# Thermal expansion

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Thermal expansion is the tendency of matter to change its shape, area, and volume in response to a change in temperature. [1]

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material type.

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space.

## Contents

Temperature is a monotonic function of the average molecular kinetic energy of a substance. When a substance is heated, the kinetic energy of its molecules increases. Thus, the molecules begin vibrating/moving more and usually maintain a greater average separation. Materials which contract with increasing temperature are unusual; this effect is limited in size, and only occurs within limited temperature ranges (see examples below). The relative expansion (also called strain) divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies with temperature.

In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

In physics, the kinetic energy (KE) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

## Overview

### Predicting expansion

If an equation of state is available, it can be used to predict the values of the thermal expansion at all the required temperatures and pressures, along with many other state functions.

In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

In thermodynamics, a state function or function of state or point function is a function defined for a system relating several state variables or state quantities that depends only on the current equilibrium state of the system, for example a gas, a liquid, a solid, crystal, or emulsion. State functions do not depend on the path by which the system arrived at its present state. A state function describes the equilibrium state of a system and thus also describes the type of system. For example, a state function could describe an atom or molecule in a gaseous, liquid, or solid form; a heterogeneous or homogeneous mixture; and the amounts of energy required to create such systems or change them into a different equilibrium state.

### Contraction effects (negative thermal expansion)

A number of materials contract on heating within certain temperature ranges; this is usually called negative thermal expansion, rather than "thermal contraction". For example, the coefficient of thermal expansion of water drops to zero as it is cooled to 3.983 °C and then becomes negative below this temperature; this means that water has a maximum density at this temperature, and this leads to bodies of water maintaining this temperature at their lower depths during extended periods of sub-zero weather. Also, fairly pure silicon has a negative coefficient of thermal expansion for temperatures between about 18 and 120 kelvins. [2]

Negative thermal expansion (NTE) is an unusual physicochemical process in which some materials contract upon heating, rather than expand as most other materials do. Materials which undergo NTE have a range of potential engineering, photonic, electronic, and structural applications. For example, if one were to mix a negative thermal expansion material with a "normal" material which expands on heating, it could be possible to make a zero expansion composite material.

The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume:

The kelvin is the base unit of temperature in the International System of Units (SI), having the unit symbol K. It is named after the Belfast-born, Glasgow University engineer and physicist William Thomson, 1st Baron Kelvin (1824–1907).

### Factors affecting thermal expansion

Unlike gases or liquids, solid materials tend to keep their shape when undergoing thermal expansion.

Thermal expansion generally decreases with increasing bond energy, which also has an effect on the melting point of solids, so, high melting point materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids. The thermal expansion of glasses is higher compared to that of crystals. [3] At the glass transition temperature, rearrangements that occur in an amorphous material lead to characteristic discontinuities of coefficient of thermal expansion and specific heat. These discontinuities allow detection of the glass transition temperature where a supercooled liquid transforms to a glass. [4]

The melting point of a substance is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at a standard pressure such as 1 atmosphere or 100 kPa.

Glass is a non-crystalline, amorphous solid that is most often transparent and has widespread practical, technological, and decorative uses in, for example, window panes, tableware, optics and optoelectronics. The most familiar, and historically the oldest, types of manufactured glass are "silicate glasses" based on the chemical compound silica (silicon dioxide, or quartz), the primary constituent of sand. The term glass, in popular usage, is often used to refer only to this type of material, which is familiar from use as window glass and in glass bottles. Of the many silica-based glasses that exist, ordinary glazing and container glass is formed from a specific type called soda-lime glass, composed of approximately 75% silicon dioxide (SiO2), sodium oxide (Na2O) from sodium carbonate (Na2CO3), calcium oxide (CaO), also called lime, and several minor additives.

Absorption or desorption of water (or other solvents) can change the size of many common materials; many organic materials change size much more due to this effect than due to thermal expansion. Common plastics exposed to water can, in the long term, expand by many percent.

Sorption is a physical and chemical process by which one substance becomes attached to another. Specific cases of sorption are treated in the following articles:

## Coefficient of thermal expansion

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. The choice of coefficient depends 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.

The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient, and the most relevant for fluids. In general, substances expand or contract when their temperature changes, with expansion or contraction occurring in all directions. Substances that expand at the same rate in every direction are called isotropic. For isotropic materials, the area and volumetric thermal expansion coefficient are, respectively, approximately twice and three times larger than the linear thermal expansion coefficient.

Mathematical definitions of these coefficients are defined below for solids, liquids, and gases.

