Thermal expansion is the tendency of matter to change in volume in response to a change in temperature, through heat transfer. The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in size per degree change in temperature at a constant pressure. Several types of coefficients have been developed: volumetric, area, and linear. This is used depending on the particular application and which dimensions are considered important. For solids, one might only be concerned with the change along a length, or over some area. Over small temperature ranges, the linear nature of thermal expansion leads to expansion relationships for length, area, and volume in terms of the linear expansion coefficient.
The relationship governing the linear expansion of a long thin rod can be reasoned out as follows:
General volumetric thermal expansion coefficient
In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal expansion is given by
The subscript p indicates that the pressure is held constant during the expansion, and the subscript "V" stresses that it is the volumetric (not linear) expansion that enters this general definition. In the case of a gas, the fact that the pressure is held constant is important, because the volume of a gas will vary appreciably with pressure as well as temperature. For a gas of low density this can be seen from the ideal gas law.
The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write:
where is some area of interest on the object, and is the rate of change of that area per unit change in temperature. The change in the area can be estimated as:
This equation works well as long as the area expansion coefficient does not change much over the change in temperature . If it does, the equation must be integrated. Volume expansion
For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written:
where is the volume of the material, and is the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 °C. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 K, or 0.004% K−1. If we already know the expansion coefficient, then we can calculate the change in volume
where is the fractional change in volume (e.g., 0.002) and is the change in temperature (50 °C). The above example assumes that the expansion coefficient did not change as the temperature changed. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, then the above equation will have to be integrated:
where is the starting temperature and is the volumetric expansion coefficient as a function of temperature T. Expansion in gases
For an ideal gas, the volumetric thermal expansion (i.e., relative change in volume due to temperature change) depends on the type of process in which temperature is changed. Two simple cases are isobaric change, where pressure is held constant, and adiabatic change, where no heat is exchanged with the environment. In an isobaric process, the volumetric thermal expansively, which we denote , is given by the ideal gas law:
The index denotes an isobaric process....
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