Dependence of body sizes on temperature - Knowledge Hypermarket. Taking into account the thermal expansion of bodies

When bodies are heated, the average kinetic energy translational motion of molecules and the average distance between molecules. Therefore, all substances expand when heated and contract when cooled. A distinction is made between linear and volumetric expansion.

The change in one specific size of a solid with changes in temperature is called linear expansion (or compression).

Where is the length of the rod at 0 0,

Linear expansion coefficient. Dimension = O C -1.

Body length at any temperature t: ;

With volumetric expansion volume increases: , where: – volume of the body at 0 0 C.

Body volume at any temperature t: , where:

Volume expansion coefficient;

It has been experimentally established that. That's why .

Likewise for surface area solid: .

There is one remarkable exception in liquids: water contracts when heated from 0 0 C to +4 0 C, and expands when cooled from +4 0 C to 0 0 C. The coefficient of volumetric expansion of water changes greatly with temperature changes.

Examples of thermal expansions:

When water freezes, it expands and breaks rocks, metal pipes and other technical structures.

In automation, bimetallic plates are used, using the difference in the linear expansion coefficients of each of the two plates. When heated, the bimetallic plate loses stability and presses the switch, causing the actuator to operate.

Thermal expansion is important to consider when laying rails, stringing wires, building bridges, etc. Terminals from electric lamps and radio lamps are made from a material whose linear expansion coefficient is close to the linear expansion coefficient of glass.

Melting and crystallization.
Diagram phase states

The transition of a substance from a solid state to a liquid state called melting and the transition from a liquid to a solid state is hardening or crystallization. Melting and solidification occur at the same temperature, called melting temperature. Pressure has virtually no effect on the melting temperature. The melting point of a substance at normal atmospheric pressure is called melting point.

When a solid melts, the distances between the particles forming the crystal lattice increase and the lattice itself is destroyed. For the vast majority of substances, the volume increases when melting and decreases when solidifying.

A region in which a substance is homogeneous in all physical and chemical properties, called phase state of this substance. The liquid and solid phases of a substance at the same temperature can remain in equilibrium indefinitely for a long time(ice and water at 0 0 C). Therefore, until the entire substance melts, its temperature remains unchanged, equal to the melting point.

Heat of fusion is the amount of heat that must be supplied to a body of mass m located at melting point to melt it.

Where - specific heat melting.

1 J/kg.

Figure 34 shows graphs of changes in the temperature of a substance during melting and solidification. The segment (Figure 34a) expresses the amount of heat, received a substance when heated in the solid state (from to T PL), a segment - when melting, and a segment - when heated in a liquid state. The segment (Figure 34b) expresses the amount of heat, given away a substance when cooled in a liquid state (from to), a segment - during solidification, and a segment - when cooled in a solid state.

Figure 34. Graphs of changes in the temperature of a substance during melting and solidification

Many solids have an odor. This proves that solids can change into a gaseous state without passing through a liquid state. Evaporation solids called sublimation or sublimation(from the Latin “sublimate” - to exalt). IN Food Industry“dry ice” (CO 2), which has this property, is used. The reverse process is also possible - the growth of crystals from gaseous substance(ice on windows, overgrowing of ROM jumpers).

For each substance you can make state diagram in coordinates P and T (Figure 35), on the basis of which you can easily determine what state this substance will be in under certain external conditions. Each point in the diagram corresponds to the equilibrium state of a substance, in which it can remain for as long as desired.

KC curve – dependence of saturation steam pressure on temperature. Point K is a critical point.

The CA curve is the temperature dependence of the pressure of saturating vapors in equilibrium with the surface of a solid.

The KC curve is the line of equilibrium between the liquid and gaseous phases. Straight line BC is the line of equilibrium of the liquid and solid phases. The AC curve is the line of equilibrium between the solid and gaseous phases.

Point C represents the equilibrium between all three phases and is called the triple point. Helium does not have a triple point.

Control questions:

1. Explain the thermal expansion of solids.

2. What is melting and crystallization? What is the heat of fusion?

3. What is sublimation of a substance?

4. Explain the state diagram of matter.

The change in the linear dimensions of a 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 speed of thermal movement of free molecules ( cm. Boyle-Marriott's law, Charles's law, Equation of state of an ideal gas).