### General volumetric thermal expansion coefficient

In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal expansion is given by

${\displaystyle \alpha _{V}={\frac {1}{V}}\,\left({\frac {\partial V}{\partial T}}\right)_{p}}$

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.

## Expansion in solids

When calculating thermal expansion it is necessary to consider whether the body is free to expand or is constrained. If the body is free to expand, the expansion or strain resulting from an increase in temperature can be simply calculated by using the applicable coefficient of Thermal Expansion.

If the body is constrained so that it cannot expand, then internal stress will be caused (or changed) by a change in temperature. This stress can be calculated by considering the strain that would occur if the body were free to expand and the stress required to reduce that strain to zero, through the stress/strain relationship characterised by the elastic or Young's modulus. In the special case of solid materials, external ambient pressure does not usually appreciably affect the size of an object and so it is not usually necessary to consider the effect of pressure changes.

Common engineering solids usually have coefficients of thermal expansion that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, practical calculations can be based on a constant, average, value of the coefficient of expansion.

### Linear expansion

Linear expansion means change in one dimension (length) as opposed to change in volume (volumetric expansion). To a first approximation, the change in length measurements of an object due to thermal expansion is related to temperature change by a "Coefficient of linear thermal expansion" CLTE. It is the fractional change in length per degree of temperature change. Assuming negligible effect of pressure, we may write:

${\displaystyle \alpha _{L}={\frac {1}{L}}\,{\frac {dL}{dT}}}$

where ${\displaystyle L}$ is a particular length measurement and ${\displaystyle dL/dT}$ is the rate of change of that linear dimension per unit change in temperature.

The change in the linear dimension can be estimated to be:

${\displaystyle {\frac {\Delta L}{L}}=\alpha _{L}\Delta T}$

This estimation works well as long as the linear-expansion coefficient does not change much over the change in temperature ${\displaystyle \Delta T}$, and the fractional change in length is small ${\displaystyle \Delta L/L\ll 1}$. If either of these conditions does not hold, the exact differential equation (using ${\displaystyle dL/dT}$) must be integrated.

#### Effects on strain

For solid materials with a significant length, like rods or cables, an estimate of the amount of thermal expansion can be described by the material strain, given by ${\displaystyle \epsilon _{\mathrm {thermal} }}$ and defined as:

${\displaystyle \epsilon _{\mathrm {thermal} }={\frac {(L_{\mathrm {final} }-L_{\mathrm {initial} })}{L_{\mathrm {initial} }}}}$

where ${\displaystyle L_{\mathrm {initial} }}$ is the length before the change of temperature and ${\displaystyle L_{\mathrm {final} }}$ is the length after the change of temperature.

For most solids, thermal expansion is proportional to the change in temperature:

${\displaystyle \epsilon _{\mathrm {thermal} }\propto \Delta T}$

Thus, the change in either the strain or temperature can be estimated by:

${\displaystyle \epsilon _{\mathrm {thermal} }=\alpha _{L}\Delta T}$

where

${\displaystyle \Delta T=(T_{\mathrm {final} }-T_{\mathrm {initial} })}$

is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit, degrees Rankine, degrees Celsius, or kelvins, and ${\displaystyle \alpha _{L}}$ is the linear coefficient of thermal expansion in "per degree Fahrenheit", "per degree Rankine", “per degree Celsius”, or “per kelvin”, denoted by °F−1, R−1, °C−1, or K−1, respectively. In the field of continuum mechanics, the thermal expansion and its effects are treated as eigenstrain and eigenstress.

### Area expansion

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:

${\displaystyle \alpha _{A}={\frac {1}{A}}\,{\frac {dA}{dT}}}$

where ${\displaystyle A}$ is some area of interest on the object, and ${\displaystyle dA/dT}$ is the rate of change of that area per unit change in temperature.

The change in the area can be estimated as:

${\displaystyle {\frac {\Delta A}{A}}=\alpha _{A}\Delta T}$

This equation works well as long as the area expansion coefficient does not change much over the change in temperature ${\displaystyle \Delta T}$, and the fractional change in area is small ${\displaystyle \Delta A/A\ll 1}$. If either of these conditions does not hold, 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: [5]

${\displaystyle \alpha _{V}={\frac {1}{V}}\,{\frac {dV}{dT}}}$

where ${\displaystyle V}$ is the volume of the material, and ${\displaystyle dV/dT}$ 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 K. 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

${\displaystyle {\frac {\Delta V}{V}}=\alpha _{V}\Delta T}$

where ${\displaystyle \Delta V/V}$ is the fractional change in volume (e.g., 0.002) and ${\displaystyle \Delta T}$ is the change in temperature (50 °C).

