃 (17.9)

where ∆𝜃 Is the change in temperature. Thus we can quantify the linear thermal expansion of a given
solid ∆𝐿 by combining the above and defining a coefficient of thermal expansion, 𝛼, (in K-1 or oC-1):

∆𝐿 = 𝛼𝐿0 ∆𝜃 = 𝛼𝐿0 ∆𝑇 (17.10)

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 Thermal 4

Heat Transfer
We have already noted that heat will spontaneously flow from hot objects to cold objects. In this section
we shall quantify two major heat transfer mechanisms, namely Radiation and Conduction (omitting
Convection, as a quantitative treatment is beyond the scope of this course).

Radiation
This is the transfer of energy (heat) via electromagnetic radiation. It is in general dependent on the
surface of a body, although we shall use the concept of a “black body” which is an idealised “perfect”
emitter or absorber of radiation (a practical “black body” can be made by drilling a hole in an actual solid
and studying the radiation emitted from the hole). Our experience tells us that as objects heat up they
emit at increasingly shorter wavelengths. Thus a piece of metal at room temperature will emit primarily
in the infrared (IR) and as it is heated up it will begin emitting in the red part of the visible spectrum
(“red hot”) and if further heated will emit at ever shorter wavelengths (“white hot”). We note that it is
not only the peak wavelength that changes with increasing temperature, but also the absolute rate of
heat transfer by radiation (i.e. the total power radiated). These two effects can be simply quantified by
the following:

Wien’s Displacement Law: This states that the peak wavelength (in m) of radiation emitted, 𝜆𝑝 , is
related to the absolute temperature (in K) of a black body, 𝑇, by the simple relationship:

𝜆𝑝 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ≈ 2.90 × 10−3 𝑚 𝐾 (17.11)

Stefan-Bolzmann Law: This states that the absolute power 𝑃 (in W) emitted by a black body at
temperature 𝑇 (in K) and of total surface area 𝐴 (m2) is given by:
𝑑𝑄
𝑃= 𝑑𝑡
= 𝜎𝐴𝑇 4 (17.12)

where 𝜎 is the Stefan-Boltzmann constant ≈ 5.7×10-8 W m-2 K-4.

It is worth noting that this is a highly superlinear relationship between power and temperature, where
doubling of absolute temperature will result in a 16 fold increase in emitted power!

Thermal Conduction
This is the flow of heat through a medium. Some materials, such as metals are very good thermal (and
electrical) conductors as a result of the large number of free electrons within them that can mediate the
conduction. In solid electrical insulators the thermal conduction is not zero, as it can be mediated via
oscillations within the material. In liquids and gases thermal conduction is also not zero but low,
mediated via molecular collisions (although convection is the main form of heat transfer). In order to
quantify thermal conduction through a medium let us consider a solid slab of thermal conductor of
thickness 𝐿 and surface area 𝐴 between two reservoirs at temperatures 𝑇1 (or 𝜃1 ) and 𝑇2 (or 𝜃2 ) as
shown below:

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 Thermal 5

𝑇1 (or 𝜃1 ) 𝑇2 (or 𝜃2 )
A

L
𝑑𝑄
If 𝑇1 > 𝑇2 heat will flow from the right to the left at a rate (power) 𝑃 = 𝑑𝑡

Intuitively, we expect the rate of heat flow (the Power, in W) to:

increase as the area increases

increase as the temperature difference increases

decrease as the thickness increases

Putting all these considerations together we can quantify the rate of heat flow for thermal conduction
as:
𝑑𝑄 𝑄 𝑘𝐴∆𝑇
𝑃= 𝑑𝑡
= 𝑡
= 𝐿
(17.13)

where 𝑘 is the coefficient of thermal conductivity (in W m-1 K-1) and ∆𝑇 = 𝑇1 − 𝑇2 = 𝜃1 − 𝜃2 .

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Mechanics and Materials School of Physics and Astronomy