VI. Motion of rigid bodies Moment of force Try to turn a heavy flywheel by hand. Pull on the needle. It will be difficult for you if you grab your hand too close to the axis. Move your hand to the rim, and things will go easier. What has changed? After all, the force in both cases

What thermal motion looks like The interaction between molecules can be of greater or lesser importance in the "life" of molecules. The three states of matter - gaseous, liquid and solid - differ from one another in the role that interaction plays in them

TURN ELECTRICITY INTO MOTION Faraday noticed one small detail in Oersted's experiments that seemed to hold the key to understanding the problem. He guessed that the magnetism of electric current always deflects the compass needle in one direction. For example, if

With uniformly accelerated motion, the following equations are valid, which we give without derivation:

As you understand, the vector formula on the left and the two scalar formulas on the right are equal. From the point of view of algebra, scalar formulas mean that with uniformly accelerated motion, the projections of displacement depend on time according to a quadratic law. Compare this with the nature of the instantaneous velocity projections (see § 12-h).

Knowing that  sx = x – xo  u   sy = y – yo  (see § 12-e), from the two scalar formulas from the upper right column we obtain equations for the coordinates:

Since the acceleration during uniformly accelerated motion of the body is constant, the coordinate axes can always be arranged so that the acceleration vector is directed parallel to one axis, for example, the Y axis. Consequently, the equation of motion along the X axis will be noticeably simplified:

x  =  xo + υox t  + (0) and y  =  yo + υoy t  + ½ ay t²

Please note that the left equation coincides with the equation of uniform rectilinear motion (see § 12-g). This means that uniformly accelerated motion can be "composed" of uniform motion along one axis and uniformly accelerated motion along the other. This is confirmed by the experience with the cannonball on a yacht (see § 12-b).

Task. Stretching out her arms, the girl tossed the ball. He rose to 80 cm and soon fell at the girl's feet, flying 180 cm. With what speed was the ball thrown and what speed did the ball have when it hit the ground?

Let us square both sides of the equation for the projection onto the Y-axis of the instantaneous velocity: υy  =  υoy + ay t  (see § 12-i). We get the equality:

υy²  =  ( υoy + ay t )²  =  υoy² + 2 υoy ay t + ay² t²

Let us take the factor  2 ay  out of brackets only for two right-hand terms:

υy²  =  υoy² + 2 ay ( υoy t + ½ ay t² )

Note that in parentheses we get a formula for calculating the displacement projection:  sy = υoy t + ½ ay t². Replacing it with sy , we get:

Solution. Let's make a drawing: point the Y axis up, and place the origin on the ground at the girl's feet. Let's apply the formula we derived for the square of the velocity projection first at the top point of the ball's ascent:

0 = υoy² + 2 (–g) (+h) ⇒ υoy = ±√¯2gh = +4 m/s

Then, at the beginning of the movement from the top point down:

υy² = 0 + 2 (–g) (–H) ⇒ υy = ±√¯2gh = –6 m/s

Answer: the ball was thrown upwards with a speed of 4 m/s, and at the moment of landing it had a speed of 6 m/s directed against the Y axis.

Note. We hope you understand that the formula for the square of the instantaneous velocity projection will be true by analogy for the X axis:

If the movement is one-dimensional, that is, it occurs only along one axis, you can use either of the two formulas in the framework.

Rectilinear motion with constant acceleration is called uniformly accelerated if the modulus of speed increases with time, or uniformly decelerated if it decreases.

An example of accelerated movement would be the fall of a flower pot from the balcony of a low house. At the beginning of the fall, the speed of the pot is zero, but in a few seconds it manages to grow to tens of m/s. An example of slow motion is the movement of a stone thrown vertically upwards, the speed of which is initially high, but then gradually decreases to zero at the top of the trajectory. If we neglect the force of air resistance, then the acceleration in both these cases will be the same and equal to the acceleration of gravity, which is always directed vertically downwards, denoted by the letter g and is approximately 9.8 m/s2.