The above example assumes that the expansion coefficient did not change as the temperature changed and the increase in volume is small compared to the original volume. 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, or the increase in volume is significant, then the above equation will have to be integrated:

${\displaystyle \ln \left({\frac {V+\Delta V}{V}}\right)=\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,dT}$
${\displaystyle {\frac {\Delta V}{V}}=\exp \left(\int _{T_{i}}^{T_{f}}\alpha _{V}(T)\,dT\right)-1}$

where ${\displaystyle \alpha _{V}(T)}$ is the volumetric expansion coefficient as a function of temperature T, and ${\displaystyle T_{i}}$,${\displaystyle T_{f}}$ are the initial and final temperatures respectively.

#### Isotropic materials

For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:

${\displaystyle \alpha _{V}=3\alpha _{L}}$

This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length L. The original volume will be ${\displaystyle V=L^{3}}$ and the new volume, after a temperature increase, will be

${\displaystyle V+\Delta V=(L+\Delta L)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}\approx L^{3}+3L^{2}\Delta L=V+3V{\Delta L \over L}.}$

We can easily ignore the terms as change in L is a small quantity which on squaring gets much smaller.

So

${\displaystyle {\frac {\Delta V}{V}}=3{\Delta L \over L}=3\alpha _{L}\Delta T.}$

The above approximation holds for small temperature and dimensional changes (that is, when ${\displaystyle \Delta T}$ and ${\displaystyle \Delta L}$ are small); but it does not hold if we are trying to go back and forth between volumetric and linear coefficients using larger values of ${\displaystyle \Delta T}$. In this case, the third term (and sometimes even the fourth term) in the expression above must be taken into account.

Similarly, the area thermal expansion coefficient is two times the linear coefficient:

${\displaystyle \alpha _{A}=2\alpha _{L}}$

This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just ${\displaystyle L^{2}}$. Also, the same considerations must be made when dealing with large values of ${\displaystyle \Delta T}$.

Thermal stress : We might surely think of a condition when we prevent a body from getting elongated. You might realise that the body might acquire some kind of stree, force restoring it per unit area. This is called thermal stress.

We take the case of a steel rail of length 5 m and area of cross section 40 sq.cm which we prevent from expanding bwhile the temperature rises by 10 Kelvin.The coefficient of linear expansion =1.2*10-5 K-1

Thus compressive strain=1.2*10-5*10=1.2*10-4 (unitless)

Let, Young Modulus of steel=2*1011 N/m-2

Hence, thermal stresses=2.4*107 Nm-2

Hence, The applied external force= 2.4*107*40*10-4 N

= 9.6*104 N

Anisotropic substances:

#### Anisotropic materials

Materials with anisotropic structures, such as crystals (with less than cubic symmetry, for example martensitic phases) and many composites, will generally have different linear expansion coefficients ${\displaystyle {\frac {}{}}\alpha _{L}}$ in different directions. As a result, the total volumetric expansion is distributed unequally among the three axes. If the crystal symmetry is monoclinic or triclinic, even the angles between these axes are subject to thermal changes. In such cases it is necessary to treat the coefficient of thermal expansion as a tensor with up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by x-ray powder diffraction. The thermal expansion coefficient tensor for the materials possessing cubic symmetry (for e.g. FCC, BCC) is isotropic [6] .

## Isobaric 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 constant pressure (an isobaric process ) and constant volume (an isochoric process ).

The derivative of the ideal gas law, ${\displaystyle PV=T}$, is

${\displaystyle PdV+VdP=dT}$

where ${\displaystyle P}$ is the pressure, ${\displaystyle V}$ is the specific volume, and ${\displaystyle T}$ is temperature measured in energy units.

By definition of an isobaric thermal expansion, we have ${\displaystyle dP=0}$, so that ${\displaystyle PdV=dT}$, and the isobaric thermal expansion coefficient is

${\displaystyle \alpha _{P}\equiv {\frac {1}{V}}\left({\frac {dV}{dT}}\right)={\frac {1}{V}}\left({\frac {1}{P}}\right)={\frac {1}{PV}}={\frac {1}{T}}}$.

