Description
1 Thermal expansion
The linear thermal expansion coefficient describes the increase in length of a homogenous
material as its temperature changes and is defined by
α1 :=
1
L
∂L
∂T
where L is the length of the material. The subscript in α1 is not standard notation; it
is included to distinguish the linear thermal expansion coefficient from the volume thermal
expansion coefficient.
a) For copper, α1 = 1.65×10−5 K−1. Determine the amount by which a copper rod of length
10 m will expand if it is heated from −10◦ C to 30◦ C, assuming that α is independent
of temperature.
b) A rectangular sheet of any material will expand as its temperature increases. Here the
coefficient of area expansion is defined by
γ :=
1
A
∂A
∂T
where A is the area of the material. Show that
γ = 2α1.
Determine the amount by which the area of a rectangular copper roof, whose sides are
10 m and 4 m increases if it is heated from −10◦ C to 30◦ C.
2 Thermal expansion coefficient for a van der Waals gas
Determine an expression for the thermal expansion coefficient for a van der Waals gas. Hint:
note that differentiation of V w.r.t. T will be difficult. There is an identity that we encountered
in class that will make this easier.
3 Isothermal compressibility of an ideal gas
Determine an expression for the isothermal compressibility of an ideal gas, in terms of N,P
and T. Show that it is positive.
1
4 Equation of state for a solid
The state of a solid material can be described by the same variables as for a gas. Suppose
that the equation of state of the solid is
V = V0 (1 + aT − bP)
where V0 is a constant equal to the volume when pressure and temperature are zero and a
and b are constants that are very small.
a) Determine expressions for the isothermal compressibility and the isobaric expansion
coefficient.
b) Suppose that a and b are so small that at typical temperatures aT # 1 and bP # 1.
Determine approximate expressions for the isothermal compressibility and the isobaric
thermal expansion coefficient in this case. Are they approximately constant or not?