The free fall acceleration, g, is caused by the Earth's gravity. This force accelerates all bodies moving towards the earth and slows down those moving away from it.

where v is the speed of the body at time t, whence, after simple transformations, we obtain equation for speed when moving with constant acceleration: v = v0 + at

8. Equations of motion with constant acceleration.

To find the equation for the velocity in a rectilinear motion with constant acceleration, we assume that at the time t=0 the body had an initial velocity v0. Since the acceleration a is constant, the following equation is true for any time t:

where v is the speed of the body at time t, from which, after simple transformations, we obtain the equation for the speed when moving with constant acceleration: v = v0 + at

To derive an equation for the path traveled during rectilinear motion with constant acceleration, we first construct a graph of the speed versus time (5.1). For a>0, the graph of this dependence is shown on the left in Fig. 5 (blue line). As we established in §3, the displacement made in time t can be determined by calculating the area under the velocity-time curve between t=0 and t. In our case, the figure under the curve, bounded by two vertical lines t=0 and t, is a trapezoid OABC, whose area S, as you know, is equal to the product of half the sum of the lengths of the bases OA and CB and the height OC:

As seen in Figure 5, OA = v0, CB= v0 + at, and OC = t. Substituting these values ​​into (5.2), we obtain the following equation for the displacement S completed in time t during rectilinear motion with constant acceleration a at initial speed v0:

It is easy to show that formula (5.3) is valid not only for motion with acceleration a>0, for which it was derived, but also in cases where a<0. На рис.5 справа красными линиями показаны графики зависимости S при положительных (верх) и отрицательных (низ) значениях a, построенные по формуле (5.3) для различных величин v0. Видно, что в отличие от равномерного движения (см. рис. 3), график зависимости перемещения от времени является параболой, а не прямой, показанной для сравнения пунктирной линией.

9. Free fall of bodies. Motion with constant free fall acceleration.

Free fall of bodies is called the fall of bodies to the Earth in the absence of air resistance (in a void)

The acceleration with which bodies fall to the Earth is called free fall acceleration. The gravitational acceleration vector is indicated by the symbol, it is directed vertically down. At different points on the globe, depending on the geographical latitude and height above sea level, the numerical value of g turns out to be unequal, varying from approximately 9.83 m/s2 at the poles to 9.78 m/s2 at the equator. At the latitude of Moscow, g = 9.81523 m/s2. Usually, if high accuracy is not required in the calculations, then the numerical value of g at the Earth's surface is taken equal to 9.8 m/s2 or even 10 m/s2.

A simple example of free fall is the fall of a body from a certain height h without initial velocity. Free fall is a rectilinear motion with constant acceleration.

Ideal free fall is possible only in a vacuum, where there is no air resistance force, and regardless of mass, density and shape, all bodies fall equally quickly, i.e. at any moment of time, bodies have the same instantaneous velocities and accelerations.

All formulas for uniformly accelerated motion are applicable to the free fall of bodies.

The value of the free fall speed of a body at any given time:

body movement:

In this case, instead of acceleration a, the acceleration of free fall g = 9.8 m/s2 is introduced into the formulas for uniformly accelerated motion.

10. Movement of bodies. TRANSLATIONAL MOTION OF A RIGID BODY

The translational motion of a rigid body is such a motion in which any straight line, invariably connected with the body, moves parallel to itself. For this, it is sufficient that two non-parallel lines connected with the body move parallel to themselves. In translational motion, all points of the body describe the same, parallel trajectories and have the same velocities and accelerations at any time. Thus, the translational motion of a body is determined by the motion of one of its points O.

In the general case, translational motion occurs in three-dimensional space, but its main feature - the preservation of the parallelism of any segment to itself, remains in force.

Progressively moves, for example, the elevator car. Also, in the first approximation, the cabin of the Ferris wheel performs forward movement. However, strictly speaking, the movement of the Ferris wheel cabin cannot be considered progressive. If the body moves forward, then to describe its movement it is sufficient to describe the movement of its arbitrary point (for example, the movement of the center of mass of the body).