The basic law of thermal expansion states that a body with linear size L in the corresponding dimension when its temperature increases by Δ T expands by an 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 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.

For engineers, thermal expansion is a vital phenomenon. When designing a steel bridge across a river in a city with a continental climate, it is impossible not to take into account possible temperature changes 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 tensile loads in the winter, designers compose the bridge from separate sections, connecting them with special thermal buffer joints, which are rows of teeth that engage, but are not rigidly connected, that close tightly in the heat and diverge quite widely in the cold. On a long bridge there may be quite a few of these buffers.

However, not all materials, especially crystalline solids, expand uniformly in all directions. And not all materials expand equally at different temperatures. Most shining example last kind- water. When water cools, it first contracts, like most substances. However, from +4°C to the freezing point of 0°C, water begins to expand when cooled and contract 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 than +4°C becomes less dense than warmer water and floats to the surface, displacing water with a temperature above +4°C to the bottom.

The fact that ice has a specific density lower than the density 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 sink to the bottom of rivers, lakes and oceans, and they, again, would freeze to the bottom, killing all living things.

Lesson typology: lesson in learning new knowledge and ways of doing things

Lesson type: combined

Lesson objectives:

• didactic:
explain the physical nature of the thermal expansion of bodies; teach students to calculate linear and volumetric changes in solid and liquid bodies when their temperature changes;
• educational:
improve students’ abilities to apply acquired theoretical knowledge to solutions practical problems; arouse interest in the process being studied;
• developing:
develop students' thinking about the use and significance of thermal expansion in nature and technology; be able to explain the mechanism of thermal expansion of bodies based on molecular kinetic theory.

Lesson Plan

1. Organization of the beginning of the lesson
2. Repetition of learned material
3. Learning new material
4. Intermediate fastening of material
5. Learning new material (continued) Annex 1
6. Reinforcing the material learned Appendix 2,
7. Homework assignment Appendix 4

Topic study plan.

Equipment: ball with ring; bimetallic plate; thermal relay; a flask with a rubber and glass tube inserted into a stopper; G – cut glass tube with a drop of water; uncolored water; electric stove; transformer; wire.

Demos:

1. Thermal expansion of solids.
2. Thermal expansion of liquids.
3. Action and purpose of a bimetallic thermal regulator.

Message:

Features of thermal expansion of water.

Motivation of cognitive activity of students

It is well known that a substance usually expands when heated and contracts when cooled, i.e. thermal deformation of the body occurs under the influence of molecular forces during heating and cooling. This phenomenon is explained by the fact that an increase in temperature is associated with an increase in the speed of movement of molecules, and this leads to an increase in intermolecular distances and, in turn, to expansion of the body.

Thermal expansion must be taken into account during heat treatment and during the thermal method of manufacturing parts and equipment, during the construction of machines, pipelines, electrical lines, bridges, and buildings subject to temperature changes.

PROGRESS OF THE CLASS

I. Organization of the beginning of the lesson

Greeting, topic statement, lesson objectives, an indication of the amount of work ahead. Motivation for cognitive activity.

II. Repetition of learned material

1. Checking homework

Check the solution of qualitative physical problems on the topic “Solid bodies and their properties” (frontal survey of students).

2. Preparation for the perception of new material

1. Repeat the formulas from the mathematics course (a+c) 3, a 3 +c 3;
2. Repeat the topic “Thermal expansion of gases” (Gay-Lussac’s law)
3. Repeat the topic “Deformation of solids.”

III. Learning new material

1. Students are asked to answer the following questions:
1. What happens to bodies when they cool and expand?
2. Why do bodies expand? What changes in a body during the process of expansion?

During the discussion, the concept of thermal expansion of bodies, examples of expansion of bodies, and types of thermal expansion are introduced.

Thermal expansion is an increase in the linear dimensions of a body and its volume that occurs with increasing temperature.

When the body expands, its volume increases, and they speak of volumetric expansion of the body. But sometimes we are only interested in changing one dimension, such as the length of railroad tracks or a metal rod. In that case they talk about linear expansion. Car designers are interested in expanding the surface area of ​​the metal sheets used to build the car. Here the question is about superficial expansion.

