Expansion coefficient
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:
The temperature-dependent length of a rod can be calculated by solving this differential equation , it is:
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 :
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 :
The change in length in a linear approximation is thus:
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.
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
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:
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:
With a room expansion coefficient that is independent of the temperature, this results together with :
As for the coefficient of linear expansion, the linearization can be used here as an approximation for small temperature changes:
With a Maxwell relation it is possible to relate the spatial expansion coefficient to the entropy :
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:
If the expansion coefficient is known as a function of temperature, the density results from:
Here, any temperature for. B. where the density is known.
Eduard Grüneisen has shown that the quotient between the thermal expansion coefficient and the specific heat capacity is approximately independent of the temperature.
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 .
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:
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.
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).
- The final values after thermal expansion are: and .
The following applies: and .
For the change in length due to thermal expansion applies: .
The volume of the cube after the expansion ,, is given by:
- .
After multiplying the cubic binomial it follows:
- .
By subtracting the initial volume, the change in volume of the cube resulting from the thermal expansion follows:
- and thus:
- .
Now, in the definition equation of the cubic expansion coefficient, the difference volume is substituted (exchanged) by this equation:
- .
It follows by substituting of and :
- .
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 :
- .
As a limit value, this equation shows the known equation for the case that the temperature difference approaches zero.
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 .
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:
Inserted into the definition of the room expansion coefficient results in:
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:
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:
- and
- .
The following applies to all solids and liquids that do not show any density anomaly :
- and for .
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:
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:
- and
- .
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:
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)
designation | α in 10 ^{−6} K ^{−1} |
---|---|
aluminum | 23.1 |
lead | 28.9 |
iron | 11.8 |
nickel | 13.0 |
gold | 14.2 |
iridium | 7th |
copper | 16.5 |
lithium | 58 |
magnesium | 24.8 |
sodium | 7.1 |
platinum | 8.8 |
silver | 18.9 |
Tantalum | 6.6 |
titanium | 8.6 |
zinc | 30.2 |
tin | 22.0 |
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)
designation | α in 10 ^{−6} K ^{−1} |
---|---|
diamond | 1.18 |
Germanium | 5.8 |
graphite | 1.9 ... 2.9 |
white phosphorus | 125 |
rhombic sulfur | 74 |
silicon | 2.6 |
Metal alloys
designation | α in 10 ^{−6} K ^{−1} |
---|---|
Aluminum bronze | 15 to 16 |
bronze | 17.5 |
"Indilatans Extra" (Krupp) (36Ni, XX) at 12 to 100 ° C | −0.04 |
Invar | 0.55 ... 1.2 |
Constantan (at −191 to 16 ° C) | 12.22 |
Brass | 18.4 to 19.3 |
Platinum iridium | 8.9 |
steel | 11… 13 |
Building materials
designation | α in 10 ^{−6} K ^{−1} |
---|---|
concrete | 12 |
Wood (oak) | 8th |
Clinker (hard-fired brick) | 2.8 to 4.8 |
Brick | 5 |
Plastics
designation | α in 10 ^{−6} K ^{−1} |
---|---|
Soft rubber | 17 to 28 |
Hard rubber | 80 |
Polyamide (PA) | 60 to 150 |
Polycarbonate (PC) | 60 to 70 |
Polyethylene (HD-PE) | 150 to 200 |
Polypropylene (PP) | 100 to 200 |
Polyoxymethylene (POM) | 70 to 130 |
Polytetrafluoroethylene (PTFE) | 100 to 160 |
Polyvinyl chloride (rigid PVC) | 70 to 100 |
Polymethyl methacrylate (PMMA, plexiglass ) | 75 to 80 |
Glass and ceramics
designation | α in 10 ^{−6} K ^{−1} |
---|---|
Borosilicate glass | 3.3 |
German melting jar (for connections with platinum or invar) | 9.0 |
Duran glass / Pyrex glass | 3.6 |
Enamel (enamel coatings) | 8.0 to 9.5 |
Window glass | 10 |
Jena device glass "No.20" | 4.8 |
Porcelain , Berliner | 4 to 6 |
Porcelain , Meissner | 3 to 5 |
Quartz glass (silicon dioxide) | 0.54 |
Technical ceramics | 2… 13 |
Zerodur (glass ceramic) | 0 ± 0.007 |
For other substances from which ceramic products (workpieces) are manufactured, see "Compounds and chemicals".