If the bodies that make up a closed mechanical system interact with each other only through the forces of gravity and elasticity, then the work of these forces is equal to the change in the potential energy of the bodies, taken with the opposite sign: A \u003d - (E p2 - E p1).

According to the kinetic energy theorem, this work is equal to the change in the kinetic energy of bodies

Hence

Or E k 1 + E p 1 = E k 2 + E p 2 .

The sum of the kinetic and potential energy of the bodies that make up a closed system and interact with each other through the forces of gravity and elastic forces remains unchanged.

This statement expresses the law of conservation of energy in mechanical processes. It is a consequence of Newton's laws. The sum E = E k + E p is called the total mechanical energy. The law of conservation of mechanical energy is fulfilled only when bodies in a closed system interact with each other by conservative forces, that is, forces for which the concept of potential energy can be introduced.

The mechanical energy of a closed system of bodies does not change if only conservative forces act between these bodies. Conservative forces are those forces whose work along any closed trajectory is equal to zero. Gravity is one of the conservative forces.

In real conditions, almost always moving bodies, along with gravitational forces, elastic forces and other conservative forces, are affected by friction forces or resistance forces of the medium.

The friction force is not conservative. The work of the friction force depends on the length of the path.

If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy of bodies (heating).

In any physical interactions, energy does not arise and does not disappear. It only changes from one form to another.

One of the consequences of the law of conservation and transformation of energy is the assertion that it is impossible to create a “perpetual motion machine” (perpetuum mobile) - a machine that could do work indefinitely without consuming energy.

History keeps a considerable number of "perpetual motion" projects. In some of them the errors of the "inventor" are obvious, in others these errors are masked by the complex design of the device, and it can be very difficult to understand why this machine will not work. Fruitless attempts to create a "perpetual motion machine" continue in our time. All these attempts are doomed to failure, since the law of conservation and transformation of energy "forbids" getting work without spending energy.

31. Basic provisions of the molecular-kinetic theory and their substantiation.

All bodies consist of molecules, atoms and elementary particles, which are separated by gaps, move randomly and interact with each other.

Kinematics and dynamics help us describe the movement of a body and determine the force that causes this movement. However, mechanics cannot answer many questions. For example, what are bodies made of? Why do many substances become liquid when heated and then evaporate? And, in general, what is temperature and heat?

The ancient Greek philosopher Democritus tried to answer such questions 25 centuries ago. Without making any experiments, he came to the conclusion that the bodies only seem to be solid to us, but in fact they consist of the smallest particles separated by emptiness. Considering that it is impossible to split these particles, Democritus called them atoms, which in Greek means indivisible. He also suggested that atoms can be different and are in constant motion, but we do not see this, because. they are very small.

A great contribution to the development of molecular kinetic theory was made by M.V. Lomonosov. Lomonosov was the first to suggest that heat reflects the motion of the atoms of a body. In addition, he introduced the concept of simple and complex substances, the molecules of which consist of the same and different atoms, respectively.

Molecular physics or molecular kinetic theory is based on certain ideas about the structure of matter

Thus, according to the atomistic theory of the structure of matter, the smallest particle of a substance that retains all its chemical properties is a molecule. The dimensions of even large molecules consisting of thousands of atoms are so small that they cannot be seen with a light microscope. Numerous experiments and theoretical calculations show that the size of atoms is about 10 -10 m. The size of a molecule depends on how many atoms it consists of and how they are located relative to each other.

Molecular-kinetic theory is the study of the structure and properties of matter based on the idea of ​​the existence of atoms and molecules as the smallest particles of chemical substances.

The molecular kinetic theory is based on three main provisions:

1. All substances - liquid, solid and gaseous - are formed from the smallest particles - molecules, which themselves consist of atoms ("elementary molecules"). Molecules of a chemical substance can be simple or complex, i.e. be made up of one or more atoms. Molecules and atoms are electrically neutral particles. Under certain conditions, molecules and atoms can acquire an additional electrical charge and turn into positive or negative ions.

