- Although the linear dimensions and volumes of bodies change little with temperature changes, nevertheless this change often has to be taken into account in practice; at the same time, this phenomenon is widely used in everyday life and technology.
Taking into account the thermal expansion of bodies
A change in the size of solids due to thermal expansion leads to the appearance of enormous elastic forces if other bodies prevent this change in size. For example, a steel bridge beam with a cross section of 100 cm2, when heated from -40 °C in winter to +40 °C in summer, if the supports prevent its elongation, creates pressure on the supports (tension) of up to 1.6 10 8 Pa, i.e. on supports with a force of 1.6 10 6 N.
The given values can be obtained from Hooke's law and formula (9.2.1) for the thermal expansion of bodies.
According to Hooke's law, mechanical stress is where is the relative elongation, and E is Young's modulus. According to (9.2.1). Substituting this value of relative elongation into the formula of Hooke's law, we obtain
For steel, Young's modulus E = 2.1 10 11 Pa, temperature coefficient of linear expansion α 1 = 9 10 -6 K -1. Substituting these data into expression (9.4.1), we find that at Δt = 80 °C the mechanical stress is σ = 1.6 10 8 Pa.
Since S = 10 -2 m 2, then the force F = σS = 1.6 10 6 N.
To demonstrate the forces that appear when a metal rod cools, you can do the following experiment. Let's heat an iron rod with a hole at the end into which a cast iron rod is inserted (Fig. 9.5). Then we insert this rod into a massive metal stand with grooves. When cooled, the rod contracts, and such great elastic forces arise in it that the cast iron rod breaks.
Rice. 9.5
The thermal expansion of bodies must be taken into account when designing many structures. Care must be taken to ensure that bodies can freely expand or contract as temperatures change.
For example, it is forbidden to pull telegraph wires tightly, as well as power line wires between supports. In summer, the sagging of wires is noticeably greater than in winter.
Metal steam pipelines, as well as water heating pipes, have to be equipped with bends (compensators) in the form of loops (Fig. 9.6).
Rice. 9.6
Internal stresses can arise when a homogeneous body is heated unevenly. For example, a glass bottle or glass made of thick glass may burst if you pour hot water. First of all, heating occurs in the internal parts of the vessel in contact with hot water. They expand and put a lot of pressure on the outer cold parts. Therefore, vessel destruction may occur. A thin glass does not burst when hot water is poured into it, since its inner and outer parts heat up equally quickly.
Quartz glass has a very low temperature coefficient of linear expansion. Such glass can withstand uneven heating or cooling without cracking. For example, cold water can be poured into a red-hot quartz glass flask, while a flask made of ordinary glass will burst during such an experiment.
Dissimilar materials subject to periodic heating and cooling should be joined together only if their dimensions change equally with temperature changes. This is especially important for large product sizes. For example, iron and concrete expand equally when heated. That is why reinforced concrete has become widespread - hardened concrete mortar poured into a steel lattice - reinforcement (Fig. 9.7). If iron and concrete expanded differently, then as a result of daily and annual temperature fluctuations, the reinforced concrete structure would soon collapse.
Rice. 9.7
A few more examples. Metal conductors soldered into glass cylinders of electric lamps and radio lamps are made of an alloy (iron and nickel) that has the same coefficient of expansion as glass, otherwise the glass would crack when the metal was heated. The enamel used to cover the dishes and the metal from which these dishes are made must have the same coefficient of linear expansion. Otherwise, the enamel will burst when the dishes coated with it heat and cool.
Significant forces can also be developed by a liquid if it is heated in a closed vessel that does not allow the liquid to expand. These forces can lead to the destruction of vessels that contain fluid. Therefore, this property of the liquid also has to be taken into account. For example, hot water heating pipe systems are always equipped with an expansion tank connected to the top of the system and exposed to the atmosphere. When water is heated in a pipe system, a small part of the water passes into the expansion tank, and this eliminates the stressed state of the water and pipes. For the same reason, an oil-cooled power transformer has an oil expansion tank at the top. As the temperature rises, the oil level in the tank increases, and as the oil cools, it decreases.