Setting up experiments:

1. expansion of liquids when heated (increasing the water level in a flask with a tube);
2. expansion of solids when heated (a ball with a ring, an increase in the length of stretched wires);
3. action of a bimetallic regulator (thermal relay).

Question: Do bodies expand equally when heated by the same number of degrees?

Answer: no, because different substances have molecules with different masses. A change in temperature by the same number of degrees characterizes the same root mean square speed of molecules. E k = There will be fewer molecules with lower mass than molecules with higher mass. Therefore, the intermolecular spaces of different substances change differently at the same temperature, which leads to unequal expansion.

2. Consider the linear expansion of solids and its features

The expansion of a rigid body along one of its dimensions is called linear.

To characterize the degree of linear expansion of various solids, the concept of linear expansion coefficient is introduced.

The value showing by what fraction of the initial length, taken at 0 0 C, the length of a body increases when it is heated by 1 0 C is called linear expansion coefficient and is denoted by .

K -1 = or = 0 C -1 =

Let us introduce the following notation: t 0 – initial temperature; t – final temperature; l 0 – body length at t 0 =0 0 C; l t – body length at t 0 C; l – change in body length; t – temperature change.

Let us assume that the wire was heated to 60 0 C. At the beginning, the wire had a length of 100 cm, and when heated, its length increased by 0.24 cm.

From here, we can calculate the increase in the length of the wire when heated by 1 0 C.

We divide the total elongation (0.024 cm) by the length of the wire and the temperature change: =0.000004 0 C -1 =(4*10 -6) 0 C -1.

Then = or = (1)

3. a) To calculate the length of a body depending on temperature t, we transform formula (2)

l t -l 0 = l 0 t l t =l 0 + l 0 t l t =l 0 (1+ t)

The binomial (1+t) is called linear expansion binomial . It shows how many times the length of the body increased when it was heated from 0 0 to t 0 C.

So, the final length of the body is equal to the initial length multiplied by the binomial of linear expansion.

The formula l t =l 0 (1+? t) is approximate and can be used at not very high temperatures (200 0 C-300 0 C).

For large temperature changes, this formula cannot be used.

b) When solving problems, they often use another approximate formula that simplifies the calculations. For example, if it is necessary to calculate the length of a body when heated from temperature t 1 to temperature t 2, then use the formula:

l 2 ~ l 1, linear expansion coefficient ~

IV. Intermediate fastening of material

Let's go for a walk along the railway track. If the weather is cold, we will notice that the ends of two adjacent rails are separated from each other by intervals of 0.6-1.2 cm; in hot weather, these ends almost meet closely. Hence the conclusion is that rails expand when heated and contract when cooled. Consequently, if the road was built in winter, then it was necessary to leave some reserve to allow the rails to expand freely during the hot season. The question arises, how much reserve should be left for this expansion?

Let’s assume that in our area the annual temperature changes from -30 0 C to -35 0 C and the length of the rail is 12.5 m. What gap should be left between the rails?

Answer: so it is necessary to leave a gap of 1 cm if the rails are laid at low temperatures, or to lay the rails together in a joint if the rails are laid in the hottest weather.

V. Studying new material (continued)

4. Let us consider the volumetric expansion of solids and its features

The increase in volume of bodies when heated is called volumetric expansion.

Volumetric expansion is characterized by the coefficient of volumetric expansion and is denoted by? .

Assignment: by analogy with linear expansion, define the coefficient of volumetric expansion and derive the formula =.

Students independently implement a solution to this issue and introduce the following notation: V 0 – initial volume at 0 0 C; V t – final volume at t 0 C; V – change in body volume; t 0 – initial temperature; t – final temperature.

The value showing by what fraction of the initial volume taken at 0 0 C the volume of a body increases from heating by 1 0 C is called coefficient of volumetric expansion .

a) Find the dependence of the volume of a solid on temperature. From the formula = we find the final volume V t .

V t -V 0 = V 0 t, V t =V 0 + V 0 t, V t =V 0 (1+ t).

The binomial (1+? t) is called volume expansion binomial . It shows how many times the volume of a body increased when it was heated from 0 to t 0 C.

So, the final volume of the body is equal to the initial volume multiplied by the binomial of volumetric expansion.

