Expansion coefficient

This article was registered in the quality assurance of the physics editorial team . If you are familiar with the topic, you are welcome to participate in the review and possible improvement of the article. The exchange of views about this is currently not taking place on the article discussion page , but on the quality assurance side of physics.

The coefficient of expansion or thermal expansion coefficient , also known colloquially as the expansion factor , is a characteristic value that describes the behavior of a substance with regard to changes in its dimensions when there are changes in temperature - therefore it is often called the coefficient of thermal expansion . The effect responsible for this is thermal expansion . The thermal expansion depends on the substance used, so it is a substance-specific material constant. Since the thermal expansion of many substances does not take place uniformly over all temperature ranges, the coefficient of thermal expansion itself is temperature-dependent and is therefore specified for a certain reference temperature or a certain temperature range.

A distinction is made between the thermal linear expansion coefficient α (also linear thermal expansion coefficient ) and the thermal spatial expansion coefficient γ (also spatial expansion coefficient or volume expansion coefficient or cubic expansion coefficient ).

Coefficient of linear expansion

The coefficient of linear expansion of a solid with length is the constant of proportionality between the change in temperature and the relative change in length . It is used to describe the relative change in length with a change in temperature. It is a substance-specific quantity that has the unit ("per Kelvin ") and is defined by the following equation: ${\displaystyle \alpha }$ ${\displaystyle L}$ ${\displaystyle \mathrm {d} T}$ ${\displaystyle {\frac {\mathrm {d} L}{L}}}$ ${\displaystyle \mathrm {K} ^{-1}}$

${\displaystyle \alpha L={\frac {\mathrm {d} L}{\mathrm {d} T}}}$

The temperature-dependent length of a rod can be calculated by solving this differential equation , it is:

${\displaystyle L(T)=L(T_{0})\cdot \exp \left(\int _{T_{0}}^{T}\alpha (T)\ dT\right)}$

With an expansion coefficient that is independent of the temperature , this, together with the original length, with uniform heating or cooling by the temperature difference : ${\displaystyle \alpha (T)=\alpha (T_{0})}$ ${\displaystyle L_{0}=L(T_{0})}$ ${\displaystyle \Delta T=T-T_{0}}$

${\displaystyle L=L_{0}\cdot \exp(\alpha \cdot \Delta T)}$

For most applications it is sufficient to use the following approximation, in which the exponential function was approximated by the first two terms of its Taylor series :

${\displaystyle L\approx L_{0}(1+\alpha \cdot \Delta T)}$

The change in length in a linear approximation is thus: ${\displaystyle \Delta L=L-L_{0}}$

${\displaystyle \Delta L\approx \alpha \cdot L_{0}\cdot \Delta T}$

In the case of anisotropic solids, the coefficient of linear expansion is also direction-dependent. This is particularly important when using tabular values ​​from the literature.

Examples

Aluminum has a coefficient of thermal expansion . This means that a 1000 meter long piece of aluminum expands by 2.31 cm when the temperature increases by one Kelvin. ${\displaystyle \alpha =23{,}1\cdot 10^{-6}\,\mathrm {K} ^{-1}={\frac {23{,}1}{10^{6}\,\mathrm {K} }}}$

If another piece of aluminum is 8 meters long and the temperature increase is 70 Kelvin, then this piece of aluminum expands by 1.3 centimeters, because ${\displaystyle \Delta L=\alpha \cdot L\cdot \Delta T=23{,}1\cdot 10^{-6}\,\mathrm {K} ^{-1}\cdot 8\,\mathrm {m} \cdot 70\,\mathrm {K} \approx 0{,}0129\,\mathrm {m} \approx 1{,}3\,\mathrm {cm} }$

The latter example describes e.g. B. eight laterally screwed solar modules with aluminum frames and their approximate maximum temperature difference between summer (sun-drenched aluminum) and winter (air temperature at night). It can be seen that the thermal expansion must be taken into account in the design of the fastening and frame components, e.g. B. by flexible or slidable fasteners.

Room expansion coefficient

The room expansion coefficient has the same unit  as the linear expansion coefficient . It indicates the relationship between the relative increase in volume and the change in temperature of a body . Mathematically it is defined by: ${\displaystyle \gamma }$ ${\displaystyle \alpha }$ ${\displaystyle \mathrm {K} ^{-1}}$ ${\displaystyle {\frac {\Delta V}{V}}}$ ${\displaystyle \Delta T}$

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

where the quantities pressure and particle number, which follow the partial derivatives as an index, must be kept constant. The temperature-dependent solution for this is analogous to above: ${\displaystyle p}$ ${\displaystyle N}$

${\displaystyle V(T)=V(T_{0})\cdot \exp \left(\int _{T_{0}}^{T}\gamma (T)\ \mathrm {d} T\right)}$

With a room expansion coefficient that is independent of the temperature, this results together with : ${\displaystyle \gamma (T)=\gamma (T_{0})}$ ${\displaystyle V(T_{0})=V_{0}}$

${\displaystyle V=V_{0}\cdot \exp(\gamma \cdot \Delta T)}$

As for the coefficient of linear expansion, the linearization can be used here as an approximation for small temperature changes:

${\displaystyle V\approx V_{0}(1+\gamma \cdot \Delta T)}$

With a Maxwell relation it is possible to relate the spatial expansion coefficient to the entropy : ${\displaystyle S}$

${\displaystyle \gamma ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p,N}=-{\frac {1}{V}}\left({\frac {\partial S}{\partial p}}\right)_{T,N}}$

Since the mass is independent of the temperature because of the conservation of mass, the spatial expansion coefficient results from the density as a function of the temperature: ${\displaystyle m=\rho (T)\cdot V(T)}$ ${\displaystyle \rho (T)}$

${\displaystyle \gamma =-{\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial T}}\right)_{p}}$

If the expansion coefficient is known as a function of temperature, the density results from:

${\displaystyle \rho (T)=\rho (T_{0})\cdot \exp \left(-\int _{T_{0}}^{T}\gamma (T)\ \mathrm {d} T\right)}$

Here, any temperature for. B. where the density is known. ${\displaystyle T_{0}}$ ${\displaystyle T_{0}=298{,}15\,\mathrm {K} =25\,^{\circ }{\text{C}}}$ ${\displaystyle \rho (T_{0})}$

Eduard Grüneisen has shown that the quotient between the thermal expansion coefficient and the specific heat capacity is approximately independent of the temperature. ${\displaystyle \alpha /c_{p}}$ ${\displaystyle \alpha }$ ${\displaystyle c_{p}}$

In general, the coefficient of thermal expansion is a positive quantity. Because of the law of conservation of mass , an increase in temperature is associated with a decrease in density for most substances . However, some substances, such as water between and , show the behavior known as density anomaly in certain temperature ranges , in which a negative expansion coefficient is observed. There are also materials, such as some types of glass-ceramic , whose coefficient of thermal expansion is almost zero . ${\displaystyle 0}$ ${\displaystyle 4^{\circ }{\text{C}}}$

The coefficient of thermal expansion can be determined empirically through measurements and is only valid for the substance and for the temperature range at which or in which the measurement was made.