Chemical compounds
designation | α in 10 ^{−6} K ^{−1} |
---|---|
Aluminum oxide, crystalline (corundum) | 5.6 to 7.0 |
Ice (−5 ° C to 0 ° C) | 51 to 71 |
Mica (magnesium silicate) | 13.5 |
Magnesium oxide | 13.1 |
Silicon dioxide (quartz) | 12 ... 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
designation | γ in 10 ^{−3} K ^{−1} |
---|---|
bromine | 1.11 or 1.13 |
Galinstan (eutectic thermometer fluid) | 0.126 |
NaK (eutectic alloy) | 0.16 |
mercury | 0.1811 |
Nitric acid (100%) | 1.24 |
hydrochloric acid | 0.30 |
Carbon disulfide | 1.18 |
Sulfuric acid (about 99%) | 0.57 |
Water at 0 ° C | −0.068 |
Water at approx. 20 ° C | 0.2064 |
Water at 100 ° C | 0.782 |
Organic liquids
designation | γ in 10 ^{−3} K ^{−1} | chemical group |
---|---|---|
Petrol (at 0 ° C) | 1.0 | Paraffins |
n-heptane | 1.09 | Paraffins |
Heating oil / diesel fuel | 0.96 | Paraffins |
n-hexane | 1.35 | Paraffins |
Mineral oil, hydraulic oil | 0.7 | Paraffins |
Paraffin oil | 0.764 | Paraffins |
n-pentane | 1.6 | Paraffins |
petroleum | 0.9 to 1 | Paraffins |
Lubricating oil | 0.6 to 0.7 | Paraffins |
Trichloromethane | 1.21 | halogenated paraffin |
Carbon tetrachloride | 1.21 | halogenated paraffin |
Methanol | 1.49 | monohydric alcohols |
Ethanol | 1.10 | monohydric alcohols |
Propanetriol | 0.520 | trihydric alcohols |
Ethanoic acid | 1.08 | Paraffinic acids |
Diethyl ether | 1.62 | Ether |
Propanone | 1.46 | Ketones |
olive oil | 0.72 | Fatty acid esters |
Benzene | 1.14 | Aromatic hydrocarbons |
Turpentine oil | 1 | Pines , terpenes |
Methylbenzene | 1.11 | Aromatic hydrocarbons |
Gases
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:
- and for the ideal gas applies:
The expansion coefficient of the ideal gas at 0 ° C (reference temperature) is therefore:
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:
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 | in K ^{−1} | (Saturated steam) in K ^{−1} | Hints |
---|---|---|---|---|
0.01 | 0.0006112 | −0.0000855 | 0.003669 | Density anomaly of water |
10 | 0.0012271 | 0.0000821 | 0.003544 | |
20th | 0.0023368 | 0.0002066 | 0.003431 | |
30th | 0.0042417 | 0.0003056 | 0.003327 | |
40 | 0.0073749 | 0.0003890 | 0.003233 | |
50 | 0.012335 | 0.0004624 | 0.003150 | |
60 | 0.019919 | 0.0005288 | 0.003076 | |
70 | 0.031161 | 0.0005900 | 0.003012 | |
80 | 0.047359 | 0.0006473 | 0.002958 | |
90 | 0.070108 | 0.0007019 | 0.002915 | |
100 | 0.101325 | 0.0007547 | 0.002882 | |
150 | 0.47597 | 0.001024 | 0.002897 | |
200 | 1.5551 | 0.001372 | 0.003291 | |
250 | 3.9776 | 0.001955 | 0.004321 | |
300 | 8.5917 | 0.003293 | 0.007117 | |
350 | 16,537 | 0.01039 | 0.02175 | |
360 | 18.674 | 0.01928 | 0.03899 | |
370 | 21.053 | 0.09818 | 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:
- with .
The mean room expansion coefficient between the selected temperatures results from:
- .
Alternatively, values of the mass-specific volumes or the molar volumes can also be used:
- .
The specific volumes are inversely proportional to the densities.
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 .
