Thermal Expansion

When you heat a solid or a liquid, the substance always expands, which leads to a decrease in density. This phenomenon can be described in simplified terms with the increasing kinetic energy of the particles, which results in larger distances between the atoms or molecules. Experimentally, a linear correlation between temperature and length changes of a body of length $l_0$ can be assumed for temperature changes that are not too large: $$\Delta l = l_0 \alpha \Delta T$$ where the thermal expansion coefficient $\alpha$ is the corresponding constant of proportionality. For example, if you consider a pipe whose temperature is increased by $\Delta T$, then the length increases to $$l = l_0 + \Delta l$$ which leads to the following relationship for calculating the new length $l$ after inserting the formula for calculating $\Delta T$: $$\boxed{l = l_0(1+\alpha \Delta T)}$$ If one considers the volume of a three-dimensional body instead of the length of an idealized wire, the term in brackets must be multiplied by itself three times: $$V = V_0(1+\alpha \Delta T)^3$$ As long as the changes are small enough, i.e. $\alpha \Delta T \ll 1$ is assumed, the following relation is obtained: $$\boxed{V \approx V_0(1+3 \alpha \Delta T)}$$ which is an approximation, but much easier to handle with pocket calculators. This approximation applies to both solids and liquids. Since liquids spread evenly in all directions, the volume expansion coefficient $\gamma$ must be specified here, which results from the product of the prefactor 3 with the linear expansion coefficient: $$\boxed{\gamma = 3 \alpha }$$ The coefficient of volumetric expansion for most liquids is between 0 and $2\cdot 10^{-3}\,\mathrm{K}^{-1}$. In the case of water, it takes on a negative value at temperatures below $4^\circ$C due to the density anomaly, since the volume here decreases at higher temperature values. At room temperature, on the other hand, it is $\gamma~=~0.21\cdot 10^{-3}\,\mathrm{K}^{-1}$. For comparison, $\gamma$ for ethanol has the value $1.1\cdot 10^{-3}\,\mathrm{K}^{-1}$.

The volume expansion of liquids is used in liquid thermometers to indicate the current temperature or temperature changes on a given temperature scale by the rise of the liquid column in a tube. The tube must be very thin so that even small changes in volume result in a clearly visible change in expansion, thereby minimizing reading errors. In this case, ethanol is mostly used because the coefficient of volumetric expansion of this substance is relatively large. In the past, mercury was often used. Although the expansion coefficient of mercury is almost one order of magnitude smaller than for most of the other liquids, it has the advantage to be quite stable between around $0^\circ\mathrm{C}$ and $180^\circ\mathrm{C}$ (deviation from linearity below 2%). However, in many countries, it is forbidden to be used in thermometers for reasons of health hazards and was replaced by ethanol or other liquids.