Relationship between linear and spatial expansion coefficients

For isotropic solids, three times the linear expansion coefficient can be used to calculate the volume expansion:

${\displaystyle \gamma =3\cdot \alpha }$

However, this only applies approximately to small temperature differences. See the following subsections.

The mentioned (limit value) formula for small temperature differences shows a relative percentage error of approx. −0.1% when calculating the volume expansion coefficient of aluminum if the temperature difference of the expansion attempt is 50 K. At 200 K the relative error of the cubic expansion coefficient reaches almost −0.5%. A value that is slightly too low for the cubic expansion coefficient is calculated with this formula for large temperature differences. ${\displaystyle \gamma =3\cdot \alpha }$

Derivation of the temperature (difference) dependency

From the expansion of a cube, the equation of the temperature (difference) dependence of the link between the two mean expansion coefficients, i.e. the linear and the cubic, of an expansion test can be derived:

• The initial values ​​of the experiment are (initial edge length) and (initial volume). ${\displaystyle l_{0}}$ ${\displaystyle V_{0}}$
• The final values ​​after thermal expansion are: and . ${\displaystyle l_{2}}$ ${\displaystyle V_{2}}$

The following applies: and . ${\displaystyle V_{0}={l_{0}}^{3}}$ ${\displaystyle V_{2}={l_{2}}^{3}}$

For the change in length due to thermal expansion applies: . ${\displaystyle \Delta {l}=l_{2}-l_{0}=\Delta {T}\cdot l_{0}\cdot {\bar {\alpha }}}$

The volume of the cube after the expansion ,, is given by: ${\displaystyle V_{2}}$

${\displaystyle V_{2}=(l_{0}+\Delta {l})^{3}}$.

After multiplying the cubic binomial it follows:

${\displaystyle V_{2}={l_{0}}^{3}+(\Delta {l})^{3}+3\cdot {l_{0}}^{2}\cdot \Delta {l}+3\cdot l_{0}\cdot (\Delta {l})^{2}}$.

By subtracting the initial volume, the change in volume of the cube resulting from the thermal expansion follows: ${\displaystyle V_{0}={l_{0}}^{3}}$

${\displaystyle \Delta {V}=[V_{2}]-V_{0}=[{l_{0}}^{3}+(\Delta {l})^{3}+3\cdot {l_{0}}^{2}\cdot \Delta {l}+3\cdot l_{0}\cdot (\Delta {l})^{2}]-{l_{0}}^{3}}$ and thus:

${\displaystyle \Delta {V}=(\Delta {l})^{3}+3\cdot {l_{0}}^{2}\cdot \Delta {l}+3\cdot l_{0}\cdot (\Delta {l})^{2}}$.

Now, in the definition equation of the cubic expansion coefficient, the difference volume is substituted ${\displaystyle \Delta {V}}$ (exchanged) by this equation:

${\displaystyle {\bar {\gamma }}={\frac {\Delta {V}}{V_{0}\cdot \Delta {T}}}={\frac {(\Delta {l})^{3}+3\cdot {l_{0}}^{2}\cdot \Delta {l}+3\cdot l_{0}\cdot (\Delta {l})^{2}}{V_{0}\cdot \Delta {T}}}}$.

It follows by substituting of and : ${\displaystyle V_{0}={l_{0}}^{3}}$ ${\displaystyle \Delta {l}=\Delta {T}\cdot l_{0}\cdot {\bar {\alpha }}}$

${\displaystyle {\bar {\gamma }}={\frac {(\Delta {T}\cdot l_{0}\cdot {\bar {\alpha }})^{3}+3\cdot {l_{0}}^{2}\cdot (\Delta {T}\cdot l_{0}\cdot {\bar {\alpha }})+3\cdot l_{0}\cdot (\Delta {T}\cdot l_{0}\cdot {\bar {\alpha }})^{2}}{{l_{0}}^{3}\cdot \Delta {T}}}}$.

Shortening below and above the fraction line and shortening ultimately leads to the following equation, which describes the dependency of the two expansion coefficients in an expansion attempt with real (finite) non-differential temperature differences : ${\displaystyle {l_{0}}^{3}}$ ${\displaystyle \Delta {T}}$ ${\displaystyle \Delta {T}}$

${\displaystyle {\bar {\gamma }}=(\Delta {T})^{2}\cdot ({\bar {\alpha }})^{3}+3\cdot \Delta {T}\cdot ({\bar {\alpha }})^{2}+3\cdot {\bar {\alpha }}}$.

As a limit value, this equation shows the known equation for the case that the temperature difference approaches zero. ${\displaystyle \gamma =3\cdot \alpha }$

Note: by shortening the temperature difference (below the fraction line), the exponent (power number) of the temperature differences (above the fraction line) in this equation is reduced by the value 1 and is therefore always smaller than that of the mean linear expansion coefficient . For “real” temperature differences (up to several thousand Kelvin), the left additive term in the above equation is not practically relevant, since the linear expansion coefficient as a cube number (third power) makes practically no relevant increase to the cubic expansion coefficient . ${\displaystyle {\bar {\alpha }}}$ ${\displaystyle \alpha }$ ${\displaystyle \gamma }$

Special case of differential temperature differences in the expansion attempt

For isotropic solids, the change in length is the same in all three spatial directions. The volume of a cuboid is given by the product of its edge lengths . The full differential of the volume is then: ${\displaystyle V=L_{1}\cdot L_{2}\cdot L_{3}}$

${\displaystyle \mathrm {d} V=L_{1}L_{2}\,\mathrm {d} L_{3}+L_{1}L_{3}\,\mathrm {d} L_{2}+L_{2}L_{3}\,\mathrm {d} L_{1}}$

Inserted into the definition of the room expansion coefficient results in:

${\displaystyle \gamma ={\frac {1}{V}}{\frac {\mathrm {d} V}{\mathrm {d} T}}={\frac {1}{L_{3}}}{\frac {\mathrm {d} L_{3}}{\mathrm {d} T}}+{\frac {1}{L_{2}}}{\frac {\mathrm {d} L_{2}}{\mathrm {d} T}}+{\frac {1}{L_{1}}}{\frac {\mathrm {d} L_{1}}{\mathrm {d} T}}}$

Due to the assumed isotropy, the three terms on the right-hand side are each equal to the coefficient of linear expansion, so the following applies:

${\displaystyle \gamma =3\cdot \alpha }$

For isotropic solids, three times the coefficient of linear expansion can be used to calculate the volume expansion when the temperature differences are small.

Determination from real temperature, volume or density differences

In practice it is not easy to determine the expansion coefficient with small temperature differences. Greater differences are used. Otherwise you will quickly reach the limits of measurement technology / measurement accuracy.

The two basic equations of expansion follow from the equations for the coefficients of linear expansion and volume expansion:

${\displaystyle l_{2}=l_{0}(1+{\bar {\alpha }}(T_{2}-T_{0}))}$ and

${\displaystyle V_{2}=V_{0}(1+{\bar {\gamma }}(T_{2}-T_{0}))}$ .