The use of thermal expansion in technology
Thermal expansion of bodies is widely used in technology. Let's give just a few examples. Two dissimilar plates (for example, iron and copper), welded together, form a so-called bimetallic plate (Fig. 9.8).
Rice. 9.8
When heated, such plates bend due to the fact that one expands more than the other. The one of the strips (copper) that expands more is always on the convex side (Fig. 9.9). This property of bimetallic strips is widely used for temperature measurement and regulation.
Rice. 9.9
Thermostat
Figure 9.10 schematically shows the design of one type of temperature controller. Bimetallic arc 1 changes its curvature when the temperature changes. A metal plate 2 is attached to its free end, which, when the arc unwinds, touches contact 3, and when twisted, moves away from it. If, for example, contact 3 and plate 2 are connected to the ends 4, 5 of an electrical circuit containing a heating device, then when the contact and plate come into contact electrical circuit closes: the device will begin to heat the room. Bimetallic arc 1, when heated, will begin to twist and at a certain temperature will disconnect plate 2 from contact 3: the circuit will break and heating will stop.
Rice. 9.10
When cooling, arc 1, unwinding, will again force the heating device to turn on. Thus, the room temperature will be maintained at this level. A similar thermostat is installed in incubators where it is necessary to maintain a constant temperature. In everyday life, thermostats are installed in refrigerators, electric irons, etc. The rim (bandage) of a railway car wheel is made of steel, the rest of the wheel is made of a cheaper metal - cast iron. Tires are put on the wheels when heated. After cooling, they shrink and therefore hold firmly.
Also, when heated, they put pulleys, bearings on shafts, iron hoops on wooden barrels, etc. The property of liquids to expand when heated and contract when cooled is used in instruments used to measure temperature - thermometers. Mercury, alcohol, etc. are used as liquids for making thermometers.
When bodies expand or contract, enormous mechanical stresses arise if other bodies prevent the change in size. The technique uses bimetallic plates that change their shape when heated.
Differential expansionhas great practical significance. Sometimes it is very difficult to open metal screw caps on glass or plastic bottles. If the top of the bottle is held under running hot water, the metal will expand more than glass or plastic, and the cap will open easily.
A glass stopper that fits tightly into the neck of a glass bottle can also be removed by holding the neck under running hot water. Although the expansion coefficient of the neck is the same as that of the cork, but the glass is very high, and the neck will expand before the cork becomes hot, and the cork can be easily removed.
Expanding glass is often a source of trouble at home. When glassware is filled with hot liquid, it often breaks. The reason is that the part of the glass in contact with the hot liquid very quickly acquires the temperature of the liquid and expands, while the rest remains cold, since glass is a poor conductor.
As a result, tension is established inside the glass, and the dishes burst. When making jam, a prudent cook will preheat the vessel in the oven before filling it with jam. This ensures that both the glass and the jam are heated to approximately the same temperature. Your valuable cut glassware will be preserved if you consider putting it in hot water.
Various thermal expansion in everyday life
The period of a pendulum depends on the length of the pendulum itself. When the temperature rises, the length of the pendulum increases and the period of its oscillation increases. The pendulum swings more slowly. The figure shows two types of compensated pendulum. In Figure 1, a the rod is made of invar, and the body of the lentil pendulum is made of steel.
The downward expansion of the invar is compensated by the upward expansion of the lentils. In this case, the position of the center of gravity, and therefore, remains unchanged. To set the desired period of oscillation of the pendulum, the position of the lentil is adjusted with a screw. Once installed in the desired position, such a pendulum is self-compensating.
Figure 1, b shows a more complex pendulum. The unshaded rods are larger and expand enough to compensate for the expansion of the longer shaded rods. Nowadays, when most buildings are centrally heated, they are kept at a more or less constant temperature, but it is still important to compensate for thermal effects.
The gas oven thermostat (Fig. 2) uses different thermal expansion of metals. The gas is supplied through the inlet pipe and passes through ports D, E and F to the burners. Cylinder B is made of brass and rod A is made of invar. As the oven temperature rises, the brass expands much more than the invar, causing valve C to move to the left and close holes E and F.