Similarly, if the volume is held constant, that is if ${\displaystyle dV=0}$, we have ${\displaystyle VdP=dT}$, so that the isochoric thermal expansion coefficient is

${\displaystyle \alpha _{V}\equiv {\frac {1}{P}}\left({\frac {dP}{dT}}\right)={\frac {1}{P}}\left({\frac {1}{V}}\right)={\frac {1}{PV}}={\frac {1}{T}}}$.

## Expansion in liquids

Theoretically, the coefficient of linear expansion can be found from the coefficient of volumetric expansion (αV  3αL). For liquids, αL is calculated through the experimental determination of αV. Liquids, unlike solids have no definite shape and they take the shape of the container. Consequently, liquids have no definite length and area, so linear and areal expansions of liquids have no significance.

Liquids in general, expand on heating. However water is an exception to this general behaviour: below 4 °C it contracts on heating. For higher temperature it shows the normal positive thermal expansion. The thermal expansion of liquids is usually higher than in solids because of weak intermolecular forces present in liquids.

Thermal expansion of solids usually shows little dependence on temperature, except at low temperatures, whereas liquids expand at different rates at different temperatures.

### Apparent and absolute expansion of a liquid

The expansion of liquids is usually measured in a container. When a liquid expands in a vessel, the vessel expands along with the liquid. Hence the observed increase in volume of the liquid level is not actual increase in its volume. The expansion of the liquid relative to the container is called its apparent expansion, while the actual expansion of the liquid is called real expansion or absolute expansion. The ratio of apparent increase in volume of the liquid per unit rise of temperature to the original volume is called its coefficient of apparent expansion.

For small and equal rises in temperature, the increase in volume (real expansion) of a liquid is equal to the sum of the apparent increase in volume (apparent expansion) of the liquid and the increase in volume of the containing vessel. Thus a liquid has two coefficients of expansion.

Measurement of the expansion of a liquid must account for the expansion of the container as well. For example, when a flask with a long narrow stem, containing enough liquid to partially fill the stem itself, is placed in a heat bath, the height of the liquid column in the stem will initially drop, followed immediately by a rise of that height until the whole system of flask, liquid and heat bath has warmed through. The initial drop in the height of the liquid column is not due to an initial contraction of the liquid, but rather to the expansion of the flask as it contacts the heat bath first. Soon after, the liquid in the flask is heated by the flask itself and begins to expand. Since liquids typically have a greater expansion over solids, the expansion of the liquid in the flask eventually exceeds that of the flask, causing the level of liquid in the flask to rise. A direct measurement of the height of the liquid column is a measurement of the apparent expansion of the liquid. The absolute expansion of the liquid is the apparent expansion corrected for the expansion of the containing vessel. [7]

## Examples and applications

The expansion and contraction of materials must be considered when designing large structures, when using tape or chain to measure distances for land surveys, when designing molds for casting hot material, and in other engineering applications when large changes in dimension due to temperature are expected.

Thermal expansion is also used in mechanical applications to fit parts over one another, e.g. a bushing can be fitted over a shaft by making its inner diameter slightly smaller than the diameter of the shaft, then heating it until it fits over the shaft, and allowing it to cool after it has been pushed over the shaft, thus achieving a 'shrink fit'. Induction shrink fitting is a common industrial method to pre-heat metal components between 150 °C and 300 °C thereby causing them to expand and allow for the insertion or removal of another component.

There exist some alloys with a very small linear expansion coefficient, used in applications that demand very small changes in physical dimension over a range of temperatures. One of these is Invar 36, with αL approximately equal to 0.6×106 K−1. These alloys are useful in aerospace applications where wide temperature swings may occur.

Pullinger's apparatus is used to determine the linear expansion of a metallic rod in the laboratory. The apparatus consists of a metal cylinder closed at both ends (called a steam jacket). It is provided with an inlet and outlet for the steam. The steam for heating the rod is supplied by a boiler which is connected by a rubber tube to the inlet. The center of the cylinder contains a hole to insert a thermometer. The rod under investigation is enclosed in a steam jacket. One of its ends is free, but the other end is pressed against a fixed screw. The position of the rod is determined by a micrometer screw gauge or spherometer.

To determine the coefficient of linear thermal expansion of a metal, a pipe made of that metal is heated by passing steam through it. One end of the pipe is fixed securely and the other rests on a rotating shaft, the motion of which is indicated by a pointer. A suitable thermometer records the pipe's temperature. This enables calculation of the relative change in length per degree temperature change.

