The first thermometers used the change in the volume of a gas or liquid with a change in temperature. It is this property that made it possible to ascribe to any body a certain temperature, expressed by a number. In this chapter, we will look at how the linear dimensions of solids change, as well as volumes, solids and liquids, depending on temperature. Enough has been said about the dependence of gas volume on temperature.

§ 9.1. Thermal expansion of bodies

When the temperature changes, the dimensions of the bodies change: when heated, as a rule, they increase, when cooled, they decrease. Why is this happening?

The increase in the size of the small body is small and difficult to see. But if you take an iron wire 1.5-2 m long and heat it with an electric current, then the elongation can be detected by eye without special devices. To do this, one end of the wire must be secured, and the other is thrown over the block. A weight must be attached to this end, pulling the wire downward (Figure 9.1). According to the pointer connected to the load, it is judged about the change in the length of the wire during its heating or cooling.

The expansion of a small steel ball heated on a gas burner can be seen as it passes through the ring. A cold ball easily passes through the ring, while a heated one gets stuck in it. When the ball cools down, it goes through the ring again.

How to explain why bodies expand when heated?

Molecular picture of thermal expansion

The dependence of the potential energy of interaction of molecules on the distance between them makes it possible to find out the reason for the occurrence of thermal expansion. As seen in Figure 9.2, the potential energy curve is highly asymmetric. It rises very quickly (steeply) from the minimum value. E p0 (at the point r 0) when decreasing G and grows relatively slowly with increasing r.

At absolute zero, in a state of equilibrium, the molecules would be at a distance from each other r 0 corresponding to the minimum potential energy E p0 . As they heat up, the molecules begin to vibrate around the equilibrium position. The swing is determined by the average energy E. If the potential curve were symmetric, then the average position of the molecule would still correspond to the distance r 0. This would mean the general invariability of the average distances between molecules when heated and, consequently, the absence of thermal expansion. In fact, the curve is asymmetrical. Therefore, with an average energy equal to , the average position of the vibrating molecule corresponds to the distance r 1 > r 0 .

A change in the average distance between two adjacent molecules means a change in the distance between all molecules in the body. Therefore, the size of the body increases.

Further heating of the body leads to an increase in the average energy of the molecule to a certain value , and so on. In this case, the average distance between the molecules also increases, since now oscillations are performed with a greater amplitude around the new equilibrium position: r 2 > r 1 , r 3 > r 2 etc.

When the body heats up, the average distance between the vibrating molecules increases, so the size of the body also increases.

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 cm 2, when heated from -40 ° C in winter to +40 ° C in summer, if the supports prevent it from lengthening, creates a pressure on the supports (stress) up to 1.6 10 8 Pa, that is, it acts 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, where is the relative elongation, a E- Young's modulus. According to (9.2.1). Substituting this value of the relative elongation in the formula for Hooke's law, we obtain

Steel has 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 obtain that at Δ t= 80 ° C mechanical stress σ = 1.6 10 8 Pa.

Because 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 is cooled, the following experiment can be done. We heat an iron rod with a hole at the end, into which a cast iron rod is inserted (Figure 9.5). Then we will insert this rod into a massive metal stand with grooves. When cooled, the rod shrinks, and elastic forces are so strong in it that the cast iron rod breaks.

Thermal expansion of bodies must be taken into account in the design of many structures. Care must be taken to ensure that bodies can expand or contract freely when the temperature changes.

It is impossible, for example, to pull tight telegraph wires, as well as wires of power lines (power lines) between the supports. In summer, the sagging of the wires is noticeably greater than in winter.

Metal steam pipelines, as well as water heating pipes, have to be supplied with bends (compensators) in the form of loops (Figure 9.6).

Internal stresses can arise when a homogeneous body is unevenly heated. For example, a glass bottle or thick glass can burst if hot water is poured into it. First of all, the internal parts of the vessel, which come into contact with hot water, are heated. They expand and put strong pressure on the outer cold parts. Therefore, destruction of the vessel may occur. A thin glass does not burst when hot water is poured into it, since its inner and outer parts warm up equally quickly.

Quartz glass has a very low temperature coefficient of linear expansion. Such glass withstands, without cracking, uneven heating or cooling. For example, cold water can be poured into a red-hot quartz glass cone, while an ordinary glass flask bursts during this experiment.

