Thermal expansion of liquids and solids

References: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 1.7 – 1.8

The volume thermal expansion coefficient of a substance as its temperature is increased at constant pressure is defined as the fractional change in volume per degree kelvin, that is

\displaystyle  \beta\equiv\frac{\Delta V/V}{\Delta T} \ \ \ \ \ (1)

Example 1 For mercury, {\beta=1.80\times10^{-4}\mbox{ K}^{-1}} so if we have a typical mercury thermometer with a cylindrical bulb {h=1\mbox{ cm}} long and with a radius of {r=0.2\mbox{ cm}}, and the scale on the thermometer is 1 mm per degree, then we can work out the inside diameter {2\rho} of the tube. We get

\displaystyle   V \displaystyle  = \displaystyle  \pi r^{2}h=1.26\times10^{-7}\mbox{ m}^{3}\ \ \ \ \ (2)
\displaystyle  \Delta V \displaystyle  = \displaystyle  \beta V\Delta T\ \ \ \ \ (3)
\displaystyle  \displaystyle  = \displaystyle  \left(1.80\times10^{-4}\right)\left(1.26\times10^{-7}\right)\left(1\right)\ \ \ \ \ (4)
\displaystyle  \displaystyle  = \displaystyle  2.27\times10^{-11}\mbox{ m}^{3}\ \ \ \ \ (5)
\displaystyle  \rho \displaystyle  = \displaystyle  \sqrt{\frac{\Delta V}{\pi\left(10^{-3}\mbox{ m}\right)}}\ \ \ \ \ (6)
\displaystyle  \displaystyle  = \displaystyle  8.5\times10^{-5}\mbox{ m} \ \ \ \ \ (7)

The diameter is therefore about 0.2 mm.

Example 2 For water, {\beta} varies a lot in the liquid region. At {100^{\circ}\mbox{C}}, it is {7.5\times10^{-4}\mbox{ K}^{-1}} and decreases to zero at {4^{\circ}\mbox{ C}}. Between the freezing point at {0^{\circ}\mbox{C}} and {4^{\circ}\mbox{ C}}, {\beta} is actually negative, with its largest negative value of {\beta=-0.68\times10^{-4}\mbox{ K}^{-1}} at the freezing point. That is, melting ice actually contracts (becomes denser) as its temperature increases to {4^{\circ}\mbox{ C}} which is the reason that ice floats. If {\beta} were positive over the entire liquid range of water, a cooling lake would start to freeze from the bottom up rather than from the top down as it does in nature.

Incidentally, you might think that because {\beta>0} for temperatures between {4^{\circ}\mbox{ C}} and {100^{\circ}\mbox{ C}}, ice might sink in hot water (before it melts, of course). However, at standard pressure, the density of boiling water is {0.9584\mbox{ g cm}^{-3}} while the density of ice at {0^{\circ}\mbox{ C}} is {0.9167\mbox{ g cm}^{-3}} so ice floats even in boiling water.

For solids, we can define a linear thermal expansion coefficient as the fractional change of length per degree of increase in temperature:

\displaystyle  \alpha\equiv\frac{\Delta L/L}{\Delta T} \ \ \ \ \ (8)

Example 3 For steel, {\alpha=1.1\times10^{-5}\mbox{ K}^{-1}}. Assuming this value is constant over the range of outdoor air temperatures, we can estimate the change in length of a 1 km steel bridge between winter and summer. In Dundee, the temperature doesn’t vary as much as in more continental locations, but we’ll take a cold day in Dundee to be {0^{\circ}\mbox{ C}} and a hot day to be {25^{\circ}\mbox{ C}}, so {\Delta T=25}. The change in length is therefore

\displaystyle  \Delta L=\left(1.1\times10^{-5}\right)\left(25\right)\left(10^{3}\right)=0.275\mbox{ m} \ \ \ \ \ (9)

Thus the change in length is far from negligible, which is the reason why long bridges are built in sections with expansion joints in between.

Example 4 One type of thermometer consists of a spiral consisting of two different metal strips (with different values of {\alpha}) bonded together. Since the metals expand at different rates, the spiral winds and unwinds as the temperature changes. A dial can be attached to the end of the spiral to measure its position and thus give a measure of temperature.

Example 5 If a solid is not isotropic, it has different values of {\alpha} in each direction, so in rectangular coordinates we have {\alpha_{x}}, {\alpha_{y}} and {\alpha_{z}} defined as {\Delta x/\left(x\Delta T\right)} and so on. For a rectangular solid we have

\displaystyle   V \displaystyle  = \displaystyle  xyz\ \ \ \ \ (10)
\displaystyle  \Delta V \displaystyle  = \displaystyle  \left(x+\Delta x\right)\left(y+\Delta y\right)\left(z+\Delta z\right)-xyz\ \ \ \ \ (11)
\displaystyle  \displaystyle  = \displaystyle  yz\Delta x+xz\Delta y+xy\Delta z+\mathcal{O}\left(\Delta x^{2}\right)\ \ \ \ \ (12)
\displaystyle  \frac{\Delta V}{V} \displaystyle  = \displaystyle  \frac{\Delta x}{x}+\frac{\Delta y}{y}+\frac{\Delta z}{z}+\mathcal{O}\left(\Delta x^{2}\right)\ \ \ \ \ (13)
\displaystyle  \frac{\Delta V}{V\Delta T} \displaystyle  = \displaystyle  \alpha_{x}+\alpha_{y}+\alpha_{z}+\mathcal{O}\left(\Delta x^{2}\right)\ \ \ \ \ (14)
\displaystyle  \displaystyle  = \displaystyle  \beta \ \ \ \ \ (15)

Thus to first order in changes in length

\displaystyle  \beta=\alpha_{x}+\alpha_{y}+\alpha_{z} \ \ \ \ \ (16)

4 thoughts on “Thermal expansion of liquids and solids

  1. Pingback: Measuring heat capacity at constant volume | Physics pages

    1. gwrowe Post author

      That’s right. {\mathcal{O}} stands for ‘order’ and {\mathcal{O}\left(x^{2}\right)} means terms of order {x^{2}} (and higher) in an expansion such as a Taylor series, where {x} is small.

  2. Pingback: Heat capacities using Maxwell relations | Physics pages

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