Thus, the gas supply to the oven is reduced and the gas burns weakly. Hole D is necessary to receive gas to prevent the burners from going out when the valve is closed. As cylinder B cools, it contracts and valve C moves to the right, allowing large quantity gas to the burners. The external regulator G allows you to tighten or loosen valve C, thus reducing or increasing the gas flow and reducing or increasing the temperature in the oven.
Ticket No. 3
“Thermal expansion of bodies. Thermometer. Temperature scales. The significance of thermal expansion of bodies in nature and technology. Features of thermal expansion of water"
Thermal expansion- change in the linear dimensions and shape of a body when its temperature changes.
Cause: the temperature of the body increases -> the speed of movement of the molecules increases -> the amplitude of vibrations increases -> the distance between the molecules increases, and therefore the size of the body.
Different bodies expand differently when heated, because the masses of the molecules are different, therefore, the kinetic energy and intermolecular distances change differently.
Quantitatively, the thermal expansion of liquids and gases at constant pressure is characterized by volumetric coefficient of thermal expansion (β).
V=V0(1+β(tfinal-tinitial))
Where V is the volume of the body at the final temperature, V0 is the volume of the body at the initial temperature
To characterize thermal expansion solids additionally enter a coefficient linear thermal expansion (α)
l=l0 (1+α(tfinal-tinitial))
Where l is the length of the body at the final temperature, l0 is the length of the body at the initial temperature
Thermometer- temperature measuring device
The action of the thermometer is based on the thermal expansion of the liquid.
Invented by Galileo in 1597.
Types of thermometers:
· mercury (from -35 to 750 degrees Celsius)
alcohol (from -80 to 70 degrees Celsius)
· pentane (from -200 to 35 degrees Celsius)
Scales:
Fahrenheit. Fahrenheit in 1732 - filled pipes with alcohol, later switched to mercury. The zero of the scale is the temperature of the mixture of snow with ammonia or table salt. Freezing point of water is 32°F. A healthy person's temperature is 96°F. Water boils at 212°F.
Celsius. Swedish physicist Celsius in 1742. The freezing point of a liquid is 0°C and the boiling point is 100°C
Kelvin scale. In 1848, the English physicist William Thomson (Lord Kelvin). The reference point is “absolute zero” - -273.15°C. At this temperature, the thermal movement of molecules stops. 1°K=1°C
In fact, absolute zero is not reachable.
In everyday life and technology thermal expansion is very great importance. On electric railways It is necessary to maintain constant tension in the wire that supplies electric locomotives in winter and summer. To do this, the tension in the wire is created by a cable, one end of which is connected to the wire, and the other is thrown over a block and a load is suspended from it.
When constructing a bridge, one end of the truss is placed on rollers. If this is not done, then when it expands in summer and contracts in winter, the truss will loosen the abutments on which the bridge rests.
When making incandescent lamps, part of the wire running inside the glass must be made of a material whose expansion coefficient is the same as that of glass, otherwise it may crack.
Power line wires are never tensioned to avoid breaking.
Steam pipelines are equipped with bends and compensators.
Thermal expansion of air plays a big role role in natural phenomena. Thermal expansion of air creates the movement of air masses in the vertical direction (heated, less dense air rises up, cold and less dense air goes down). Uneven heating of air in different parts the ground gives rise to wind. Uneven heating of water creates currents in the oceans.
When rocks are heated and cooled due to daily and annual temperature fluctuations (if the composition of the rock is heterogeneous), cracks form, which contributes to the destruction of rocks.
The most abundant substance on the Earth's surface is water- has a feature that distinguishes it from most other liquids. It expands when heated only above 4 °C. From 0 to 4 °C, the volume of water, on the contrary, decreases when heated. Thus, water has its greatest density at 4 °C. These data refer to fresh (chemically pure) water. U sea water the highest density is observed at approximately 3 °C. An increase in pressure also lowers the temperature of the highest density of water.