Dissimilar materials that undergo periodic heating and cooling should be joined together only when their dimensions change equally with temperature changes. This is especially important for large product sizes. So, for example, iron and concrete expand in the same way when heated. That is why reinforced concrete has become widespread - a hardened concrete solution poured into a steel lattice - reinforcement (Figure 9.7). If iron and concrete expanded in different ways, 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 tubes are made of an alloy (iron and nickel), which has the same coefficient of expansion as glass, otherwise the glass would crack when the metal was heated. The enamel used to cover the cookware and the metal from which the cookware is made must have the same coefficient of linear expansion. Otherwise, the enamel will burst when the cookware covered with it is heated and cooled.

Significant forces can also develop in 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 the vessels that contain the fluid. Therefore, this property of the liquid also has to be reckoned with. For example, hot water pipe systems are always equipped with an expansion vessel connected to the top of the system and communicating with the atmosphere. When the water in the pipe system is heated, a small part of the water passes into the expansion tank, and this eliminates the stress state of the water and pipes. For the same reason, the oil-cooled power transformer has an oil expansion tank at the top. When the temperature rises, the oil level in the tank rises, and when the oil cools down, it goes down.

The change in the linear dimensions of the body when heated is proportional to the change in temperature.

The vast majority of substances expand when heated. This is easily explained from the standpoint of the mechanical theory of heat, since when heated, the molecules or atoms of a substance begin to move faster. In solids, atoms begin to vibrate with greater amplitude around their average position in the crystal lattice, and they require more free space. As a result, the body expands. Likewise, liquids and gases, for the most part, expand with increasing temperature due to an increase in the rate of thermal motion of free molecules ( cm. Boyle's Law — Mariotte, Charles's Law, Ideal Gas Equation of State).

The basic law of thermal expansion says that a body with a linear size L in the corresponding dimension with an increase in its temperature by Δ T expands by the amount Δ L equal to:

Δ L = αLΔ T

where α — so-called coefficient of linear thermal expansion. Similar formulas are available for calculating changes in body area and volume. In the given simplest case, 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.

For engineers, thermal expansion is vital. When designing a steel bridge across a river in a city with a continental climate, one must take into account the possible temperature differences ranging from -40 ° C to + 40 ° C throughout the year. Such differences will cause a change in the total length of the bridge up to several meters, and so that the bridge does not heave in the summer and does not experience powerful breaking loads in winter, the designers compose the bridge from separate sections, connecting them with special thermal buffer joints, which are meshing, but not rigidly connected rows of teeth that close tightly in the heat and diverge quite widely in the cold. There can be quite a few of these buffers on a long bridge.

However, not all materials, especially crystalline solids, expand uniformly in all directions. And not all materials expand in the same way at different temperatures. The most striking example of the latter kind is water. When cooled, water first contracts, like most substances. However, starting from + 4 ° C and up to the freezing point of 0 ° C, the water begins to expand when cooled and shrink when heated (from the point of view of the above formula, we can say that in the temperature range from 0 ° C to + 4 ° C the coefficient of thermal expansion water α takes a negative value). It is thanks to this rare effect that the earth's seas and oceans do not freeze to the bottom even in the most severe frosts: water colder + 4 ° C becomes less dense than warmer, and floats to the surface, displacing water with a temperature higher than + 4 ° C to the bottom.

The fact that ice has a specific gravity lower than that of water is another (although not related to the previous one) anomalous property of water, to which we owe the existence of life on our planet. If not for this effect, the ice would go to the bottom of rivers, lakes and oceans, and they, again, would freeze to the bottom, killing all life.

Thermal expansion- change in the linear dimensions and shape of the 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, then due to the vibration of atoms in the lattice, the process of absorption of heat by it occurs. In this case, the vibrations of atoms become more intense, i.e. their amplitude and frequency increase. With an increase in the distance between atoms, the potential energy also increases, which is characterized by the interatomic potential.

The latter is expressed by the sum of the potentials of the forces of repulsion and attraction. The forces of repulsion between atoms change faster than the forces of attraction with a change in the interatomic distance; As a result, the shape of the energy minimum curve turns out to be asymmetric, and the equilibrium interatomic distance increases. This phenomenon corresponds to thermal expansion.

