# Expansion coefficient

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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}}$

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

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
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
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:
${\ displaystyle \ gamma _ {\ text {ideal gas}} ^ {(T _ {\ text {abs.}})} = {\ frac {1} {T _ {\ text {abs.}}}}}$

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

${\ displaystyle \ gamma _ {\ text {Ideal gas}} ^ {\ text {(0 ° C)}} = {\ frac {1} {\ text {273.15 K}}} \ approx 0 {,} 003661 }$

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 ${\ displaystyle \ gamma _ {\ text {Water}} ^ {(T)}}$in K −1 ${\ displaystyle \ gamma _ {\ text {Water vapor}} ^ {(T)}}$(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: ${\ displaystyle \ rho}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$

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

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

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

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

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

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 {avg.}}}$${\ displaystyle T}$${\ 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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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 ${\ displaystyle T _ {\ text {medium}}}$ ${\ displaystyle \ gamma _ {\ text {avg.}}}$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}}$

## 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. 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. 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. 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. 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. 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. 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. 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. 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.