Numerical values of liquids at air pressure
substance | / in ° C | / in g / cm³ | in K | mean temp. in ° C | in K ^{−1} | swell |
---|---|---|---|---|---|---|
water | 0/1 | 0.999840 / 0.999899 | 1 | 0.5 | −0.000059006 | |
3/5 | 0.999964 / 0.999964 | 2 | 4th | 0 | ||
0/20 | 0.999840 / 0.998203 | 20th | 10 | 0.0000820 | ||
17/19 | 0.998773 / 0.998403 | 2 | 18th | 0.0001853 | ||
19/21 | 0.998403 / 0.997991 | 2 | 20th | 0.0002064 | ||
24/26 | 0.997295 / 0.996782 | 2 | 25th | 0.0002573 | ||
20/100 | 0.998203 / 0.95835 | 80 | 60 | 0.0005198 | ||
90/100 | 0.96532 / 0.95835 | 10 | 95 | 0.0007273 | ||
mercury | −20 / −18 | 13.6446 / 13.6396 | 2 | −19 | 0.0001833 | |
−2 / 2 | 13.6000 / 13.5901 | 4th | 0 | 0.00018212 | ||
0/20 | 13.5951 / 13.5457 | 20th | 10 | 0.0001823 | ||
16/20 | 13.5556 / 13.5457 | 4th | 18th | 0.00018271 | ||
18/22 | 13.5507 / 13.5408 | 4th | 20th | 0.00018278 | ||
24/26 | 13.5359 / 13.5310 | 2 | 25th | 0.00018107 | ||
20/100 | 13.5457 / 13.3512 | 80 | 60 | 0.0001821 | ||
90/100 | 13.3753 / 13.3512 | 10 | 95 | 0.0001805 | ||
240/260 | 13.018 / 12.970 | 20th | 250 | 0.00018504 | ||
Propanetriol (glycerine) | 20/60 | 1.260 / 1.239 | 40 | 40 | 0.0004237 | |
80/100 | 1.224 / 1.207 | 20th | 90 | 0.0007042 | ||
140/160 | 1.167 / 1.143 | 20th | 150 | 0.001050 | ||
180/200 | 1.117 / 1.090 | 20th | 190 | 0.001239 | ||
220/240 | 1.059 / 1.025 | 20th | 230 | 0.001659 | ||
" Baysilone M10" ® silicone oil | −40 / 0 | 0.990 / 0.950 | 40 | −20 | 0.00105 | |
0/40 | 0.950 / 0.920 | 40 | 20th | 0.000815 | ||
40/80 | 0.920 / 0.880 | 40 | 60 | 0.00114 | ||
80/120 | 0.880 / 0.850 | 40 | 100 | 0.000882 | ||
120/160 | 0.850 / 0.810 | 40 | 140 | 0.00123 | ||
160/200 | 0.810 / 0.770 | 40 | 180 | 0.00130 | ||
200/240 | 0.770 / 0.740 | 40 | 220 | 0.00101 |
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 | / in ° C | / in g / cm³ | in K | mean temperature in ° C | in K ^{−1} | swell |
---|---|---|---|---|---|---|
boiling superheated water | 95/100 | 0.96172 / 0.95813 | 5 | 97.5 | 0.00074938 | |
90/110 | 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 | / in ° C | / in g / cm³ | in K | mean temperature in ° C | 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 | / in ° C | / in g / cm³ | in K | mean temperature in ° C | 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 | / in ° C | / in g / l | in K | mean temperature in ° C | 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 | / in ° C | / in g / l | in K | mean temperature in ° C | 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:
- .
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 :
- .
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.
See also
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
- ↑ The coefficients of thermal expansion of wood and wood products (PDF; 5.1 MB) Retrieved on May 10, 2012.
- ↑ ^{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 .
- ↑ ^{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 .
- ↑ ^{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.
- ↑ 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.
- ↑ ^{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 .
- ↑ ^{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).
- ↑ ^{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
- ↑ 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 .
- ↑ ^{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.
- ↑ Wolfgang Kaiser: Kunststoffchemie für Ingenieure: From synthesis to application. 2nd Edition. Carl Hanser, 2007, ISBN 978-3-446-41325-2 , p. 228
- ↑ Technical Glasses Data Sheet (PDF) schott.com.
- ↑ Product information page of the manufacturer Heraeus-Quarzglas. at heraeus-quarzglas.de.
- ↑ Ceramic dressing thermal properties. Retrieved May 29, 2018 .
- ↑ 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.).
- ^ 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.
- ↑ 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
- ^ 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).
- ↑ ^{a } ^{b } ^{c } ^{d } ^{e} Fritz Dietzel: Technische Wärmelehre, Vogel Verlag Würzburg, 1990, ISBN 3-8023-0089-0 , p. 159ff.
- ↑ ^{a } ^{b } ^{c} Physikalisches Praktikum, W01 - Thermische Ausdehnung b-tu.de, accessed January 12, 2019.
- ↑ ^{a } ^{b } ^{c} Walther Bierwerth: Table book Chemietechnik, Europa-Lehrmittel KG, 2005, ISBN 3-8085-7085-7 , length and volume change p. 61 ff.
- ↑ ^{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.
- ↑ 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.