2. Atoms and molecules are in continuous chaotic motion.

3. Particles interact with each other by forces that are electrical in nature. The gravitational interaction between particles is negligible.

The most striking experimental confirmation of the ideas of the molecular kinetic theory about the random motion of atoms and molecules is Brownian motion. This is the thermal movement of the smallest microscopic particles suspended in a liquid or gas. It was discovered by the English botanist R. Brown in 1827. Brownian particles move under the influence of random collisions of molecules. Due to the chaotic thermal motion of the molecules, these impacts never balance each other. As a result, the speed of a Brownian particle randomly changes in magnitude and direction, and its trajectory is a complex zigzag curve.

The constant chaotic movement of the molecules of a substance also manifests itself in another easily observed phenomenon - diffusion. Diffusion is the phenomenon of penetration of two or more adjoining substances into each other. The process proceeds most rapidly in a gas.

The random random motion of molecules is called thermal motion. The kinetic energy of thermal motion increases with increasing temperature.

A mole is the amount of a substance containing as many particles (molecules) as there are atoms in 0.012 kg of carbon 12 C. A carbon molecule consists of one atom.

32. Mass of molecules, relative molecular mass of molecules. 33. Molar mass of molecules. 34. Amount of substance. 35. Avogadro's constant.

In molecular kinetic theory, the amount of a substance is considered to be proportional to the number of particles. The unit of quantity of a substance is called a mole (mole).

A mole is the amount of a substance containing as many particles (molecules) as there are atoms in 0.012 kg (12 g) of carbon 12 C. A carbon molecule consists of one atom.

One mole of a substance contains the number of molecules or atoms equal to the Avogadro constant.

Thus, one mole of any substance contains the same number of particles (molecules). This number is called the Avogadro constant N A: N A \u003d 6.02 10 23 mol -1.

The Avogadro constant is one of the most important constants in molecular kinetic theory.

The amount of substance ν is defined as the ratio of the number N of particles (molecules) of the substance to the Avogadro constant N A:

The molar mass, M, is the ratio of the mass m of a given sample of a substance to the amount n of the substance contained in it:

which is numerically equal to the mass of the substance taken in the amount of one mole. Molar mass in the SI system is expressed in kg/mol.

Thus, the relative molecular or atomic mass of a substance is the ratio of the mass of its molecule and atom to 1/12 the mass of a carbon atom.

36. Brownian motion.

Many natural phenomena testify to the chaotic movement of microparticles, molecules and atoms of matter. The higher the temperature of the substance, the more intense this movement. Therefore, the heat of the body is a reflection of the random movement of its constituent molecules and atoms.

The proof that all atoms and molecules of a substance are in constant and random motion can be diffusion - the interpenetration of particles of one substance into another.

So, the smell quickly spreads around the room even in the absence of air movement. A drop of ink quickly turns the entire glass of water uniformly black.

Diffusion can also be detected in solids if they are pressed tightly together and left for a long time. The phenomenon of diffusion demonstrates that the microparticles of a substance are able to spontaneously move in all directions. Such movement of microparticles of a substance, as well as its molecules and atoms, is called their thermal movement.

BROWNIAN MOVEMENT - random movement of the smallest particles suspended in a liquid or gas, occurring under the influence of impacts of environmental molecules; discovered by R. Brown in 1827

Observations show that Brownian motion never stops. In a drop of water (if you do not let it dry) the movement of grains can be observed for many days, months, years. It does not stop either in summer or winter, day or night.

The reason for Brownian motion is the continuous, never-ending motion of the molecules of the liquid in which the grains of the solid are located. Of course, these grains are many times larger than the molecules themselves, and when we see the movement of grains under a microscope, we should not think that we see the movement of the molecules themselves. Molecules cannot be seen with an ordinary microscope, but we can judge their existence and movement by the impacts they produce, pushing grains of a solid body and making them move.

