LAB # 7 - Thermal Expansion
Introduction
It is a common observation that materials expand when heated. In fact, there is only one known exception
to this, a ceramic material called zirconium tungstate (discovered in 1968). It is the only material known to
contract when heated over a large range. Some materials cheat and contract in one direction while
expanding in others, while other materials contract, but only over a small range. But by far the majority of
materials expand upon heating. In some cases, this thermal expansion is desirable. In the vast majority of
cases, measurements of temperature depend on thermal expansion to function. It may be the expansion of
mercury in a standard thermometer or the expansion of a bimetallic strip in a household thermostat. In
either case, the thermal expansion is used to infer the change in temperature.
In other cases, thermal expansion is a problem. It is necessary to leave gaps in concrete sidewalks and
bridges to ensure that they don't crack or buckle as the temperature rises. Engineers go to great lengths to
find materials that don't change their size as the temperature changes. One common example is Invar, an
alloy of iron and nickel (with trace amounts of other stuff thrown in), which expands very little as the
temperature rises. Part of the excitement surrounding zirconium tungstate is that perhaps by mixing it with
other materials that expand upon heating, materials can be constructed that don't expand at all.
Why do they expand?
The reason that things expand upon heating can be understood by considering the potential function for the
forces between atoms. By considering that (experimentally) it is easier to pull things apart than to
compress them, it is reasonable to conclude that the potential function for an atom near another atom in a
solid (or a liquid or gas, for that matter) should look roughly like:
5
0
V( x )
5
10
0
0.5
1
1.5
2
x
where I show the potential energy an atom (or molecule) in a material has by virtue of the forces it feels
from nearby atoms (or molecules). To see that this makes sense, recall that the force is the negative
gradient of the potential energy. That is:
F (x ) = −
dV ( x )
dx
(1)
So the force is the negative slope of the potential energy. Looking at the potential function, we see first
that the slope is negative and very large for small distances. That means that the force is very large and
positive (in the +x direction), which says that as I try to compress a material, the atoms push back (against
each other and against me) so that it is VERY hard to compress the material. At the other extreme, we can
see that for larger distances, the slope is positive and not so large. That says that as I pull a material apart,
the atoms pull pack and resist me tearing the material apart, but not as hard as it resisted me pushing them
together. (But note that if I pull them TOO far apart, the slope, and thus the force, goes to zero. This is
when the solid or liquid becomes gaseous and the particles don't stay close together.) Finally, we see that at
the bottom of the potential function, the slope is zero (that's what we mean by "bottom"). That is the point
at which an atom is comfortable and feels no force in either direction (we have called this a point of
equilibrium). This distance, along with the mass of the atoms, determines the density of the material (at
least at zero temperature).
The last point to realize about thermal expansion is to recall that temperature is a measure of the average
kinetic energy of the atoms or molecules comprising the material. If an object is "hot", we mean that the
atoms inside are moving very fast and if an object is "cold" we mean that the atoms are moving
comparatively slowly. (But note: "slowly" is a relative term here - in water around freezing, the molecules
are moving about 1500 m/s or about 3000 mph). So consider that a molecule or atom in the above potential
has some amount of kinetic energy. As it vibrates back and forth, the kinetic energy will be converted to
potential energy and back again. And the distance a particle travels before rolling back again will be
determined by the amount of energy it has:
5
0
V( x )
B
5
A
10
0
0.5
1
1.5
2
x
If a particle has enough energy to reach the level of A, it will have maximum speed (i.e., maximum kinetic
energy) when it passes the minimum of potential energy and zero kinetic energy where the line A crosses
the potential energy curve. Thus, a particle with energy A will only reach as far as the points where A
crosses the potential energy curve. But if a particle has energy B, it will oscillate farther before turning and
falling back into the potential curve. And note also that with energy B a particle oscillates farther toward
LARGE distances than toward smaller ones (again - it is easier for atoms and molecules to get further apart
than it is to get closer together). So, as I heat up any material, the particles of which it is composed will be,
ON AVERAGE, farther away from each other. And note that the conclusion we've drawn does NOT
depend on the exact shape of the potential curve we've drawn. All that is necessary is that it be
ASSYMETRIC (that is - it must be harder to push the atoms one way than the other).
So, we've seen that for higher energies (meaning higher temperatures), atoms and molecules move further
from each other. As they move further from each other, the object they comprise will get longer. And
clearly, that increase in length will depend on the temperature. It is straightforward, then, to expand the
length of the object in a Taylor series expansion and obtain:
∆L ≡ L(T ) − L(T0 ) ∝ (T − T0 )
(2)
That is, the length as L(T0) at the original temperature T0 will increase to L(T) as the temperature is
increased to T, and the change in length will be proportional to the increase in temperature (and this will
always be true if the temperature change is not too big). In addition, the increase in length will depend on
the original length. (This can be seen by considering a row of people all standing shoulder to shoulder.
Now let them all increase the distance to their neighbor by an arm length; the total length of the row of
people will have increased by far more than an arm length, and will depend on the number of people in the
row). Then, introducing a constant of proportionality, α, we can write the change in length as:
∆L ≡ L(T ) − L0 = αL0 (T − T0 )
(3)
where I have written L(T0) as L0. The constant of proportionality is called the coefficient of thermal
expansion, and varies for different materials. For example, for copper, α is 17·10-6 /K whereas for Invar, it
is about 1.5·10-6 /K. The goal of the present lab is to test this expression (i.e., determine if the change in
length is in fact proportional to the change in temperature) and if it holds, to determine the constant of
proportionality. It should be clear that the coefficient depends on the shape of the potential energy function
described above. Therefore, it should be possible to learn the shape of the interatomic potential energy by
measuring the thermal expansion coefficient.
The Lab
In the experiment, you will measure the change in length of a rod of aluminum and one of copper as a
function of temperature. The heating will be accomplished by flowing steam around the rod. There is a
provision for measuring the temperature of the rod while simultaneously measuring its length. This will
give you data of the type specified in Eq. 3; change in length as a function of change in temperature.
The rod is placed in a sleeve that is designed to permit the flow of steam to pass. The steam will heat the
rod to something near 100°C. A mercury thermometer mounted through a cork in the sleeve will allow the
temperature of the rod to be monitored. After the rod reaches the highest temperature, disconnect the tube
providing the steam. As the rod cools, you can make periodic measurements of the length using a
micrometer located at the end.
Conclusions
First and foremost, you will have to decide how to plot what data in order to test Eq. 3. Having done so,
you must then decide A) how to decide whether or not Eq. 3 is correct and B) how to extract the coefficient
of thermal expansion from your data if it is. If you can extract α, you must determine an error bound on it
using the standard methods of error propagation. Reference to tables of such coefficients should allow you
to assess the extent to which you are in agreement with those tabulated values. In your error analysis, what
significance is there to the fact that you are able to measure ∆L with an accuracy of a fraction of a micron
(using the micrometer) but you are only able to measure L0 to an accuracy of a fraction of a millimeter?
Finally, by postulating some function for the potential energy function, you can relate the macroscopic
property you measure (fractional change in length per unit temperature) to the microscopic properties of the
material.