The following applies to all solids and liquids that do not show any density anomaly :

${\displaystyle V_{2}>V_{0}}$and for . ${\displaystyle l_{2}>l_{0}}$ ${\displaystyle T_{2}>T_{0}}$

The expansion coefficients are mean values ​​for the temperature range from the initial temperature to the final temperature of the experiment. Now, one can define the equation of the cubic volume or as a volume, or as the edge length ( or insert) in one of the two equations. Then the final volume or the final length of both equations is set equal to one another . Dividing by the initial volume or initial length gives the quotients of length and volume. The densities are inversely proportional to the volumes; specific volumes are directly proportional to the volumes. This leads to the following relation between lengths , volumes , specific volumes and densities for real (non-differential) temperature differences of an expansion test: ${\displaystyle T_{0}}$ ${\displaystyle T_{2}}$ ${\displaystyle V_{0}=(l_{0})^{3}}$ ${\displaystyle V_{2}=(l_{2})^{3}}$ ${\displaystyle l_{0}={\sqrt[{3}]{V_{0}}}}$ ${\displaystyle l_{2}={\sqrt[{3}]{V_{2}}}}$ ${\displaystyle V_{2}}$ ${\displaystyle l_{2}}$ ${\displaystyle V_{0}}$ ${\displaystyle l_{0}}$ ${\displaystyle l}$ ${\displaystyle V}$ ${\displaystyle v}$ ${\displaystyle \rho }$

${\displaystyle \left({\frac {l_{2}}{l_{0}}}\right)^{3}={\frac {V_{2}}{V_{0}}}={\frac {v_{2}}{v_{0}}}={\frac {\rho _{0}}{\rho _{2}}}=1+{\bar {\gamma }}(T_{2}-T_{0})=(1+{\bar {\alpha }}(T_{2}-T_{0}))^{3}}$

As you can see, the mean linear expansion coefficient and the mean volume expansion coefficient for finite temperature differences can only be (exactly) converted into one another if the temperature difference is known:

${\displaystyle {\bar {\alpha }}={\frac {{\sqrt[{3}]{(1+{\bar {\gamma }}(T_{2}-T_{0}))}}-1}{(T_{2}-T_{0})}}}$ and

${\displaystyle {\bar {\gamma }}={\frac {(1+{\bar {\alpha }}(T_{2}-T_{0}))^{3}-1}{(T_{2}-T_{0})}}}$.

If the temperature difference in the experiment is exactly 1 K, the above three equations are considerably simplified.

Alternative definition equations for real temperature differences

for and (analogously also for lengths and volumes) applies: ${\displaystyle T_{2}>T_{0}}$ ${\displaystyle \Delta {T}=T_{2}-T_{0}}$

${\displaystyle \alpha _{\text{mittl.}}={\bar {\alpha }}={\frac {\Delta {l}}{l_{0}\cdot \Delta {T}}}={\frac {({\frac {l_{2}}{l_{0}}})-1}{\Delta {T}}}={\frac {{\sqrt[{3}]{({\frac {V_{2}}{V_{0}}})}}-1}{\Delta {T}}}={\frac {{\sqrt[{3}]{({\frac {v_{2}}{v_{0}}})}}-1}{\Delta {T}}}={\frac {{\sqrt[{3}]{({\frac {\rho _{0}}{\rho _{2}}})}}-1}{\Delta {T}}}}$

${\displaystyle \gamma _{\text{mittl.}}={\bar {\gamma }}={\frac {\Delta {V}}{V_{0}\cdot \Delta {T}}}={\frac {({\frac {l_{2}}{l_{0}}})^{3}-1}{\Delta {T}}}={\frac {({\frac {V_{2}}{V_{0}}})-1}{\Delta {T}}}={\frac {({\frac {v_{2}}{v_{0}}})-1}{\Delta {T}}}={\frac {({\frac {\rho _{0}}{\rho _{2}}})-1}{\Delta {T}}}}$

The density quotients are each indirectly proportional to the volume quotient.

Numerical values ​​of expansion coefficients

Solids

Linear expansion coefficients are usually used for solids. Since many materials are isotropic, they can, as described above, also be used to describe the volume expansion. For anisotropic substances there are different expansion coefficients for the different spatial directions. Some composite materials , such as the natural product wood, show strong anisotropy : the expansion across the fiber is about ten times greater than along the fiber. The behavior of carbon fibers , which even have a slightly negative expansion coefficient in the direction of the fibers , is also strongly anisotropic . Using CFRP , it is possible to manufacture components that have no or only minimal changes in size in the preferred directions when there are temperature changes.

The Invar alloy was specially developed to achieve a low coefficient of expansion. Due to small deviations in the composition, the expansion coefficient for this substance fluctuates relatively strongly.

Plastics ( polymers ) are very diverse in structure and properties and usually consist of a mixture of different pure substances. The expansion coefficient fluctuates accordingly with the actual composition, but is usually significantly higher than for metals, i.e. greater than 50 · 10 ⁻⁶  K ⁻¹ . Below their glass transition , polymers, or generally amorphous solids, generally have a significantly smaller coefficient of expansion than above.

Pure metals (elements)

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
aluminum	023.1
lead	028.9
iron	011.8
nickel	013.0
gold	014.2
iridium	007th
copper	016.5
lithium	058
magnesium	024.8
sodium	007.1
platinum	008.8
silver	018.9
Tantalum	006.6
titanium	008.6
zinc	030.2
tin	022.0

The “Chemistry Tabular Book” (author collective Kaltofen, GDR, thick version), see literature recommendation, names the expansion coefficients for many other metals.

Non-metals and semi-metals (elements)

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
diamond	0001.18
Germanium	0005.8
graphite	0001.9 ... 2.9
white phosphorus	0125
rhombic sulfur	0074
silicon	0002.6

Metal alloys

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
Aluminum bronze	015 to 16
bronze	017.5
"Indilatans Extra" (Krupp) (36Ni, XX) at 12 to 100 ° C	0−0.04
Invar	000.55 ... 1.2
Constantan (at −191 to 16 ° C)	012.22
Brass	018.4 to 19.3
Platinum iridium	008.9
steel	011… 13

Building materials

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
concrete	012
Wood (oak)	008th
Clinker (hard-fired brick)	002.8 to 4.8
Brick	005

Plastics

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
Soft rubber	0017 to 28
Hard rubber	0080
Polyamide (PA)	0060 to 150
Polycarbonate (PC)	0060 to 70
Polyethylene (HD-PE)	0150 to 200
Polypropylene (PP)	0100 to 200
Polyoxymethylene (POM)	0070 to 130
Polytetrafluoroethylene (PTFE)	0100 to 160
Polyvinyl chloride (rigid PVC)	0070 to 100
Polymethyl methacrylate (PMMA, plexiglass )	0075 to 80

Glass and ceramics

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
Borosilicate glass	003.3
German melting jar (for connections with platinum or invar)	009.0
Duran glass / Pyrex glass	003.6
Enamel (enamel coatings)	008.0 to 9.5
Window glass	010
Jena device glass "No.20"	004.8
Porcelain , Berliner	004 to 6
Porcelain , Meissner	003 to 5
Quartz glass (silicon dioxide)	000.54
Technical ceramics	002… 13
Zerodur (glass ceramic)	000 ± 0.007

For other substances from which ceramic products (workpieces) are manufactured, see "Compounds and chemicals".