The discovery of Brownian motion was of great importance for the study of the structure of matter. It showed that bodies really do consist of separate particles - molecules and that the molecules are in continuous random motion.

An explanation of Brownian motion was given only in the last quarter of the 19th century, when it became obvious to many scientists that the motion of a Brownian particle is caused by random impacts of the molecules of the medium (liquid or gas) that make thermal motion. On average, the molecules of the medium act on the Brownian particle from all sides with equal force, however, these impacts never exactly balance each other, and as a result, the speed of the Brownian particle randomly changes in magnitude and direction. Therefore, a Brownian particle moves along a zigzag path. In this case, the smaller the size and mass of a Brownian particle, the more noticeable its motion becomes.

Thus, the analysis of Brownian motion laid the foundations for the modern molecular-kinetic theory of the structure of matter.

37. Forces of interaction of molecules. 38. The structure of gaseous substances. 39. The structure of liquid substances. 40. The structure of solids.

The distance between molecules and the forces acting between them determine the properties of gaseous, liquid and solid bodies.

We are accustomed to the fact that liquid can be poured from one vessel to another, and gas quickly fills the entire volume provided to it. Water can only flow along the riverbed, and the air above it knows no boundaries.

Intermolecular attractive forces act between all molecules, the magnitude of which decreases very quickly with the distance of the molecules from each other, and therefore, at a distance equal to several diameters of the molecules, they do not interact at all.

Thus, between the molecules of a liquid, located almost close to each other, attractive forces act, preventing these molecules from scattering in different directions. On the contrary, the negligible forces of attraction between gas molecules are not able to hold them together, and therefore gases can expand, filling the entire volume provided to them. The existence of intermolecular forces of attraction can be verified by setting up a simple experiment - to press two lead bars against each other. If the contact surfaces are smooth enough, then the bars will stick together and it will be difficult to separate them.

However, intermolecular forces of attraction alone cannot explain all the differences between the properties of gaseous, liquid, and solid substances. Why, for example, is it very difficult to reduce the volume of a liquid or a solid, but it is relatively easy to compress a balloon? This is explained by the fact that between molecules there are not only attractive forces, but also intermolecular repulsive forces that act when the electron shells of atoms of neighboring molecules begin to overlap. It is these repulsive forces that prevent one molecule from penetrating into a volume already occupied by another molecule.

When external forces do not act on a liquid or solid body, the distance between their molecules is such that the resultant forces of attraction and repulsion are equal to zero. If you try to reduce the volume of the body, then the distance between the molecules decreases, and from the side of the compressed body, the resultant of the increased repulsive forces begins to act. On the contrary, when a body is stretched, the elastic forces that arise are associated with a relative increase in the forces of attraction, since As molecules move apart, the repulsive forces decrease much faster than the attractive forces.

Gas molecules are located at distances tens of times greater than their size, as a result of which these molecules do not interact with each other, and therefore gases are much easier to compress than liquids and solids. Gases do not have any specific structure and are a collection of moving and colliding molecules.

A liquid is a collection of molecules that are almost closely adjacent to each other. Thermal motion allows a liquid molecule to change its neighbors from time to time, jumping from one place to another. This explains the fluidity of liquids.

Atoms and molecules of solids do not have the ability to change their neighbors, and their thermal motion is only small fluctuations relative to the position of neighboring atoms or molecules. The interaction between atoms can lead to the fact that a solid becomes a crystal, and the atoms in it occupy positions at the nodes of the crystal lattice. Since the molecules of solids do not move relative to their neighbors, these bodies retain their shape.

41. Ideal gas in molecular kinetic theory.

An ideal gas is a model of a rarefied gas in which the interaction between molecules is neglected. The forces of interaction between molecules are quite complex. At very small distances, when molecules fly close to each other, large repulsive forces act between them. At large or intermediate distances between molecules, relatively weak forces of attraction act. If the distances between molecules are on average large, which is observed in a sufficiently rarefied gas, then the interaction manifests itself in the form of relatively rare collisions of molecules with each other when they fly up close. In an ideal gas, the interaction of molecules is generally neglected.