The control of thermal expansion in brittle materials is a key concern for a wide range of reasons. For example, both glass and ceramics are brittle and uneven temperature causes uneven expansion which again causes thermal stress and this might lead to fracture. Ceramics need to be joined or work in concert with a wide range of materials and therefore their expansion must be matched to the application. Because glazes need to be firmly attached to the underlying porcelain (or other body type) their thermal expansion must be tuned to 'fit' the body so that crazing or shivering do not occur. Good example of products whose thermal expansion is the key to their success are CorningWare and the spark plug. The thermal expansion of ceramic bodies can be controlled by firing to create crystalline species that will influence the overall expansion of the material in the desired direction. In addition or instead the formulation of the body can employ materials delivering particles of the desired expansion to the matrix. The thermal expansion of glazes is controlled by their chemical composition and the firing schedule to which they were subjected. In most cases there are complex issues involved in controlling body and glaze expansion, so that adjusting for thermal expansion must be done with an eye to other properties that will be affected, and generally trade-offs are necessary.

Thermal expansion can have a noticeable effect on gasoline stored in above-ground storage tanks, which can cause gasoline pumps to dispense gasoline which may be more compressed than gasoline held in underground storage tanks in winter, or less compressed than gasoline held in underground storage tanks in summer. [9]

Heat-induced expansion has to be taken into account in most areas of engineering. A few examples are:

• Metal-framed windows need rubber spacers.
• Rubber tires need to perform well over a range of temperatures, being passively heated or cooled by road surfaces and weather, and actively heated by mechanical flexing and friction.
• Metal hot water heating pipes should not be used in long straight lengths.
• Large structures such as railways and bridges need expansion joints in the structures to avoid sun kink.
• One of the reasons for the poor performance of cold car engines is that parts have inefficiently large spacings until the normal operating temperature is achieved.
• A gridiron pendulum uses an arrangement of different metals to maintain a more temperature stable pendulum length.
• A power line on a hot day is droopy, but on a cold day it is tight. This is because the metals expand under heat.
• Expansion joints absorb the thermal expansion in a piping system. [10]
• Precision engineering nearly always requires the engineer to pay attention to the thermal expansion of the product. For example, when using a scanning electron microscope small changes in temperature such as 1 degree can cause a sample to change its position relative to the focus point.
• Liquid thermometers contain a liquid (usually mercury or alcohol) in a tube, which constrains it to flow in only one direction when its volume expands due to changes in temperature.
• A bi-metal mechanical thermometer uses a bimetallic strip and bends due to the differing thermal expansion of the two metals.

## Thermal expansion coefficients for various materials

This section summarizes the coefficients for some common materials.

For isotropic materials the coefficients linear thermal expansion α and volumetric thermal expansion αV are related by αV = 3α. For liquids usually the coefficient of volumetric expansion is listed and linear expansion is calculated here for comparison.

For common materials like many metals and compounds, the thermal expansion coefficient is inversely proportional to the melting point. [11] In particular for metals the relation is:

${\displaystyle \alpha \approx {\frac {0.020}{M_{P}}}}$

for halides and oxides

${\displaystyle \alpha \approx {\frac {0.038}{M_{P}}}-7.0\cdot 10^{-6}\,\mathrm {K} ^{-1}}$

In the table below, the range for α is from 10−7 K−1 for hard solids to 10−3 K−1 for organic liquids. The coefficient α varies with the temperature and some materials have a very high variation ; see for example the variation vs. temperature of the volumetric coefficient for a semicrystalline polypropylene (PP) at different pressure, and the variation of the linear coefficient vs. temperature for some steel grades (from bottom to top: ferritic stainless steel, martensitic stainless steel, carbon steel, duplex stainless steel, austenitic steel). The highest linear coefficient in a solid has been reported for a Ti-Nb alloy [12] .

(The formula αV  3α is usually used for solids.) [13]