Although the linear dimensions and volumes of bodies change little with temperature changes, nevertheless this change often has to be taken into account in practice; at the same time, this phenomenon is widely used in everyday life and technology.
Taking into account the thermal expansion of bodies
A change in the size of solids due to thermal expansion leads to the appearance of enormous elastic forces if other bodies prevent this change in size. For example, a steel bridge beam with a cross-section of 100 cm2, when heated from -40 °C in winter to +40 °C in summer, if the supports prevent its elongation, creates pressure on the supports (tension) of up to 1.6 108 Pa, i.e. acts on the supports with a force of 1.6 106N.
The given values can be obtained from Hooke's law and formula (9.2.1) for the thermal expansion of bodies.
F
According to Hooke's law, mechanical stress a = ^ = Ee,
Where? = y- - relative elongation, a E - Young's modulus, "o
According to (9.2.1) y1 = e = Substituting this value relative to
strong extension into the formula of Hooke's law, we get
For steel, Young's modulus E = 2.1 1011 Pa, temperature coefficient of linear expansion a1 = 9 10-6 K-1. Substituting these data into expression (9.4.1), we find that at At = 80 °C the mechanical stress is a = 1.6 108 Pa.
Since S = 10~2 m2, then the force F = aS = 1.6 106 N.
To demonstrate the forces that appear when a metal rod cools, you can do the following experiment. Let's heat an iron rod with a hole at the end into which a cast iron rod is inserted (Fig. 9.5). Then we insert this rod into a massive metal stand with grooves. When cooled, the rod contracts, and such great elastic forces arise in it that the cast iron rod breaks.
Rice. 9.5
The thermal expansion of bodies must be taken into account when designing many structures. It is necessary to take measures to ensure that bodies can freely expand or contract when temperature changes.
For example, it is forbidden to pull telegraph wires tightly, as well as power line wires between supports. In summer, the sagging of wires is noticeably greater than in winter.
Metal steam pipelines, as well as water heating pipes, have to be equipped with bends (compensators) in the form of loops (Fig. 9.6).
Internal stresses may ^^
disappear with uneven heating
homogeneous body. For example, glass - I I
A thick glass bottle or glass may burst if hot water is poured into it. First of all, what happened was Fig. 9.6 1. The internal parts of the vessel in contact with hot water are heated. They expand and put strong pressure on the outer cold parts. Therefore, vessel destruction may occur. A thin glass does not burst when hot water is poured into it, since its inner and outer parts heat up equally quickly.
Quartz glass has a very low temperature coefficient of linear expansion. Such glass can withstand uneven heating or cooling without cracking. For example, cold water can be poured into a red-hot quartz glass flask, while a flask made of ordinary glass will burst during such an experiment.
Dissimilar materials subject to periodic heating and cooling should be joined together only if their dimensions change equally with temperature changes. This is especially important for large product sizes. For example, iron and concrete expand equally when heated. That is why reinforced concrete has become widespread - hardened concrete mortar poured into a steel lattice - reinforcement (Fig. 9.7). If iron and concrete expanded differently, then as a result of daily and annual temperature fluctuations, the reinforced concrete structure would soon collapse.
A few more examples. Metal conductors soldered into glass cylinders of electric lamps and radio lamps are made of an alloy (iron and nickel) that has the same coefficient of expansion as glass, otherwise the glass would crack when the metal was heated. The enamel used to cover the dishes and the metal from which these dishes are made must have the same coefficient of linear expansion. Otherwise, the enamel will burst when the dishes coated with it heat and cool.
Significant forces can also be developed by a liquid if it is heated in a closed vessel that does not allow the liquid
expand. These forces can lead to the destruction of vessels that contain liquid. Therefore, this property of the liquid also has to be taken into account. For example, hot water heating pipe systems are always equipped with an expansion tank connected to the top of the system and exposed to the atmosphere. When water is heated in a pipe system, a small part of the water passes into the expansion tank, and this eliminates the stressed state of the water and pipes. For the same reason, an oil-cooled power transformer has an oil expansion tank at the top. When the temperature rises, the oil level in the tank increases, and when the oil cools, it decreases.
