Topic 2.1.6 Linear and volumetric expansion of solids when heated.

1. Thermal expansion.

2. Linear expansion.

3. Volume expansion.

4. Thermal expansion of liquids.

Literature: Dmitrieva V.F. Physics: Basic textbook for students of initial degrees of the 1st and 2nd levels of accreditation. – K: Tekhnika, 2008. – 648 p. (§81)

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

In the process of heating a solid, the average distances between atoms increase.

2. The value equal to the ratio of the relative elongation of the body to the change in its temperature by ∆T = T – T 0 is called the temperature expansion coefficient:

From this formula we determine the dependence of the length of a solid on temperature:

l = l 0 (1+α∆T)

3. As temperature increases, the volume of the body also changes. Within a not very large temperature range, the volume increases in proportion to the temperature. Volumetric expansion of solids is characterized by the temperature coefficient of volumetric expansion β - a value equal to the ratio of the relative increase in volume ∆V/V 0 of the body to the change in temperature ∆T:

; V = V 0 (1+ β∆Т).

4. In the process of heating a liquid, the average kinetic energy of the chaotic movement of its molecules increases. This leads to an increase in the distance between the molecules, and therefore to an increase in volume. The thermal expansion of liquids, like solids, is characterized by the temperature coefficient of volumetric expansion. The volume of liquid when heated is determined by the formula: V = V 0 (1+ β∆T). If the volume of bodies increases, then their density decreases: ρ = ρ 0 /(β∆T)

The volume of most bodies increases during the melting process, and decreases during the solidification process, while the density of the substance also changes.

The density of a substance decreases when melting, and increases when solidifying. But there are substances, such as silicon, germanium, bismuth, whose density increases when melted and decreases when solidified. Ice (water) also belongs to such substances.

Test questions and tasks

1 When does thermal expansion of bodies occur?

2 What is the temperature coefficient of expansion?

3 What characterizes the volumetric expansion of solids?

4 What is the thermal expansion of liquids characterized by?

5 Why is it that when reinforced concrete structures are heated and cooled, the iron in them is not separated from the concrete?

Change in body size or volume when heated

Animation

Description

Thermal expansion is the effect of changing the size of a body with a change in temperature at constant pressure. This phenomenon for solids is due to the asymmetry of the interaction potential of the atoms of the substance in the lattice, which leads to anharmonicity of the vibrations of the atoms relative to the average position. For gases, this is due to an increase in the kinetic energy of molecules and atoms.

Quantitatively, thermal expansion at constant pressure P is characterized by an isobaric expansion coefficient (volumetric or linear).

The coefficient of volumetric expansion a is defined as the relative change in volume V when a body (solid, liquid or gaseous) is heated by 1 K.

here T is the absolute body temperature.

The practical value of a is calculated using the formula:

where V 1, V 2 are the volumes of the body at temperatures T 1 and T 2, respectively (T 1<Т 2 ).

To characterize thermal expansion, along with a, the linear expansion coefficient a L is used:

where l is the size of the body in a given direction.

In the general case of polycrystalline anisotropic bodies consisting of anisotropic single crystals, a L = a x + a y + a z, and the difference or equality of linear thermal expansion coefficients a x, a y, a z along the crystallographic axes x, y, z is determined by the symmetry of the crystal. For example, for crystals of the cubic system, as well as for isotropic bodies a L = a x = a y = a z and a = 3a l. For most bodies a >0, but there are also anomalies. For example, water when heated from 0 to 40 C under normal atmospheric pressure is compressed (a<0). Зависимость a (Т ) наиболее заметна у газов (для идеального газа a =1/Т ); у жидкостей она проявляется слабее. У ряда веществ в твердом состоянии (кварца, инвара и т.д.) коэффициент a мал и практически постоянен в широком интервале температур. При Т ® 0, a® 0. Коэффициент a и a L определяются экспериментальными методами.

Timing characteristics

Initiation time (log to -1 to 3);

Lifetime (log tc from 0 to 6);

Degradation time (log td from -1 to 3);

Time of optimal development (log tk from 3 to 5).

Diagram:

Technical implementations of the effect

Thermometer

The implementation of this effect does not require any additional means other than a regular household alcohol or mercury thermometer. When it is heated, the liquid column grows, which means the volumetric expansion of the liquid.

Applying an effect

This effect is widely used in the design of technical systems operating in extreme or optimal thermal conditions with large temperature differences. The anomalous property of water to decrease in volume when the temperature increases from 0 to 40 C, on the one hand, is harmful, leading to the defrosting of “hydraulic systems”, i.e. their mechanical destruction, and on the other hand, it is the basis for a number of technological processes, for example, the destruction of rocks. In addition, so-called bimetallic plates are widely used in technical devices as temperature limit sensors, leading to automatic switching on and off of household electrical devices (irons, vacuum cleaners, refrigerators, etc.).

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, where is the relative elongation, a E- Young's modulus. According to (9.2.1). Substituting this value of relative elongation into the formula of 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 for 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 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.

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).

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 hot water is poured into it. First of all, the internal parts of the vessel in contact with hot water are heated. 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.

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.

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 the heat capacity, i.e. at low temperatures the coefficients decrease with decreasing temperature in proportion to the cube of the temperature, tending, like the 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.