42. Gas pressure in molecular-kinetic theory.

An ideal gas is a model of a rarefied gas in which the interaction between molecules is neglected.

The pressure of an ideal gas is proportional to the product of the concentration of molecules and their average kinetic energy.

Gas is all around us. In any place on earth, even under water, we carry a part of the atmosphere, the lower layers of which are compressed under the action of gravity of the upper ones. Therefore, by measuring atmospheric pressure, one can judge what is happening high above us and predict the weather.

43. The average value of the square of the speed of molecules of an ideal gas.

44. Derivation of the basic equation of the molecular-kinetic theory of gas. 45. Derivation of a formula relating pressure and average kinetic energy of gas molecules.

The pressure p on a given section of the surface is the ratio of the force F acting perpendicular to this surface to the area S of its given section

The SI unit for pressure is Pascal (Pa). 1 Pa \u003d 1 N / m 2.

Let us find the force F with which a molecule of mass m0 acts on the surface from which it rebounds. When reflected from the surface, lasting a period of time Dt, the component of the velocity of the molecule, perpendicular to this surface, vy changes to the opposite (-vy). Therefore, when reflected from the surface, the molecule acquires momentum, 2m0vy , and hence, according to Newton's third law, 2m0vy =FDt, whence:

Formula (22.2) makes it possible to calculate the force with which one gas molecule presses on the vessel wall during the interval Dt. To determine the average force of gas pressure, for example, in one second, it is necessary to find how many molecules are reflected per second from a surface area S, and it is also necessary to know the average velocity vy of molecules moving towards this surface.

Let there be n molecules per unit volume of gas. Let's simplify our task by assuming that all gas molecules move at the same speed, v. In this case, 1/3 of all molecules move along the Ox axis, and the same number move along the Oy and Oz axes (see Fig. 22c). Let half of the molecules moving along the Oy axis move towards wall C, and the rest move in the opposite direction. Then, obviously, the number of molecules per unit volume, rushing towards wall C, will be n/6.

Let us now find the number of molecules that hit the surface area S (shaded in Fig. 22c) in one second. Obviously, in 1 s, those molecules that move towards it and are at a distance not greater than v will have time to reach the wall. Therefore, 1/6 of all molecules in the rectangular parallelepiped, highlighted in Fig. 1, will hit this area of ​​the surface. 22c, the length of which is equal to v, and the area of ​​the end faces is S. Since the volume of this parallelepiped is Sv, the total number N of molecules that hit the wall surface area in 1 s will be equal to:

Using (22.2) and (22.3) it is possible to calculate the impulse, which in 1 s gave the gas molecules a section of the wall surface with an area S. This impulse will be numerically equal to the gas pressure force, F:

whence, using (22.1), we obtain the following expression relating the gas pressure and the average kinetic energy of the translational motion of its molecules:

where Е СР is the average kinetic energy of ideal gas molecules. Formula (22.4) is called the basic equation of the molecular-kinetic theory of gases.

46. ​​Thermal equilibrium. 47. Temperature. Temperature change. 48. Instruments for measuring temperature.

Thermal equilibrium between bodies is possible only when their temperature is the same.

By touching any object with our hand, we can easily determine whether it is warm or cold. If the temperature of the object is lower than the temperature of the hand, the object seems cold, and if vice versa, then it is warm. If you squeeze a cold coin in your fist, then the warmth of the hand will begin to heat the coin, and after a while its temperature will become equal to the temperature of the hand, or, as they say, thermal equilibrium will come. Therefore, temperature characterizes the state of thermal equilibrium of a system of two or more bodies having the same temperature.

