Solids, liquids, and gases all exhibit dimensional changes for changes intemperature . The molecular mechanisms at workand the methods of data presentation are quite different for the three cases.
The temperature coefficient of linear expansion αl is defined by Eq. (1),
(1) 
where l is the length of the specimen, t is the temperature, and p is the pressure.For each solid there is a Debye characteristic temperature &THgr;, below which αlis strongly dependent upon temperature and above which αl is practicallyconstant. Many common substances are near or above &THgr; at roomtemperature and follow approximate equation (2),
(2) 
where l0 is the length at 0°C and t is the temperature in °C. The total change inlength from absolute zero to the melting point has a range of approximately 2%for most substances.
So-called perfect gases follow the relation in Eq. (3),
(3) 
where p is absolute pressure, v is specific volume, T is absolute temperature, andR is the so-called gas constant. Real gases often follow this equation closely.
The coefficient of cubic expansion αv is defined by Eq.(4)
, and for a perfect gas this is found to be 1/T. The behavior of real gases is largelyaccounted for by the van der Waals equation. See Kinetic theory of matter
For liquids, αv is somewhat a function of pressure but is largely determined bytemperature. Though αv may often be taken as constant over a sizable range oftemperature (as in the liquid expansion thermometer), generally some variationmust be accounted for. For example, water contracts with temperature rise from32 to 39°F (0 to 4°C), above which it expands at an increasing rateA quantitative characterization of thermal expansion at constant pressure isprovided by the isobaric thermal expansion coefficient
which is often called the coefficient of volume, or cubical, expansion. In practicethe value of α is determined from the formula
Here, Vʹ is the volume of the gas, liquid, or solid at the temperature T2 > T1; V isthe initial volume of the substance; and the temperature difference T2 – T1 isassumed to be small.
| Table 1. Isobaric coefficients of volume expansion of some gases and liquidsat atmospheric pressure | ||
|---|---|---|
| Substance | Temperature (°C) | α [10–3(°C)–1] |
| Gases | ||
| Helium ............... | 0–100 | 3.658 |
| Hydrogen ............... | 0–100 | 3.661 |
| Oxygen ............... | 0–100 | 3.665 |
| Nitrogen ............... | 0–100 | 3.674 |
| Air (without CO2) ............... | 0–100 | 3.671 |
| Liquids | ||
| Water ............... | 10 | 0.0879 |
| 20 | 0.2066 | |
| 80 | 0.6413 | |
| Mercury ............... | 20 | 0.182 |
| Glycerol ............... | 20 | 0.500 |
| Benzene ............... | 20 | 1.060 |
| Acetone ............... | 20 | 1.430 |
| Ethyl alcohol ............... | 20 | 1.659 |
The thermal expansion of solids is characterized by, in addition to α, thecoefficient of linear expansion
where l is the initial length of the solid in some chosen direction. In the generalcase of anisotropic solids, α = αx + αy + αz, where the linear expansioncoefficients αx, αy, and αz along the x, y, and z crystallographic axes,respectively, are equal or unequal depending on the symmetry of the crystal. Forcrystals with cubic symmetry, for example, as for isotropic solids, αx = αy = αz andα ≈ 3α1.
For most substances, α > 0. Water, on the other hand, contracts when it is heatedfrom 0° to 4°C at atmospheric pressure. The dependence of α on T is mostpronounced in the cases of gases; for an ideal gas, α = 1/T. The dependence isless marked for liquids. For a number of substances, such as quartz and Invar, ais small and is virtually constant over a broad range of temperatures. As T → 0, α→ 0. Tables 1 and 2 give the isobaric coefficients of volume and linear expansionof a number of substances at atmospheric pressure.
| Table 2. Isobaric coefficients of linear expansion of some solids atatmospheric pressure | ||
|---|---|---|
| Substance | Temperature (°C) | α1 [10–6(°C)–1 |
| Carbon | ||
| diamond ............... | 20 | 1.2 |
| graphite ............... | 20 | 79 |
| Silicon ............... | 3–18 | 25 |
| Quartz | ||
| parallel to axis ............... | 40 | 78 |
| perpendicular to axis ............... | 40 | 14.1 |
| fused ............... | 0–100 | 0.384 |
| Glass | ||
| crown ............... | 0–100 | ∼9 |
| flint ............... | 0–100 | ∼7 |
| Tungsten ............... | 25 | 4.5 |
| Copper ............... | 25 | 16.6 |
| Brass ............... | 20 | 18.9 |
| Aluminum ............... | 25 | 25 |
| Iron ............... | 25 | 12 |
The thermal expansion of a gas is due to the increase in the kinetic energy of thegas particles as the gas is heated; this energy is used to perform work against theexternal pressure. In the case of solids and liquids, thermal expansion isassociated with the asymmetry (anharmonicity) of the thermal vibrations of theatoms; as a result of this asymmetry, the interatomic distances increase withincreasing T. The experimental determination of α and α1 is carried out by themethods of dilatometry. The thermal expansion of substances is taken intoaccount in the designing of all installations, devices, and machines that operateunder variable temperature conditions.