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?

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 of the behavior of different materials under high and low temperatures to design buildings, 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 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 so many 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 at the very beginning of the 19th 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 became the official result of the study.

Water vapor pressure

The thermal expansion of gases depends on their elasticity, that is, their ability to return to their original volume. Ziegler was the first to explore this issue 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.

At the end of the 18th century, the 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

The thermal expansion of gases (as a 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 highest 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. In simple words, 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, much more cumbersome calculations are used, since in reality materials behave differently than modeled 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.

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, casts 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 like that. 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 pieces 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.

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 of 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

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.

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.).