If the volume of the body V 1 at temperature t 1 is known, then the volume V 2 at temperature t 2 can be found using the approximate formula V 2 ~V 1, and the coefficient of volumetric expansion ~.

The derivation and recording of formulas is carried out by students independently.

6. What is the coefficient of volumetric expansion? very small value.

However, if we turn to the tables, we will see that the meaning? for solids it is not there. It turns out there is a relationship between the coefficients of linear and volumetric expansion? =3? .

Let's derive this ratio.

Let's assume that we have a cube whose edge length at 0 0 C is 1 cm. We heat the cube by 1 0 C, then the length of its edge will be l t = 1+? *1 0 =1+? . Volume of the heated cube V t = (1+?) 3. On the other hand, the volume of the same cube can be calculated using the formula V t =1+? *1 0 =1+? .

From the last equalities we get 1+? =(1+?) 3, hence 1+? =1+3? +3? 2+? 3.

Because numeric values? very small - on the order of parts per million, then 3? 2 and? 3 are even more so extremely small quantities. On this basis, neglecting the values ​​of 3? 2 and? 3, we get what? =3? .

The coefficient of volumetric expansion of a solid is equal to triple the coefficient of linear expansion.

7. Let’s find out how the density of bodies changes with temperature changes. Body density at 0 0 C.

p, from where m=p 0 *V 0, where m is body mass; V 0 – volume at 0 0 C;

m = const when the temperature changes, but the volume of the body changes, which means the density also changes.

On this basis, we can write that the density of the body at temperature t = 0 0 C, because V t = V 0 (1+? t), then .

When making calculations, you need to take into account that the tables indicate the density of the substance at 0 0 C. Density at other temperatures is calculated using the formula? t.

When heated, p t decreases, when cooled, p t increases.

1. Talk about the design, purpose and principle of operation of a bimetallic thermal relay, demonstrate its operation. Give examples of the beneficial and harmful effects of thermal deformation in technology, transport, construction, etc.
2. Briefly describe the features of thermal expansion of liquids.
3. Message “Features of thermal expansion of water.”

VI. Consolidation of the studied material.

1. A short survey-conversation is conducted for a deeper understanding and consolidation of the studied material on the issues.
2. Independent work students. Solve problems on the topic.

1. P.I. Samoilenko, A.V. Sergeev.
Physics. –M.: 2002.
2. A.A. Pinsky, G.Yu. Grakovsky.
Physics. –M.: 2002.
3. V.F. Dmitrieva.
Physics.-M.: 2000.
4. G.I. Ryabovodov, P.I. Samoilenko, E.I. Ogorodnikova.
Planning educational process in physics.-M.: graduate School, 1988.
5. A.A. Gladkova
. Collection of tasks and questions for secondary educational institutions in physics. -M.: Science. 1996.

It is known that under the influence of heat, particles accelerate their chaotic movement. If you heat a gas, the molecules that make it up simply fly apart from each other. The heated liquid will first increase in volume and then begin to evaporate. What will happen to solids? Not each of them can change its state of aggregation.

Thermal Expansion: Definition

Thermal expansion is the change in size and shape of bodies with changes in temperature. Mathematically, it is possible to calculate the volumetric expansion coefficient, which allows us to predict the behavior of gases and liquids under changing external conditions. To obtain the same results for solids, it is necessary to take into account Physicists have allocated a whole section for this kind of research and called it dilatometry.

Engineers and architects need knowledge about behavior different materials under the influence of high and low temperatures for designing buildings, laying roads and pipes.

Expansion of gases

The thermal expansion of gases is accompanied by an expansion of their volume in space. This was noticed by natural philosophers back in ancient times, but only modern physicists were able to construct mathematical calculations.

First of all, scientists became interested in the expansion of air, since it seemed to them a feasible task. They got down to business so zealously that they got quite contradictory results. Naturally, the scientific community was not satisfied with this outcome. The accuracy of the measurement depended on the type of thermometer used, the pressure, and many other conditions. Some physicists even came to the conclusion that the expansion of gases does not depend on changes in temperature. Or is this dependence not complete...

Works by Dalton and Gay-Lussac

Physicists would have continued to argue until they were hoarse or would have abandoned measurements if He and another physicist, Gay-Lussac, had not been able to obtain the same measurement results at the same time independently of each other.