Chemical compounds

Linear expansion coefficient α at 20 ° C

designation	α in 10 −6  K −1
Aluminum oxide, crystalline (corundum)	005.6 to 7.0
Ice (−5 ° C to 0 ° C)	051 to 71
Mica (magnesium silicate)	013.5
Magnesium oxide	013.1
Silicon dioxide (quartz)	012 ... 16

Temperature dependence for solids

In chemical plant construction, mean expansion coefficients are often used for the temperature range under consideration in which a plant is to work. Numerical values ​​of expansion coefficients at elevated temperatures are difficult to find in popular scientific literature. For some container materials, Dietzel gives mean expansion coefficients for two temperature ranges (0 to 100 ° C and 0 to 200 ° C), quote (table):

	0 to 100 ° C	0 to 200 ° C
Aluminum (pure)	0.0000239	0.0000246
Cast iron	0.0000104	0.0000111
techn. Glass	0.0000060	0.0000065
Brass	0.0000183	0.0000193
Steel (up to 0.5% C)	0.0000110	0.0000120

These values ​​show the increase in the mean expansion coefficient in K ⁻¹ for solids with increasing temperature. There is a 50 K temperature difference between the mean values ​​of the temperatures (50 ° C and 100 ° C) of the two temperature ranges.

liquids

The room expansion coefficient can be specified for liquids. They expand isotropically, i.e. equally in all directions. Their shape is given by the vessel containing them, which is why it is not advisable to determine the coefficient of linear expansion for them, although it can be formally calculated.

Liquids usually have a significantly higher coefficient of expansion than solids. This is why they are often given in thousandths per Kelvin instead of millionths per Kelvin for solids. The units in the tables in this section have been selected accordingly.

Inorganic liquids, elements and liquid metals / metal alloys

Room expansion coefficient γ at 20 ° C

designation	γ in 10 −3  K −1
bromine	01.11 or 1.13
Galinstan (eutectic thermometer fluid)	00.126
NaK (eutectic alloy)	00.16
mercury	00.1811
Nitric acid (100%)	01.24
hydrochloric acid	00.30
Carbon disulfide	01.18
Sulfuric acid (about 99%)	00.57
Water at 0 ° C	−0.068
Water at approx. 20 ° C	00.2064
Water at 100 ° C	00.782

Organic liquids

Room expansion coefficient γ at 20 ° C

designation	γ in 10 −3  K −1	chemical group
Petrol (at 0 ° C)	01.0	Paraffins
n-heptane	01.09	Paraffins
Heating oil / diesel fuel	00.96	Paraffins
n-hexane	01.35	Paraffins
Mineral oil, hydraulic oil	00.7	Paraffins
Paraffin oil	00.764	Paraffins
n-pentane	01.6	Paraffins
petroleum	00.9 to 1	Paraffins
Lubricating oil	00.6 to 0.7	Paraffins
Trichloromethane	01.21	halogenated paraffin
Carbon tetrachloride	01.21	halogenated paraffin
Methanol	01.49	monohydric alcohols
Ethanol	01.10	monohydric alcohols
Propanetriol	00.520	trihydric alcohols
Ethanoic acid	01.08	Paraffinic acids
Diethyl ether	01.62	Ether
Propanone	01.46	Ketones
olive oil	00.72	Fatty acid esters
Benzene	01.14	Aromatic hydrocarbons
Turpentine oil	01	Pines , terpenes
Methylbenzene	01.11	Aromatic hydrocarbons

Gases

Thermal expansion of gases, some liquids and some solids

Gases under normal pressure and well above the boiling point behave approximately like an ideal gas . This expands proportionally to the absolute temperature. This simple linear relationship between volume and temperature results in a coefficient of expansion that changes strongly with temperature and is inversely proportional to the absolute temperature: ${\displaystyle \gamma }$

${\displaystyle \gamma _{\text{Realgas}}^{(T_{\text{abs.}})}\sim {\frac {1}{T_{\text{abs.}}}}}$ and for the ideal gas applies:

${\displaystyle \gamma _{\text{Idealgas}}^{(T_{\text{abs.}})}={\frac {1}{T_{\text{abs.}}}}}$

The expansion coefficient of the ideal gas at 0 ° C (reference temperature) is therefore:

${\displaystyle \gamma _{\text{Idealgas}}^{\text{(0°C)}}={\frac {1}{\text{273,15 K}}}\approx 0{,}003661}$

The expansion coefficient for ideal gases at 20 ° C is 1 / 293.15 K ⁻¹ = 3.41 · 10 ⁻³  K ⁻¹ . In general, the coefficient of expansion can be calculated using the thermal equation of state of ideal gases as γ ( T ) or using the thermal equation of state of real gases as γ ( T , p ).

For the ideal gas (at low pressure), according to the ideal gas equation for isobaric (thermal) expansion:

${\displaystyle {\frac {V_{0}}{V_{2}}}={\frac {T_{0,{\text{abs.}}}}{T_{2,{\text{abs.}}}}}}$

The temperatures must be used as absolute temperatures in [Kelvin]. For temperatures that differ by a fixed temperature difference, for example by 1 K, the volume ratio tends towards the value 1 for ever higher temperatures. The coefficient of expansion therefore tends towards zero for ever higher temperatures. So for ideal gases it decreases with increasing temperature.

Comparison of the isobaric (differential) expansion coefficients of water and water vapor

Fratscher and Picht name the expansion coefficients in 10 ° C for boiling water and the equilibrium saturated steam (100% steam, 0% liquid water) for temperatures from 0.01 ° C to 374.15 ° C (critical temperature of water) -Steps. The associated system pressure is the respective steam pressure of water. Some of the values ​​are given here as examples:

Temperature in ° C	System pressure (vapor pressure) in MPa	${\displaystyle \gamma _{\text{Wasser}}^{(T)}}$in K −1	${\displaystyle \gamma _{\text{Wasserdampf}}^{(T)}}$(Saturated steam) in K −1	Hints
000.01	0.0006112	−0.0000855	0.003669	Density anomaly of water
010	0.0012271	00.0000821	0.003544
020th	0.0023368	00.0002066	0.003431
030th	0.0042417	00.0003056	0.003327
040	0.0073749	00.0003890	0.003233
050	0.012335	00.0004624	0.003150
060	0.019919	00.0005288	0.003076
070	0.031161	00.0005900	0.003012
080	0.047359	00.0006473	0.002958
090	0.070108	00.0007019	0.002915
100	0.101325	00.0007547	0.002882
150	0.47597	00.001024	0.002897
200	1.5551	00.001372	0.003291
250	3.9776	00.001955	0.004321
300	8.5917	00.003293	0.007117
350	16,537	00.01039	0.02175
360	18.674	00.01928	0.03899
370	21.053	00.09818	0.1709
374.15 ( critical temp. )	22.12 (critical pressure!)	> 0.1709 (original literature names "∞", *)		critical point

"*": In real terms, the expansion coefficient at the critical point (and above) cannot be infinite, it must have finitely large values, since otherwise the medium would expand to zero density (infinite volume) and, in addition, under the respective constant system conditions, pressure and temperature, then an infinite amount of isobaric displacement work would have to be performed.

