# Thermal Expansion

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## Thermal expansion

Solids, liquids, and gases all exhibit dimensional changes for changes in temperature while pressure is held constant. The molecular mechanisms at work and the methods of data presentation are quite different for the three cases.

The temperature coefficient of linear expansion α_{l} is defined by Eq. (1),

*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 α

_{l}is strongly dependent upon temperature and above which α

_{l}is practically constant. Many common substances are near or above &THgr; at room temperature and follow approximate equation (2),

*l*

_{0}is the length at 0°C and

*t*is the temperature in °C. The total change in length 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),

*p*is absolute pressure,

*v*is specific volume,

*T*is absolute temperature, and

*R*is the so-called gas constant. Real gases often follow this equation closely.

*See*Gas constant

The coefficient of cubic expansion α_{v} is defined by Eq.(4)

*T*. The behavior of real gases is largely accounted for by the van der Waals equation.

*See*Kinetic theory of matter

For liquids, α* _{v}* is somewhat a function of pressure but is largely determined by temperature. Though α

*may often be taken as constant over a sizable range of temperature (as in the liquid expansion thermometer), generally some variation must be accounted for. For example, water contracts with temperature rise from 32 to 39°F (0 to 4°C), above which it expands at an increasing rate.*

_{v}*See*Thermometer

## Thermal expansion

## Thermal Expansion

the dimensional changes exhibited by a substance when it is heated.

A quantitative characterization of thermal expansion at constant pressure is provided by the isobaric thermal expansion coefficient

which is often called the coefficient of volume, or cubical, expansion. In practice the value of α is determined from the formula

Here, *Vʹ* is the volume of the gas, liquid, or solid at the temperature *T*_{2} > *T _{1}; V* is the initial volume of the substance; and the temperature difference

*T*

_{2}–

*T*

_{1}is assumed to be small.

Table 1. Isobaric coefficients of volume expansion of some gases and liquids at 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 CO_{2}) ............... | 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, the coefficient of linear expansion

where *l* is the initial length of the solid in some chosen direction. In the general case of anisotropic solids, α = α_{x} + α_{y} + α_{z}, where the linear expansion coefficients α_{x}, α_{y}, and α_{z} along the *x, y,* and *z* crystallographic axes, respectively, are equal or unequal depending on the symmetry of the crystal. For crystals with cubic symmetry, for example, as for isotropic solids, α_{x} = α_{y} = α_{z} and *α* ≈ 3α_{1}.

For most substances, α > 0. Water, on the other hand, contracts when it is heated from 0° to 4°C at atmospheric pressure. The dependence of *α* on *T* is most pronounced in the cases of gases; for an ideal gas, *α* = 1/*T*. The dependence is less marked for liquids. For a number of substances, such as quartz and Invar, a is 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 expansion of a number of substances at atmospheric pressure.

Table 2. Isobaric coefficients of linear expansion of some solids at atmospheric pressure | ||
---|---|---|

Substance | Temperature (°C) | α_{1} [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 the gas particles as the gas is heated; this energy is used to perform work against the external pressure. In the case of solids and liquids, thermal expansion is associated with the asymmetry (anharmonicity) of the thermal vibrations of the atoms; as a result of this asymmetry, the interatomic distances increase with increasing *T*. The experimental determination of α and α_{1} is carried out by the methods of dilatometry. The thermal expansion of substances is taken into account in the designing of all installations, devices, and machines that operate under variable temperature conditions.

### REFERENCES

Novikova, S. I.*Teplovoe rasshirenie tverdykh tel*. Moscow, 1974.

Hirschfelder, J., C. Curtiss, and R. Bird.

*Molekuliarnaia teoriia gazov i zhidkostei*. Moscow, 1961. (Translated from English.)

Perry, J.

*Spravochnik inzhenera-khimika,*vol. 1. Leningrad, 1969. (Translated from English.)