Lussac tried to find the reason for such a number different results and noticed that there was water in some of the devices at the time of the experiment. Naturally, during the heating process it turned into steam and changed the amount and composition of the gases being studied. Therefore, the first thing the scientist did was to thoroughly dry all the instruments that he used to conduct the experiment, and eliminate even the minimum percentage of moisture from the gas under study. After all these manipulations, the first few experiments turned out to be more reliable.

Dalton studied this issue longer than his colleague and published the results back in early XIX century. He dried the air with sulfuric acid vapor and then heated it. After a series of experiments, John came to the conclusion that all gases and steam expand by a factor of 0.376. Lussac came up with a number of 0.375. This is what became official result research.

Water vapor pressure

The thermal expansion of gases depends on their elasticity, that is, their ability to return to their original volume. First this question Ziegler began to explore in the mid-eighteenth century. But the results of his experiments varied too much. More reliable figures were obtained by using my father’s boiler for high temperatures, and a barometer for low temperatures.

IN late XVIII century French physicist Prony attempted to derive a single formula that would describe the elasticity of gases, but it turned out to be too cumbersome and difficult to use. Dalton decided to empirically test all the calculations using a siphon barometer. Despite the fact that the temperature was not the same in all experiments, the results were very accurate. So he published them in table form in his physics textbook.

Evaporation theory

Thermal expansion of gases (as physical theory) has undergone various changes. Scientists have tried to get to the bottom of the processes that produce steam. Here again the already well-known physicist Dalton distinguished himself. He hypothesized that any space is saturated with gas vapor, regardless of whether any other gas or vapor is present in this tank (room). Therefore, it can be concluded that the liquid will not evaporate simply by coming into contact with atmospheric air.

The pressure of the air column on the surface of the liquid increases the space between the atoms, tearing them apart and evaporating, that is, it promotes the formation of vapor. But the force of gravity continues to act on the vapor molecules, so scientists believed that atmospheric pressure has no effect on the evaporation of liquids.

Expansion of liquids

The thermal expansion of liquids was studied in parallel with the expansion of gases. The same scientists were engaged in scientific research. To do this, they used thermometers, aerometers, communicating vessels and other instruments.

All experiments together and each separately refuted Dalton's theory that homogeneous liquids expand in proportion to the square of the temperature to which they are heated. Of course, the higher the temperature, the greater the volume of liquid, but there was no direct relationship between it. And the expansion rate of all liquids was different.

The thermal expansion of water, for example, begins at zero degrees Celsius and continues as the temperature decreases. Previously, such experimental results were associated with the fact that it is not the water itself that expands, but the container in which it is located that narrows. But some time later, the physicist DeLuca finally came to the idea that the cause should be sought in the liquid itself. He decided to find the temperature of its greatest density. However, he failed due to neglect of some details. Rumfort, who studied this phenomenon, found that the maximum density of water is observed in the range from 4 to 5 degrees Celsius.

Thermal expansion of bodies

In solids, the main expansion mechanism is a change in the vibration amplitude of the crystal lattice. If we talk in simple words, then the atoms that make up the material and are rigidly linked to each other begin to “tremble.”

The law of thermal expansion of bodies is formulated as follows: any body with linear size L in the process of heating by dT (delta T is the difference between the initial and final temperatures), expands by dL (delta L is the derivative of the coefficient of linear thermal expansion by the length of the object and by the difference temperature). This is the simplest version of this law, which by default takes into account that the body expands in all directions at once. But for practical work they use much more cumbersome calculations, since in reality materials behave differently than simulated by physicists and mathematicians.

Rail thermal expansion

Physics engineers are always involved in laying railway tracks, since they can accurately calculate what distance should be between the rail joints so that the tracks do not deform when heated or cooled.

As mentioned above, thermal linear expansion applies to all solids. And the rail was no exception. But there is one detail. Linear change occurs freely if the body is not affected by friction. The rails are rigidly attached to the sleepers and welded to adjacent rails, therefore the law that describes the change in length takes into account overcoming obstacles in the form of linear and butt resistances.

If the rail cannot change its length, then with a change in temperature, thermal stress increases in it, which can either stretch or compress it. This phenomenon is described by Hooke's law.