Shortly before the critical point is reached, the expansion coefficients of water and water vapor increase sharply. At the critical point, liquid and vapor become one / identical. There is therefore only one expansion coefficient. Compared to 370 ° C, its value must be higher, since the volume has again increased disproportionately.

Concentration-dependent expansion coefficients of aqueous solutions

At constant temperature, aqueous solutions show a coefficient of expansion that usually increases with the concentration of the dissolved substance.

Bierwerth names sodium chloride solution, potassium chloride solution and calcium chloride solutions of various mass concentrations as examples. For example (quote) potassium chloride solutions with a mass content of 4/10/20% have expansion coefficients of 0.00025 / 0.00031 / 0.00041 at 20 ° C each. From the examples mentioned it can be concluded that with these aqueous salt solutions the numerical value of the expansion coefficient increases by about 25% (at relatively low concentrations) to 50% (at higher concentrations) each time the mass concentration of the solution is doubled.

Calculation of the mean room expansion coefficient from values ​​of density or specific volumes

Since the change in the volume of solids and liquids results in a change in their density , the mean statistical volume expansion coefficient can also be calculated from the quotient of two densities for two temperatures: ${\displaystyle \rho }$ ${\displaystyle \gamma _{\text{mittl.}}}$

${\displaystyle {\frac {\rho _{0}}{\rho _{2}}}=1+\gamma _{\text{mittl.}}\cdot (T_{2}-T_{0})}$with . ${\displaystyle T_{2}>T_{0}}$

The mean room expansion coefficient between the selected temperatures results from:

${\displaystyle \gamma _{\text{mittl.}}={\frac {({\frac {\rho _{0}}{\rho _{2}}})-1}{(T_{2}-T_{0})}}}$.

Alternatively, values ​​of the mass-specific volumes or the molar volumes can also be used:

${\displaystyle \gamma _{\text{mittl.}}={\frac {({\frac {v_{2}}{v_{0}}})-1}{(T_{2}-T_{0})}}}$.

The specific volumes are inversely proportional to the densities. ${\displaystyle v}$

The mean statistical room expansion coefficient has advantages in application compared to the "usual" volume expansion coefficient " " related to a temperature . The usual volume expansion coefficient is only valid for one temperature. In liquids, its value usually increases with increasing temperature. Because of the density anomaly , u. a. of water and liquid ammonia , these substances also have negative coefficients of expansion in narrow temperature ranges. If you calculate the volume change with the help of the mean volume expansion coefficient from temperature to temperature , you get a correct value for the new volume - or the new density - while the calculation with the volume expansion coefficient at a fixed temperature would show a "larger" error. It is also possible to calculate the volume expansion coefficient very precisely for a specific temperature using this method. For this, the density values ​​for 1 K less and one Kelvin more are used. 2 K is used as the temperature difference. Is obtained for water at 4 ° C as from the density values of 3 ° C and 5 ° C a volume expansion coefficient of the value 0 . This is correct, as water has its density maximum at 4 ° C, its density increases from 0 ° C to 4 ° C and decreases again from 4 ° C. Consequently the volume expansion coefficient for water at 4 ° C is zero . ${\displaystyle \gamma _{\text{mittl.}}}$ ${\displaystyle T}$ ${\displaystyle \gamma ^{(T)}}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{2}}$ ${\displaystyle \gamma ^{(T)}}$

Numerical values ​​of liquids at air pressure

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / cm³ ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temp. in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
water	000/1	0.999840 / 0.999899	01	000.5	−0.000059006
	003/5	0.999964 / 0.999964	02	004th	00
	000/20	0.999840 / 0.998203	20th	010	00.0000820
	017/19	0.998773 / 0.998403	02	018th	00.0001853
	019/21	0.998403 / 0.997991	02	020th	00.0002064
	024/26	0.997295 / 0.996782	02	025th	00.0002573
	020/100	0.998203 / 0.95835	80	060	00.0005198
	090/100	0.96532 / 0.95835	10	095	00.0007273
mercury	−20 / −18	13.6446 / 13.6396	02	−19	00.0001833
	0−2 / 2	13.6000 / 13.5901	04th	000	00.00018212
	000/20	13.5951 / 13.5457	20th	010	00.0001823
	016/20	13.5556 / 13.5457	04th	018th	00.00018271
	018/22	13.5507 / 13.5408	04th	020th	00.00018278
	024/26	13.5359 / 13.5310	02	025th	00.00018107
	020/100	13.5457 / 13.3512	80	060	00.0001821
	090/100	13.3753 / 13.3512	10	095	00.0001805
	240/260	13.018 / 12.970	20th	250	00.00018504
Propanetriol (glycerine)	020/60	1.260 / 1.239	40	040	00.0004237
	080/100	1.224 / 1.207	20th	090	00.0007042
	140/160	1.167 / 1.143	20th	150	00.001050
	180/200	1.117 / 1.090	20th	190	00.001239
	220/240	1.059 / 1.025	20th	230	00.001659
" Baysilone M10" ® silicone oil	−40 / 0	0.990 / 0.950	40	−20	00.00105
	000/40	0.950 / 0.920	40	020th	00.000815
	040/80	0.920 / 0.880	40	060	00.00114
	080/120	0.880 / 0.850	40	100	00.000882
	120/160	0.850 / 0.810	40	140	00.00123
	160/200	0.810 / 0.770	40	180	00.00130
	200/240	0.770 / 0.740	40	220	00.00101

At approx. 4 ° C, water has its maximum density of 0.999975 g / cm³ ( density anomaly ) and the volume expansion coefficient is zero here.

The calculated values ​​show, for example, for a temperature increase from 0 to 20 ° C, a volume increase of + 0.164% for water and +0.365% for mercury. From 20 to 100 ° C, the volume increases by + 4.16% for water and by +1.46% for mercury.

As you can see, the volume expansion coefficient of liquids almost always only increases with increasing temperature, unless the substance has a density anomaly in a narrow temperature range, as is the case with water between 0 and 4 ° C.

Numerical values ​​of boiling liquids at the respective vapor pressure (not isobaric)

At every temperature a liquid has a different vapor pressure , according to its vapor pressure function . Therefore, temperature-related expansion or contraction of the volume does not take place isobarically .

