# Thermal expansion

Problem

Mercury in a glass thermomether expands to encompass the whole glass volume at . For how much does the pressure in mercury rise, when the temperature is raised from to ?

Assumptions:
– Compressibility of mercury is
– Glass is rigid, i.e. it does not expand

Cases:
a) Thermal expansion coefficient of mercury is temperature independent at
b) Thermal expansion coefficient of mercury is at and increases by when the temperature is raised to .

Data

Solution, case a)

In a liquid, such as mercury the equation relating its volume with pressure and temperature is

Since the glass is rigid, its volume does not change, so . Simplifying we obtain a direct relationship between a change in temperature and a change in pressure:

When a liquid is heated from to we expect the pressure to rise from to . As in this case both thermal expansion and compressibilty are independent of temperature and pressure, we can simply state

Finally, the pressure increase is

Solution, case b)

In this case, we have to take into account that the thermal expansion coefficient increases with temperature. It is at and at . We do not know, how it increases, but since the temperature difference is small, we can assume, that it increases linearly. Assuming linear dependence, we write an equation of a line:

We discovered in case a) that the relationship between a change in temperature and a change in pressure is:

As the thermal expansion is now a function of temperature, we should integrate this equation from the state to the final state :

On the pressure side of the equations, compressiblity is constant, so the integral simply transforms the pressure differential to the pressure difference . The integral on the left hand side is evaluated as follows:

Combining the integral on the left and right hand side we finally obtain

Since we assumed linear dependence of thermal expansion with temperature it is not surprising that the solution of case b) is conceptually the same as the solution in case a) but with average thermal expansion . At the same time, we notice, that the error made when neglecting thermal expansion is .