GROUNDED CORKS

Everyone knows that when heated, bodies expand.
Sometimes the ground stopper in a glass bottle is so tight that you can’t pull it out. It is dangerous to use too much force - you can break off the neck and cut your hands. Therefore, they resort to a proven method: a burning match is brought to the neck, and the bottle is turned so that the neck is evenly heated.

The flame of one match is enough for the glass of the neck to expand due to heating, and the stopper, which did not have time to heat up, can be easily removed.

NEEDLE EXTENDATION

Cut out a bow from cork, from a board, or from plywood, like the one in our picture. Insert the needle with the tip into the whole end of the bow (the left one in the picture), and place the eye loosely on the right, cut end. Choose another needle, thinner. Its tip should pass through the eye of the first, horizontal needle and also enter the wood by 2-3 mm.

This vertical needle will be the arrow of our device. To make its movement more noticeable, stick a second control next to it.

The control needle should be parallel to the arrow needle.
Now heat the horizontal needle on a candle or match.
It will lengthen, the ear will crawl to the right and deflect the vertical arrow!

THERMAL SCALES

Experience 1

To do this, take a straight piece of copper wire 1-2 millimeters thick, about 40 centimeters long. Stick the end of this wire into a hole drilled in a wooden stick of approximately the same length, and hang the resulting thermal balance beam from the middle on a thread. Balance it out.

To do this, you may need to trim a wooden stick or, conversely, hang a small weight on it, such as pieces of paper. You can achieve balance by moving the rocker arm suspension point. Light the rocker with a table lamp so that one end, such as a copper end, provides a shadow on the wall. At this point, attach white paper to the wall and mark with a pencil the position of the shadow when the rocker hangs strictly horizontally. Then take two lit candles and place them under the copper wire. When it heats up well, it will elongate and the balance will be disrupted. Because the shoulder ratio was disrupted. The end of the wire will drop a few millimeters. This will be clearly visible from the shadow on the wall.

If the candles are removed, the copper wire will cool down, become shorter, that is, the same as it was before heating, and the rocker arm of our thermal balance, or rather its shadow, will fall on its mark.

Experience 2

A beautiful experiment can be made with a steel knitting needle.
Pass it through a cork (or a carrot scrap). On both sides of the knitting needle, insert two pins into this plug, as shown in the figure. They should stand with sharp ends on the bottom of the glass.

Place carrots on the ends of the knitting needles. It’s better not in the middle, but so that the main part of each carrot is at the bottom. This will make the balance of the spoke more stable: after all, the center of gravity has dropped lower! It turned out something like a scale. By moving the carrots, make sure that the knitting needle is completely horizontal.

Happened?
Well, now place a lit candle under one shoulder of these scales.
Attention... Look: the heated shoulder has dropped! Remove the candle and after a while the balance will be restored.

What's the matter here?
Has one side of the knitting needle become heavier due to heating? Of course not. It just became longer, and the carrot “moved” further from the fulcrum. That's why she pulled it, like a bird pulled a hippopotamus! And when the knitting needle cooled down, it shortened again, and everything became the same.

SEPARATING GLASSES

All bodies expand when heated and contract when cooled - the law!
At home, we are constantly faced with manifestations of an insidious law: either a glass into which boiling water has been poured will crack, or the screw cap on a jar will be compressed by pressure so that it cannot be opened, or water pipes will burst due to severe frost (in the last example we are talking about “wrong behavior of water, because it expands when it freezes).
But it’s better to be friends with this law!

Experience

How to separate two glasses inserted one into the other?

Yesterday they were washed with hot water and left there. And they “grabbed” in such a way that they would rather break than separate. Pour cold water into the top glass and place the second glass in a bowl of hot water. A few moments - and with a magician's gesture you will separate them.

RUSTY SCREW

Heat the head of a rusty screw that cannot be removed with a screwdriver with a soldering iron. Let the screw cool and try again.

Due to the sudden expansion and then contraction, particles of rust and other foreign substances on the surface of the thread should separate. If this does not help immediately, repeat the heating.

THE BOARD IS SMART

If you would like to demonstrate your strength, that is, to show how a thick board shatters into splinters under the edge of your palm, we will reveal the secret of one circus performer: before the performance, he soaked the prepared board in water and exposed it to the cold. Then he let it thaw, soaked it again and froze again. And so on several times.

As you might guess, the freezing water tore the wood cells, and the board became loose and weak. It is not difficult to break it with a sharp blow of the palm. However, it’s not good to lie...
By the way, what should you do with a donut to make its hole bigger?

BALL EXPANSION

Let's do an experiment with expansion caused by heating a solid object. It would be nice to find a metal ball from a billiard table or from a ball bearing. Based on its size, look for some kind of metal plate with a hole. If the diameter of the hole is smaller than the ball, use a round file to widen it.

Make sure that the ball, if placed on the hole, falls through without stopping in it. But there should not be a gap between the ball and the hole. Place the ball on the hot plate. If the stove is gas, then place it on a metal circle, which every housewife has to protect some dishes from burning. When the ball is well heated, take it with pliers and quickly place it on the hole in the plate, previously fixed above the metal box. When heated, the ball will increase in size and will remain in the hole until it cools down. When it cools down, it will slip through it on its own.

COIN EXPANSION

Heat the coin and try passing it between the plates again. You won't succeed until the coin cools down and returns to its previous size.

You can do the experiment even easier using two nails driven into a board. The distance between the nails should be equal to the diameter of the unheated patch.