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / cm³ ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temperature in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
boiling superheated water	095/100	0.96172 / 0.95813	5	97.5	0.00074938
	090/110	0.96516 / 0.95066	20th	100	0.00076263
	120/130	0.94286 / 0.93458	10	125	0.00088596
	140/160	0.92584 / 0.90728	20th	150	0.0010228
	190/200	0.87604 / 0.86468	10	195	0.0013138
	190/210	0.87604 / 0.85281	20th	200	0.0013620
	200/210	0.86468 / 0.85281	10	205	0.0013919
	240/260	0.81360 / 0.78394	20th	250	0.0018915
	290/300	0.73212 / 0.71220	10	295	0.0027970
	290/310	0.73212 / 0.69061	20th	300	0.0030053
	300/310	0.71220 / 0.69061	10	305	0.0031262
	310/320	0.69061 / 0.66689	10	315	0.0035568
	320/330	0.66689 / 0.64045	10	325	0.0041283
	330/340	0.64045 / 0.61013	10	335	0.0049694
	340/350	0.61013 / 0.57448	10	345	0.0062056
	350/360	0.57448 / 0.52826	10	355	0.0087495
	360/370	0.52826 / 0.44823	10	365	0.017855
	370 / 374.15 ( critical temperature )	0.44823 / 0.3262	4.15	372.075	0.09014

Numerical values ​​of boiling liquid gases at the respective vapor pressure (not isobaric)

At every temperature a liquid / liquid gas has a different vapor pressure , according to its vapor pressure function . Therefore, temperature-related expansion or contraction of the volume does not take place isobarically.

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / cm³ ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temperature in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
liquid carbon dioxide , boiling	−50 / −40	1.1526 / 1.1136	10	−45	0.0035022
	−30 / −20	1.0727 / 1.0293	10	−25	0.0042165
	0/2	0.9285 / 0.9168	2	1	0.006381
	18/22	0.7979 / 0.7548	4th	20th	0.01428
	28/30	0.6568 / 0.5929	2	29	0.05389
	30 / 31.05 (critical temperature)	0.5929 / 0.4680	1.05	30.525	0.2542!
liquid propane , boiling	−50 / −40	0.5917 / 0.5858	10	−25	0.001007
	−30 / −20	0.5679 / 0.5559	10	−45	0.002159
	−5 / 5	0.5365 / 0.5233	10	0	0.002522
	20/30	0.5020 / 0.4866	10	25th	0.003165
	40/50	0.4684 / 0.4500	10	45	0.004089
liquid ethene (ethylene), boiling	−40 / -30	0.4621 / 0.4403	10	−35	0.004951
	−30 / −20	0.4403 / 0.4153	10	−25	0.006020
	−20 / −10	0.4153 / 0.3851	10	−15	0.007842
	−10 / 0	0.3851 / 0.3471	10	−5	0.01095
	−5 / 5	0.3671 / 0.3186	10	0	0.01522
	0/2	0.3471 / 0.3378	2	1	0.01377
	0/4	0.3471 / 0.3258	4th	2	0.01634
	4/6	0.3258 / 0.3102	2	5	0.02515
	7/8	0.2995 / 0.2858	1	7.5	0.04794
	7/9	0.2995 / 0.2646	2	8th	0.06595
	8/9	0.2858 / 0.2646	1	8.5	0.08012
	8 / 9.9 (critical temperature)	0.2858 / 0.2111	1.9	8.95	0.1862
	9 / 9.5	0.2646 / 0.2483	0.5	9.25	0.1313
	9 / 9.9 (critical temperature)	0.2646 / 0.2111	0.9	9.45	0.2816
	9.5 / 9.9 (critical temperature)	0.2483 / 0.2111	0.4	9.7	0.4405!
liquid ammonia , boiling	−70 / −68	0.72527 / 0.72036	2	−69	+0.003408
	−68 / −66	0.72036 / 0.72067	2	−67	−0.000215
	−66 / −64	0.72067 / 0.71839	2	−65	+0.001587
	-64 / -62	0.71839 / 0.71608	2	−63	+0.001613
	−50 / −48	0.70200 / 0.69964	2	−49	+0.001687
	−30 / −28	0.67764 / 0.67517	2	−29	+0.001829
	−28 / −26	0.67517 / 0.67263	2	−27	+0.001888
	−26 / −24	0.67263 / 0.67463	2	−25	−0.001482
	−24 / −22	0.67463 / 0.68587	2	−23	−0.008194
	−22 / −20	0.68587 / 0.66503	2	−21	+0.015668
	−2 / 0	0.64127 / 0.63857	2	−1	+0.002114
	−2 / 2	0.64127 / 0.63585	4th	0	+0.002131
	0/2	0.63857 / 0.63585	2	1	+0.002139
	18/20	0.61320 / 0.61028	2	19th	+0.002392
	18/22	0.61320 / 0.60731	4th	20th	+0.002425
	20/22	0.61028 / 0.60731	2	21st	+0.002445
	24/26	0.60438 / 0.60132	2	25th	+0.002544
	48/50	0.56628 / 0.56306	2	49	+0.002859

Note: Density values ​​and expansion coefficients of liquid ammonia show two density anomalies .

Numerical values ​​of metal melts

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / cm³ ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temperature in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
Sodium-potassium alloy (here: 25% Na / 75% K, mass percent)	20/100	0.872 / 0.852	80	60	0.000293
	100/200	0.852 / 0.828	100	150	0.000290
	200/300	0.828 / 0.803	100	250	0.000311
	300/500	0.803 / 0.753	200	400	0.000332
	500/600	0.753 / 0.729	100	550	0.000329
	600/700	0.729 / 0.704	100	650	0.000355
Lithium melt	200/300	0.151 / 0.505	100	250	−0.00701
	300/400	0.505 / 0.495	100	350	+0.000202
	400/600	0.495 / 0.474	200	500	+0.000222
	600/700	0.474 / 0.465	100	650	+0.000194
Tin melt	240/300	6.985 / 6.940	60	270	0.0001081
	300/400	6.940 / 6.865	100	350	0.0001093
	400/500	6.865 / 6.790	100	450	0.0001105
	500/600	6.790 / 6.720	100	550	0.0001042
	600/700	6.720 / 6.640	100	650	0.0001205
Lead melt	400/500	10.582 / 10.476	100	450	0.00010118
	500/600	10.476 / 10.360	100	550	0.00011197
	600/700	10.360 / 10.242	100	650	0.00011521
	700/800	10.242 / 10.125	100	750	0.00011556

Numerical values ​​of gases (isobaric)