MaterialLinear
coefficient CLTE α
at 20 °C
(10−6 K−1)
Volumetric
coefficient αV
at 20 °C
(10−6 K−1)
Notes
Aluminium 23.169
Aluminium nitride 4.2 a-axis, 5.3 c-axis [14] 13.7AlN is anisotropic
Benzocyclobutene 42126
Brass 1957
Carbon steel 10.832.4
CFRP – 0.8 [15] Anisotropic Fiber direction
Concrete 1236
Copper 1751
Diamond 13
Ethanol 250750 [16]
Gallium(III) arsenide 5.817.4
Gasoline 317950 [13]
Glass 8.525.5
Glass, borosilicate 3.3 [17] 9.9matched sealing partner for tungsten, molybdenum and kovar.
Glass (Pyrex)3.2 [18]
Glycerine 485 [18]
Gold 1442
Helium 36.65 [18]
Ice 51
Indium phosphide 4.613.8
Invar 1.23.6
Iron 11.835.4
Kapton 20 [19] 60DuPont Kapton 200EN
Macor 9.3 [20]
Magnesium 2678
Mercury 61182 [18] [21]
Molybdenum 4.814.4
Nickel 1339
Oak 54 [22] Perpendicular to the grain
Douglas-fir 27 [23] 75radial
Douglas-fir 45 [23] 75tangential
Douglas-fir 3.5 [23] 75parallel to grain
Platinum 927
PP 150450[ citation needed ]
PVC 52156
Quartz (fused (pure glass))0.591.77
alpha-Quartz 12-16/6-9 [24] Parallel to a-axis/c-axis T = -50 to 150 C
Rubber disputeddisputedsee Talk
Sapphire 5.3 [25] Parallel to C axis, or [001]
Silicon Carbide 2.77 [26] 8.31
Silicon 2.56 [27] 9
Silver 18 [28] 54
Sitall 0±0.15 [29] 0±0.45average for −60 °C to 60 °C
Stainless steel 10.1 ~ 17.330.3 ~ 51.9
Steel 11.0 ~ 13.033.0 ~ 39.0Depends on composition
Titanium 8.626 [30]
Tungsten 4.513.5
Turpentine 90 [18]
Water 69207 [21]
YbGaGe ≐0≐0 [31] Refuted [32]
Zerodur ≈0.007-0.1 [33] at 0...50 °C

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The volumetric heat capacity of a material is the heat capacity of a sample of the substance divided by the volume of the sample. Informally, it is the amount of energy that must be added, in the form of heat, to one unit of volume of the material in order to cause an increase of one unit in its temperature. The SI unit of specific heat is joule per kelvin per cubic meter, J/K/m3 or J/(K m3).

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection.

Thermal conduction is the transfer of heat internal energy by microscopic collisions of particles and movement of electrons within a body. The microscopically colliding particles, that include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases of including solids, liquids, gases and waves. The rate at which energy is conducted as heat between two bodies is a function of the temperature difference temperature gradient between the two bodies and the properties of the conductive interface through which the heat is transferred.

In thermodynamics, the Joule–Thomson effect describes the temperature change of a real gas or liquid when it is forced through a valve or porous plug while keeping it insulated so that no heat is exchanged with the environment. This procedure is called a throttling process or Joule–Thomson process. At room temperature, all gases except hydrogen, helium, and neon cool upon expansion by the Joule–Thomson process when being throttled through an orifice; these three gases experience the same effect but only at lower temperatures. Most liquids such as hydraulic oils will be warmed by the Joule–Thomson throttling process.

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

In thermodynamics, the internal energy of a system is the total energy contained within the system. It is the energy necessary to create or prepare the system in any given state, but does not include the kinetic energy of motion of the system as a whole, nor the potential energy of the system as a whole due to external force fields which includes the energy of displacement of the system's surroundings. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

A temperature coefficient describes the relative change of a physical property that is associated with a given change in temperature. For a property R that changes when the temperature changes by dT, the temperature coefficient α is defined by the following equation:

Thermal shock occurs when a thermal gradient causes different parts of an object to expand by different amounts. This differential expansion can be understood in terms of stress or of strain, equivalently. At some point, this stress can exceed the strength of the material, causing a crack to form. If nothing stops this crack from propagating through the material, it will cause the object's structure to fail.

The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.

The Joule expansion is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container, with the other side of the container being evacuated. The partition between the two parts of the container is then opened, and the gas fills the whole container.

Entropy is a property of thermodynamical systems. The term entropy was introduced by Rudolf Clausius who named it from the Greek word τρoπή, "transformation". He considered transfers of energy as heat and work between bodies of matter, taking temperature into account. Bodies of radiation are also covered by the same kind of reasoning.

A dilatometer is a scientific instrument that measures volume changes caused by a physical or chemical process. A familiar application of a dilatometer is the mercury-in-glass thermometer, in which the change in volume of the liquid column is read from a graduated scale. Because mercury has a fairly constant rate of expansion over ambient temperature ranges, the volume changes are directly related to temperature.

Multi-particle collision dynamics (MPC), also known as stochastic rotation dynamics (SRD), is a particle-based mesoscale simulation technique for complex fluids which fully incorporates thermal fluctuations and hydrodynamic interactions. Coupling of embedded particles to the coarse-grained solvent is achieved through molecular dynamics.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

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