The use of thermal expansion in technology
Rice. 9.8
Thermostat
Figure 9.10 schematically shows the design of one type of temperature controller. Bimetallic arc 1 changes its curvature when the temperature changes. A metal plate 2 is attached to its free end, which, when the arc unwinds, touches contact 3, and when twisted, moves away from it. If, for example, contact 3 and plate 2 are connected to the ends 4, 5 of an electrical circuit containing a heating device, then in contact
Thermal expansion of bodies is widely used in technology. Let's give just a few examples. Two dissimilar plates (for example, iron and copper), welded together, form a so-called bimetallic plate (Fig. 9.8). When heated, such plates bend due to the fact that one expands more than the other. The one of the strips (copper) that expands more is always on the convex side (Fig. 9.9). This property of bimetallic strips is widely used to measure temperature and regulate it.
Once the contact and plate are in place, the electrical circuit will close: the device will begin to heat the room. Bimetallic arc 1, when heated, will begin to twist and at a certain temperature will disconnect plate 2 from contact 3: the circuit will break and heating will stop. When cooling, arc 1, unwinding, will again force the heating device to turn on. Thus, the room temperature will be maintained at this level. A similar thermostat is installed in incubators where it is necessary to maintain a constant temperature. In everyday life, thermostats are installed in refrigerators, electric irons, etc. The rim (bandage) of a railway car wheel is made of steel, the rest of the wheel is made of a cheaper metal - cast iron. Tires are put on the wheels when heated. After cooling, they shrink and therefore hold firmly.
Also, in a heated state, pulleys, bearings are put on shafts, iron hoops on wooden barrels, etc. The property of liquids to expand when heated and contract when cooled is used in instruments used to measure temperature - thermometers. Mercury, alcohol, etc. are used as liquids for making thermometers.
When bodies expand or contract, enormous mechanical stresses arise if other bodies prevent the change in size. The technique uses bimetallic plates that change their shape when heated.
Thermal expansion- a change in the linear dimensions and shape of a body when its temperature changes. To characterize the thermal expansion of solids, the coefficient of linear thermal expansion is introduced.
The mechanism of thermal expansion of solids can be represented as follows. If thermal energy is supplied to a solid body, then due to the vibration of atoms in the lattice, the process of absorption of heat occurs. In this case, the vibrations of atoms become more intense, i.e. their amplitude and frequency increase. As the distance between atoms increases, the potential energy, which is characterized by the interatomic potential, also increases.
The latter is expressed by the sum of the potentials of the repulsive and attractive forces. The repulsive forces between atoms change faster with changes in the interatomic distance than the attractive forces; As a result, the shape of the energy minimum curve turns out to be asymmetrical, and the equilibrium interatomic distance increases. This phenomenon corresponds to thermal expansion.
The dependence of the potential energy of interaction between molecules on the distance between them makes it possible to find out the cause of thermal expansion. As can be seen from Figure 9.2, the potential energy curve is highly asymmetrical. It increases very quickly (steeply) from the minimum value E p0(at point r 0) when decreasing r and grows relatively slowly with increasing r.
Figure 2.5
At absolute zero, in a state of equilibrium, the molecules would be at a distance from each other r 0, corresponding to the minimum value of potential energy E p0 . As the molecules heat up, they begin to vibrate around their equilibrium position. The range of oscillations is determined by the average energy value E. If the potential curve were symmetrical, then the average position of the molecule would still correspond to the distance r 0 . This would mean a general invariance of the average distances between molecules when heated and, therefore, the absence of thermal expansion. In fact, the curve is asymmetrical. Therefore, with an average energy equal to , the average position of a vibrating molecule corresponds to the distance r 1> r 0.
A change in the average distance between two neighboring molecules means a change in the distance between all the molecules in the body. Therefore, body size increases. Further heating of the body leads to an increase in the average energy of the molecule to a certain value , etc. At the same time, the average distance between the molecules also increases, since now the vibrations occur with a greater amplitude around the new equilibrium position: r 2 > r 1, r 3 > r 2 etc.