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / l ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temperature in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
dry air , at 1 bar	−20 / 0	1.3765 / 1.2754	20th	−10	0.0039635
	0/20	1.2754 / 1.1881	20th	10	0.0036739
	20/40	1.1881 / 1.1120	20th	30th	0.0034218
	40/60	1.1120 / 1.0452	20th	50	0.0031956
	60/80	1.0452 / 0.9859	20th	70	0.0030074
	80/100	0.9859 / 0.9329	20th	90	0.0028406
	140/160	0.8425 / 0.8036	20th	150	0.0024204
	180/200	0.7681 / 0.7356	20th	190	0.0022091
	200/300	0.7356 / 0.6072	100	250	0.0021146
	300/400	0.6072 / 0.5170	100	350	0.0017447
	400/500	0.5170 / 0.4502	100	450	0.0014838
	500/600	0.4502 / 0.3986	100	550	0.0012945
	600/700	0.3986 / 0.3577	100	650	0.0011434
	700/800	0.3577 / 0.3243	100	750	0.0010300
	800/900	0.3243 / 0.2967	100	850	0.0009302
	900/1000	0.2967 / 0.2734	100	950	0.0008522
dry air , at 10 bar.	−25 / 0	14.16 / 12.82	25th	−12.5	0.004181
	0/25	12.82 / 11.71	25th	12.5	0.003792
	25/50	11.71 / 10.79	25th	37.5	0.003411
	50/100	10.79 / 9.321	50	75	0.003152
	100/200	9.321 / 7.336	100	150	0.002706
	200/300	7.336 / 6.053	100	250	0.002120
	300/400	6.053 / 5.153	100	350	0.001747
	400/500	5.153 / 4.487	100	450	0.001484
dry air , at 100 bar.	−25 / 0	149.5 / 131.4	25th	−12.5	0.005510
	0/25	131.4 / 117.8	25th	12.5	0.004618
	25/50	117.8 / 107.1	25th	37.5	0.003996
	50/100	107.1 / 91.13	50	75	0.003505
	100/200	91.13 / 70.92	100	150	0.002850
	200/300	70.92 / 58.37	100	250	0.002150
	300/400	58.37 / 49.71	100	350	0.001742
	400/500	49.71 / 43.55	100	450	0.001414
saturated moist air , at 100 kPa	0/2	1.2731 / 1.2634	2	1	0.003839
	8/12	1.2347 / 1.2159	4th	10	0.0038654
	16/20	1.1971 / 1.1785	4th	18th	0.003946
	18/22	1.1878 / 1.1691	4th	20th	0.003999
	24/26	1.1597 / 1.1503	2	25th	0.004086
	28/32	1.1408 / 1.1216	4th	30th	0.004280
	38/42	1.0921 / 1.0717	4th	40	0.004759
	48/50	1.0395 / 1.0282	2	49	0.005495
	55/65	0.9989 / 0.9332	10	60	0.007040
	65/75	0.9332 / 0.8552	10	70	0.009121
	75/85	0.8552 / 0.7605	10	80	0.01245
	85/95	0.7605 / 0.6442	10	90	0.01805

Note: The 100% saturation level of the humid air remains constant when heated, for example if the air is enclosed in a gas burette above the sealing liquid water while the temperature is increased.

Numerical values ​​of superheated water vapor (isobaric)

substance	${\displaystyle T_{0}}$/ in ° C ${\displaystyle T_{2}}$	${\displaystyle \rho _{0}}$/ in g / l ${\displaystyle \rho _{2}}$	${\displaystyle \Delta T}$ in K	mean temperature in ° C ${\displaystyle T_{\text{mittl.}}}$	${\displaystyle \gamma _{\text{mittl.}}}$in K −1	swell
superheated steam at 0.6 bar	100/200	0.3514 / 0.2756	100	150	0.002750
	200/300	0.2756 / 0.2272	100	250	0.002130
	300/400	0.2272 / 0.1933	100	350	0.0011754
	400/500	0.1933 / 0.1682	100	450	0.001492
superheated steam at 1 bar	100/200	0.5899 / 0.4604	100	150	0.002813
	200/300	0.4604 / 0.3791	100	250	0.002145
	300/400	0.3791 / 0.3224	100	350	0.001759
	400/500	0.3224 / 0.2805	100	450	0.001494
superheated steam at 6 bar	200/300	2.839 / 2.304	100	250	0.002322
	300/400	2.304 / 1.947	100	350	0.001834
	400/500	1.947 / 1.690	100	450	0.001521
superheated steam at 10 bar	200/300	4.850 / 3.879	100	250	0.002503
	300/400	3.879 / 3.264	100	350	0.001884
	400/500	3.264 / 2.826	100	450	0.001550

Summary

• With solids and liquids, the coefficient of expansion (positive value) almost invariably increases with increasing temperature. Some substances, liquids and solids, have density anomalies in narrow temperature ranges and then also have negative coefficients of expansion in these ranges.
• Gases have positive expansion coefficients, but their value decreases with increasing temperature.
• Liquids and liquefied gases show a strong exponential increase in the expansion coefficient shortly before the critical temperature of the liquid is reached. The calculated examples of liquid ethene and carbon dioxide clearly show this. According to Fratscher / Picht, the expansion coefficient - of the vapor which is in equilibrium with the liquid - should also show a strong exponential increase shortly before the critical temperature of the substance is reached. For the critical point for water, Fratscher calls the coefficient of expansion “infinite” as the value, but this cannot be. It must be possible to determine finite values, since otherwise an infinitely high energy (isobaric displacement work ) would have to be expended to expand the volume.

The supercritical state is neither liquid nor vapor. Therefore, the expansion coefficients of liquid and vapor must approach each other before reaching the critical point in order to finally become identical at the critical point.

Sudden changes in the density / expansion coefficient of solids and liquids indicate a change in the molecular or crystal structure under the respective conditions of pressure and temperature.

Influence of the expansion coefficient on the degree of filling of a container with temperature changes

The filling level of a container (in process engineering ) is defined as: ${\displaystyle \varphi }$

${\displaystyle \varphi ={\frac {V_{F}}{V_{B}}}}$.

If the numerical values ​​of the volume expansion coefficients of the liquid in the container and the calculable volume expansion coefficient of the container material (wall material) are not the same, every change in the temperature of the container and its contents (liquid) leads to a change in the degree of filling of the container, as the liquid and the container material differ in strength expand or contract as the temperature rises or falls. Bierwerth gives the following formula for changing the percentage filling level : ${\displaystyle \gamma }$ ${\displaystyle \gamma _{F}}$ ${\displaystyle \gamma _{B}=3\cdot \alpha _{B}}$ ${\displaystyle \varphi }$

${\displaystyle \Delta \varphi ={\frac {{V_{F}}^{0}\cdot \Delta {T}\cdot (\gamma _{F}-3\cdot \alpha _{B})}{{V_{B}}^{0}\cdot (1+3\cdot \alpha _{B}\cdot \Delta {T})}}\cdot 100\,\%}$.

