Linear Expansion
When a solid or liquid is heated, it generally expands, resulting in a decrease in density. This phenomenon can be explained by the increase in the kinetic energy of the particles, which causes greater distances between atoms or molecules. Experimentally, for small temperature changes, a linear relationship is observed between the change in temperature and the change in length of a body of initial length $l_0$:
\begin{equation}
\Delta l = l_0 \alpha \Delta T
\end{equation}
where $\alpha$ is the thermal expansion coefficient and acts as the proportionality constant. For example, if a pipe is heated by a temperature difference $\Delta T$, its length increases to:
\begin{equation}
l = l_0 + \Delta l
\end{equation}
which can be rearranged to calculate the new length $l$ as:
\begin{equation}
{l = l_0(1 + \alpha \Delta T)}
\end{equation}
Volumetric Expansion
If instead of the length of an idealized wire, we consider the volume of a three-dimensional body, the term in parentheses must be cubed:
\begin{equation}
V = V_0(1 + \alpha \Delta T)^3
\end{equation}
As long as the changes are small, i.e., $\alpha \Delta T \ll 1$, this expression can be approximated as:
\begin{equation}
{V = V_0(1 + 3\alpha \Delta T)}
\end{equation}
which is often easier to compute with a calculator. This approximation is valid for both solids and liquids. Since liquids expand uniformly in all directions, a volumetric expansion coefficient $\gamma$ must be used instead, which is defined as the product of 3 and the linear expansion coefficient:
\begin{equation}
{\gamma = 3 \alpha}
\end{equation}
The volumetric expansion coefficient for most liquids ranges from 0 to $2 \cdot 10^{-3},\mathrm{K}^{-1}$. In the case of water, it takes on a negative value at temperatures below $4^\circ$C due to its density anomaly, since the volume decreases with increasing temperature in this range. At room temperature, water’s volumetric expansion coefficient is $\gamma = 0.21 \cdot 10^{-3},\mathrm{K}^{-1}$, while for ethanol it is $1.1 \cdot 10^{-3},\mathrm{K}^{-1}$.
The volumetric expansion of liquids is used in liquid thermometers, where the rise of the liquid column in a narrow capillary tube displays the current temperature or temperature changes on a predefined scale. The capillary must be very narrow so that even small changes in volume result in a clearly visible height change, minimizing reading errors. Ethanol is often used because of its relatively high expansion coefficient. Mercury was commonly used in the past but is now banned for such use in Germany due to health concerns.
Thermal expansion can have negative consequences in engineering, for example in bridge construction. Expansion joints are typically installed to prevent damage from large temperature fluctuations.