# 3.2 Thermal expansion of solids and liquids

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• Define and describe thermal expansion.
• Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
• Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
• Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.

The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion    , the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.

What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.

What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature , an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.

## Linear thermal expansion—thermal expansion in one dimension

The change in length $\text{Δ}L$ is proportional to length $L$ . The dependence of thermal expansion on temperature, substance, and length is summarized in the equation

$\text{Δ}L=\mathrm{\alpha L}\text{Δ}T,$

where $\text{Δ}L$ is the change in length $L$ , $\text{Δ}T$ is the change in temperature, and $\alpha$ is the coefficient of linear expansion    , which varies slightly with temperature.

[link] lists representative values of the coefficient of linear expansion, which may have units of $1/\text{º}\text{C}$ or 1/K. Because the size of a kelvin and a degree Celsius are the same, both $\alpha$ and $\text{Δ}T$ can be expressed in units of kelvins or degrees Celsius. The equation $\text{Δ}L=\mathrm{\alpha L}\text{Δ}T$ is accurate for small changes in temperature and can be used for large changes in temperature if an average value of $\alpha$ is used.

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Abhi
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20/(×-6^2)
Salomon
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Salomon
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Salomon
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Salomon
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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