Liquid volume , container volume . The volumes marked with an index 0 are the values ​​before the temperature change (initial value). Coefficient of linear expansion of the container material. Volume expansion coefficient of the liquid in the container. The expansion coefficients used are the mean expansion coefficients in the respective temperature range. ${\displaystyle V_{F}}$ ${\displaystyle V_{B}}$ ${\displaystyle \alpha _{B}}$ ${\displaystyle \gamma _{F}}$

See also

• Bimetal
• Temperature coefficient
• Density anomaly

literature

• Gerhard Ondracek: Materials science. Guide to study and practice. 2nd, revised edition. Expert-Verlag, Sindelfingen 1986, ISBN 3-88508-966-1 .
• Walther Bierwerth: Chemical engineering table book, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , volume expansion coefficients of liquids and solids, p. 76; Method of determining the coefficient of differential expansion at a temperature; Expansion coefficients of many container materials in the chemical industry: steels, alloys, light metals, glasses, ceramics and plastics pp. 248–256.
• U. Hübschmann, E. Links, E. Hitzel: Tables on chemistry and analytics in training and occupation . Verlag Handwerk und Technik, 1998, ISBN 978-3-582-01234-0 .
• Fritz Dietzel: Technische Wärmelehre, Vogel Verlag Würzburg, 1990, ISBN 3-8023-0089-0 , mean expansion coefficients of aluminum, gray cast iron, glass, brass and steel (0.5% C) for two different temperature ranges: 0-100 ° C and 0-200 ° C, Appendix Table 2, p. 159.
• Collective of authors (Rolf Kaltofen et al.): Table book chemistry (thick version), VEB Verlag für Grundstoffindustrie Leipzig, GDR, 1975, 5th edition, expansion coefficients of metals, elements and alloys, of non-metals, organic liquids, various glasses and ceramics, p 389-390.
• Wolfgang Fratscher / Hans-Peter Picht: Material data and characteristics of process engineering, Deutscher Verlag für Grundstoffindindustrie, Leipzig, GDR, 1979, Stuttgart 1993, isobaric expansion coefficients of water and steam pp. 170–171; Expansion coefficients of solids (materials), inorganic and organic liquids, p. 31, ISBN 3-342-00633-1 .

Individual evidence

1. ↑ The coefficients of thermal expansion of wood and wood products (PDF; 5.1 MB) Retrieved on May 10, 2012.
2. ↑ ^{a b} Werner Martienssen, Hans Warlimont (Ed.): Springer Handbook of Condensed Matter and Material Data. Springer, Berlin a. a. 2005, ISBN 3-540-44376-2 .
3. ↑ ^{a b c d e f g h i j k l m n o p q r s t u v w x y} William M. Haynes (Ed.): CRC Handbook of Chemistry and Physics . A ready-reference book of chemical and physical data. 92nd edition. CRC Press, Boca Raton FL et al. a. 2011, ISBN 978-1-4398-5511-9 .
4. ↑ ^{a b c d e f g h i j k l m n o p} collective of authors (including: Rolf Kaltofen): Chemistry table book (thick version), VEB Verlag für Grundstoffindustrie Leipzig, GDR, 1975, 5th edition, expansion coefficients of metals, Elements and alloys, from non-metals, organic liquids, various glasses and ceramics, pp. 389–390.
5. ↑ Walther Bierwerth: Chemical engineering table book, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , coefficients of expansion of many container materials in the chemical industry: steels, alloys, light metals, glasses, ceramics and plastics, pp. 248-256.
6. ↑ ^{a b c d e f g h i j k} Wolfgang Fratscher / Hans-Peter Picht: Material data and characteristics of process engineering, Deutscher Verlag für Grundstoffindindustrie, Leipzig, GDR, 1979, Stuttgart 1993, expansion coefficients of solids (materials), inorganic and organic Liquids, p. 31, table 2.3, ISBN 3-342-00633-1 .
7. ↑ ^{a b c d e f g h} U. Hübschmann, E. Links, E. Hitzel: Tables on chemistry and analytics in training and occupation . Verlag Handwerk und Technik, 1998, ISBN 978-3-582-01234-0 (pp. 35-36: Coefficients of expansion of metal alloys (materials), glasses and inorganic chemicals).
8. ↑ ^{a b c d e} Paetec GmbH, formulas and tables for secondary levels I and II, Berlin 1996, coefficients of expansion of solids and liquids
9. ↑ Wolfgang Fratscher / Hans-Peter Picht: Material data and characteristics of process engineering, Deutscher Verlag für Grundstoffindindustrie, Leipzig, GDR, 1979, Stuttgart 1993, isobaric expansion coefficients of water and water vapor pp. 170–171; Expansion coefficients of solids (materials), inorganic and organic liquids, p. 31, Table 2.3, ISBN 3-342-00633-1 .
10. ↑ ^{a b c d e f g h i j} Walther Bierwerth: Table book Chemietechnik, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , volume expansion coefficients of liquids and solids, p. 76; Method of determining the coefficient of differential expansion at a temperature; Expansion coefficients of many container materials in the chemical industry: steels, alloys, light metals, glasses, ceramics and plastics pp. 248–256.
11. ↑ Wolfgang Kaiser: Kunststoffchemie für Ingenieure: From synthesis to application. 2nd Edition. Carl Hanser, 2007, ISBN 978-3-446-41325-2 , p. 228
12. ↑ Technical Glasses Data Sheet (PDF) schott.com.
13. ↑ Product information page of the manufacturer Heraeus-Quarzglas. at heraeus-quarzglas.de.
14. ↑ Ceramic dressing thermal properties. Retrieved May 29, 2018 . 
15. ↑ ZERODUR® glass ceramic with extremely low thermal expansion. Schott AG , accessed on February 3, 2019 (The specified value applies to Zerodur in expansion class 0 EXTREME.). 
16. ^ Coefficient of expansion of ice for −5 ° C to 0 ° C: in U. Hübschmann, E. Links, E. Hitzel: Tables for chemistry . ISBN 978-3-582-01234-0 . the volume expansion coefficient is named as 0.000213, which when converted results in a linear expansion coefficient of 0.000071.
17. ↑ Paetec GmbH, formulas and tables for secondary levels I and II, Berlin 1996, coefficient of expansion of ice at 0 ° C, whose coefficient of linear expansion is mentioned as 0.000051
18. ^ JA Kosinski, JG Gualtieri, A. Ballato: Thermal expansion of alpha quartz . In: Proceedings of the 45th Annual Symposium on Frequency Control 1991 . IEEE, Los Angeles 1991, ISBN 0-87942-658-6 , pp. 22 , doi : 10.1109 / freq.1991.145883 (American English).
19. ↑ ^{a b c d e} Fritz Dietzel: Technische Wärmelehre, Vogel Verlag Würzburg, 1990, ISBN 3-8023-0089-0 , p. 159ff.
20. ↑ ^{a b c} Physikalisches Praktikum, W01 - Thermische Ausdehnung b-tu.de, accessed January 12, 2019.
21. ↑ ^{a b c} Walther Bierwerth: Table book Chemietechnik, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , length and volume change p. 61 ff.
22. ↑ ^{a b c d e f g h i j k l m n o p q} Fratzscher / Picht: Material data and characteristic values ​​of process engineering, Verlag für Grundstoffindustrie Leipzig, GDR 1979 / BRD 1993, p. 99ff.
23. ↑ Walther Bierwerth: Table book Chemietechnik, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , length and volume change, change of the container filling level in%, p. 75.