It is well known that solids increase their volume when heated. This is thermal expansion. Let us consider the reasons that lead to an increase in body volume when heated.

It is obvious that the volume of the crystal increases with increasing average distance between the atoms. This means that an increase in temperature entails an increase in the average distance between the atoms of the crystal. What causes the increase in the distance between atoms when heated?

An increase in the temperature of a crystal means an increase in the energy of thermal motion, i.e., thermal vibrations of atoms in the lattice (see page 459), and, consequently, an increase in the amplitude of these vibrations.

But an increase in the amplitude of vibrations of atoms does not always lead to an increase in the average distance between them.

If the vibrations of atoms were strictly Harmonic, then each atom would approach one of its neighbors as much as it would move away from another, and an increase in the amplitude of its vibrations would not lead to a change in the average interatomic distance, and therefore to thermal expansion.

In reality, atoms in a crystal lattice undergo anharmonic (i.e., non-harmonic) vibrations. This is due to the nature of the dependence of the interaction forces between/atoms on the distance between them. As was indicated at the beginning of this chapter (see Fig. 152 and 153), this dependence is such that at large distances between atoms, the interaction forces between atoms manifest themselves as attractive forces, and when this distance decreases, they change their sign and become repulsive forces, quickly increasing with decreasing distance.

This leads to the fact that when the “amplitude” of atomic vibrations increases due to heating of the crystal, the growth of the repulsive forces between the atoms prevails over the growth of the attractive forces. In other words, it is “easier” for an atom to move away from its neighbor than to approach another. This, of course, should lead to an increase in the average distance between atoms, i.e., to an increase in the volume of the body when it is heated.

It follows that the cause of thermal expansion of solids is the anharmonicity of atomic vibrations in the crystal lattice.

Quantitatively, thermal expansion is characterized by linear and volumetric expansion coefficients, which are determined as follows. Let a body of length I, when the temperature changes by degrees, change its length by The coefficient of linear expansion is determined from the relation

that is, the coefficient of linear expansion is equal to the relative change in length with a change in temperature by one degree. Similarly, the coefficient of volumetric expansion is given by

i.e., the coefficient is equal to the relative change in volume per one degree.

From these formulas it follows that the length and volume at a certain temperature differing from the initial temperature by degrees are expressed by the formulas (at low

where are the initial length and volume of the body.

Due to the anisotropy of crystals, the linear expansion coefficient a can be different in different directions. This means that if a ball is cut from this crystal, then after heating it it will lose its spherical shape. It can be shown that, in the most general case, such a ball, when heated, turns into a triaxial ellipsoid, the axes of which are connected with the crystallographic axes of the crystal.

The thermal expansion coefficients along the three axes of this ellipsoid are called the principal expansion coefficients of the crystal.

If we denote them respectively by the coefficient of volumetric expansion of the crystal

For crystals with cubic symmetry, as well as for isotropic bodies,

A ball machined from such bodies remains a ball even after heating (of course, with a larger diameter).

In some crystals (for example, hexagonal)

The coefficients of linear and volumetric expansion practically remain constant if the temperature intervals in which they are measured are small and the temperatures themselves are high. In general, the coefficients of thermal expansion depend on temperature and, moreover, in the same way as heat capacity, i.e., at low temperatures coefficients decrease with decreasing temperature in proportion to the cube of temperature, tending, like heat capacity,

to zero at absolute zero. This is not surprising, since both heat capacity and thermal expansion are related to lattice vibrations: heat capacity provides the amount of heat required to increase the average energy of thermal vibrations of atoms, which depends on the vibration amplitude, while the coefficient of thermal expansion is directly related to the average distances between atoms, which also depend on the amplitude of atomic vibrations.

This implies an important law discovered by Grüneisen: the ratio of the coefficient of thermal expansion to the atomic heat capacity of a solid for a given substance is a constant value (that is, independent of temperature).

The thermal expansion coefficients of solids are usually very small, as can be seen from Table. 22. The values ​​of coefficient a given in this table refer to the temperature range between and

Table 22 (see scan) Thermal expansion coefficients of solids

Some substances have a particularly low coefficient of thermal expansion. For example, quartz has this property. Another example is an alloy of nickel and iron (36% Ni), known as invar. These substances are widely used in precision instrument making.