In relation to solids, the shape of which does not change with a change in temperature (with uniform heating or cooling), a distinction is made between a change in linear dimensions (length, diameter, etc.) - linear expansion and a change in volume - volumetric expansion. Liquids can change shape when heated (for example, in a thermometer, mercury enters a capillary). Therefore, in the case of liquids, it makes sense to talk only about volumetric expansion.
Basic law of thermal expansion of solid bodies states that a body with linear dimension L 0 when its temperature increases by ΔT expands by an amount Δ L, equal to:
Δ L = αL 0 ΔT, (2.28)
Where α - so-called coefficient of linear thermal expansion.
Similar formulas are available for calculating changes in area and volume of a body. In the simplest case presented, when the coefficient of thermal expansion does not depend on either the temperature or the direction of expansion, the substance will expand uniformly in all directions in strict accordance with the above formula.
The coefficient of linear expansion depends on the nature of the substance, as well as on temperature. However, if we consider temperature changes within not too wide limits, the dependence of α on temperature can be neglected and the temperature coefficient of linear expansion can be considered a constant value for a given substance. In this case, the linear dimensions of the body, as follows from formula (2.28), depend on the temperature change as follows:
L = L 0 ( 1 +αΔT) (2.29)
Of the solids, wax expands the most, exceeding in this respect many liquids. Depending on the type, the thermal expansion coefficient of wax is 25 to 120 times greater than that of iron. Of the liquids, ether expands the most. However, there is a liquid that expands 9 times more powerfully than ether - liquid carbon dioxide (CO3) at +20 degrees Celsius. Its expansion coefficient is 4 times greater than that of gases.
Quartz glass has the lowest coefficient of thermal expansion among solids - 40 times less than iron. A quartz flask heated to 1000 degrees can be safely lowered into ice water without fear for the integrity of the vessel: the flask will not burst. Diamond also has a low expansion coefficient, although greater than that of quartz glass.
Of the metals, the type of steel that expands the least is called Invar; its coefficient of thermal expansion is 80 times less than that of ordinary steel.
Table 2.1 below shows the coefficients of volumetric expansion of some substances.
Table 2.1 - The value of the isobaric expansion coefficient of some gases, liquids and solids at atmospheric pressure
| Volume expansion coefficient | Linear expansion coefficient | ||||
| Substance | Temperature, °C | α×10 3 , (°C) -1 | Substance | Temperature, °C | α×10 3 , (°C) -1 |
| Gases | Diamond | 1,2 | |||
| Graphite | 7,9 | ||||
| Helium | 0-100 | 3,658 | Glass | 0-100 | ~9 |
| Oxygen | 3,665 | Tungsten | 4,5 | ||
| Liquids | Copper | 16,6 | |||
| Water | 0,2066 | Aluminum | |||
| Mercury | 0,182 | Iron | |||
| Glycerol | 0,500 | Invar (36.1% Ni) | 0,9 | ||
| Ethanol | 1,659 | Ice | -10 o to 0 o C | 50,7 |
Control questions
1. Characterize the distribution of normal vibrations by frequency.
2. What is a phonon?
3. Explain the physical meaning of the Debye temperature. What determines the Debye temperature for a given substance?
4. Why does the lattice heat capacity of a crystal not remain constant at low temperatures?
5. What is called the heat capacity of a solid? How is it determined?
6. Explain the dependence of the crystal lattice heat capacity Cresh on temperature T.
7. Obtain the Dulong-Petit law for the molar heat capacity of a lattice.
8. Obtain Debye’s law for the molar heat capacity of a crystal lattice.
9. What contribution does the electronic heat capacity make to the molar heat capacity of the metal?
10. What is the thermal conductivity of a solid? How is it characterized? How does thermal conductivity occur in the cases of metal and dielectric.
11. How does the thermal conductivity of a crystal lattice depend on temperature? Explain.
12. Define the thermal conductivity of an electron gas. Compare χ el And χ solve in metals and dielectrics.
13. Give a physical explanation for the mechanism of thermal expansion of solids? Can CTE be negative? If yes, then explain the reason.
14. Explain the temperature dependence of the coefficient of thermal expansion.