The dependence of the potential energy of interaction of molecules on the distance between them makes it possible to find out the reason for the occurrence of thermal expansion. As seen in Figure 9.2, the potential energy curve is highly asymmetric. It rises very quickly (steeply) from the minimum value. E p0(at the 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 potential energy E p0. As they heat up, the molecules begin to vibrate around the equilibrium position. The swing is determined by the average energy E. If the potential curve were symmetric, then the average position of the molecule would still correspond to the distance r 0. This would mean the general invariability of the average distances between molecules when heated and, consequently, the absence of thermal expansion. In fact, the curve is asymmetrical. Therefore, with an average energy equal to , the average position of the vibrating molecule corresponds to the distance r 1> r 0.

A change in the average distance between two adjacent molecules means a change in the distance between all molecules in the body. Therefore, the size of the body increases. Further heating of the body leads to an increase in the average energy of the molecule to a certain value , and so on. In this case, the average distance between the molecules also increases, since now oscillations are performed with a greater amplitude around the new equilibrium position: r 2 > r 1, r 3> r 2 etc.

With regard to solids, the shape of which does not change when the temperature changes (with uniform heating or cooling), a change in linear dimensions (length, diameter, etc.) is distinguished - linear expansion and volume change - volumetric expansion. In liquids, when heated, the shape can change (for example, in a thermometer, mercury enters the capillary). Therefore, in the case of liquids, it makes sense to speak only of volumetric expansion.

Basic law of thermal expansion rigid bodies states that a body with a linear dimension L 0 with an increase in its temperature by ΔT expands by the amount Δ L equal to:

Δ L = αL 0 ΔT, (2.28)

where α - so-called linear thermal expansion coefficient.

Similar formulas are available for calculating changes in body area and volume. In the given simplest case, 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 constant 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 solids, wax expands most of all, exceeding in this respect many liquids. The coefficient of thermal expansion of wax, depending on the grade, is 25 - 120 times higher than that of iron. Ether expands more strongly from liquids. However, there is a liquid that expands 9 times stronger than ether - liquid carbon dioxide (CO3) at +20 degrees Celsius. Its expansion coefficient is 4 times that of gases.

The smallest coefficient of thermal expansion of solids is possessed by quartz glass - 40 times less than iron. A quartz flask heated to 1000 degrees can be safely immersed in ice water without fear for the integrity of the vessel: the flask does not burst. Diamond also has a low coefficient of expansion, although higher than that of quartz glass.

Of the metals, the least expandable grade of steel is called Invar; its coefficient of thermal expansion is 80 times less than that of ordinary steel.

The following table 2.1 shows the coefficients of volumetric expansion of some substances.

Table 2.1 - The value of the isobaric coefficient of expansion of some gases, liquids and solids at atmospheric pressure

Volumetric expansion coefficient	Linear expansion coefficient
Substance	Temperature, ° С	α × 10 3, (° C) -1	Substance	Temperature, ° С	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. Give a characteristic of the distribution of normal vibrations in frequencies.

2. What is a phonon?

3. Explain the physical meaning of the Debye temperature. What determines the value of the Debye temperature for a given substance?

4. Why does the lattice heat capacity of the 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 lattice heat capacity of the crystal Cres on temperature T.

7. Obtain the Dulong-Petit law for the molar heat capacity of the lattice.

8. Obtain Debye's law for the molar heat capacity of the crystal lattice.

9. What contribution does the electronic heat capacity make to the molar heat capacity of a metal?

10. What is called the thermal conductivity of a solid? How is it characterized? How thermal conductivity is carried out in the cases of metal and dielectric.

11. How does the coefficient of thermal conductivity of a crystal lattice depend on temperature? Explain.

12. Give a definition of the thermal conductivity of an electron gas. Compare χ email and χ res in metals and dielectrics.

13. Give a physical explanation for the mechanism of thermal expansion of solids? Can CTE be negative? If so, explain the reason.

14. Explain the temperature dependence of the coefficient of thermal expansion.

Why do most solids expand when heated? This is due to the fact that with an increase in temperature, the kinetic energy of motion of particles, which are located in the nodes of the crystal lattice, increases. An increase in kinetic energy, in turn, leads to an increase in the vibration amplitude of these particles around the equilibrium position. As a result of an increase in the vibration amplitude, the average distance between particles in the crystal lattice increases, which leads to an increase in the linear dimensions of the entire body.

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