Temperature along with the volume and pressure of a gas are macroscopic parameters. Thermometers are used to measure temperature. In some of them, a change in the volume of a liquid during heating is recorded, in others, a change in electrical resistance, etc. The most common is the Celsius temperature scale, named after the Swedish physicist A. Celsius. To obtain the Celsius temperature scale for a liquid thermometer, it is first immersed in melting ice and the position of the end of the column is noted, and then in boiling water. The segment between these two positions of the column is divided into 100 equal parts, assuming that the melting temperature of ice corresponds to zero degrees Celsius (o C), and the temperature of boiling water is 100 o C.

49. Average kinetic energy of gas molecules at thermal equilibrium.

The basic equation of the molecular kinetic theory (22.4) links the gas pressure, the concentration of molecules and their average kinetic energy. However, the average kinetic energy of molecules is, as a rule, unknown, although the results of many experiments indicate that the speed of molecules increases with increasing temperature (see, for example, Brownian motion in §20). The dependence of the average kinetic energy of gas molecules on its temperature can be obtained from the law discovered by the French physicist J. Charles in 1787.

50. Gases in a state of thermal equilibrium (describe the experience).

51. Absolute temperature. 52. Absolute temperature scale. 53. Temperature is a measure of the average kinetic energy of molecules.

The dependence of the average kinetic energy of gas molecules on its temperature can be obtained from the law discovered by the French physicist J. Charles in 1787.

According to Charles's law, if the volume of a given mass of gas does not change, its pressure pt depends linearly on temperature t:

where t is the gas temperature measured in o C, and p 0 is the gas pressure at a temperature of 0 o C (see Fig. 23b). Thus, it follows from Charles's law that the pressure of a gas occupying a constant volume is proportional to the sum (t + 273 o C). On the other hand, it follows from (22.4) that if the concentration of molecules is constant, i.e. the volume occupied by the gas does not change, then the pressure of the gas must be proportional to the average kinetic energy of the molecules. This means that the average kinetic energy, E SR of gas molecules, is simply proportional to the value (t + 273 o C):

where b is a constant coefficient, the value of which we will determine later. From (23.2) it follows that the average kinetic energy of molecules will become equal to zero at -273 ° C. Based on this, the English scientist W. Kelvin in 1848 proposed using an absolute temperature scale, the zero temperature in which would correspond to -273 ° C, and each degree of temperature would be equal to a degree Celsius. So the absolute temperature, T, is related to the temperature t, measured in Celsius, as follows:

The SI unit of absolute temperature is the Kelvin (K).

Given (23.3), equation (23.2) is transformed into:

substituting which into (22.4), we get the following:

To get rid of the fraction in (23.5), we replace 2b/3 by k, and instead of (23.4) and (23.5) we get two very important equations:

where k is the Boltzmann constant, named after L. Boltzmann. Experiments have shown that k=1.38.10 -23 J/K. Thus, the pressure of a gas and the average kinetic energy of its molecules are proportional to its absolute temperature.

54. Dependence of gas pressure on the concentration of its molecules and temperature.

In most cases, when a gas passes from one state to another, all its parameters change - temperature, volume and pressure. This happens when the gas is compressed under the piston in the cylinder of an internal combustion engine, as a result of which the temperature of the gas and its pressure increase, and the volume decreases. However, in some cases, changes in one of the gas parameters are relatively small or absent altogether. Such processes, where one of the three parameters - temperature, pressure or volume remain unchanged, are called isoprocesses, and the laws that describe them are called gas laws.

55. Measurement of the speed of gas molecules. 56. Stern's experience.

First of all, let us clarify what is meant by the speed of molecules. Recall that due to frequent collisions, the speed of each individual molecule changes all the time: the molecule moves either quickly or slowly, and for some time (for example, one second) the velocity of the molecule takes on many different values. On the other hand, at any moment in the vast number of molecules that make up the considered volume of gas, there are molecules with very different velocities. Obviously, to characterize the state of the gas, one must speak of a certain average velocity. We can assume that this is the average velocity of one of the molecules over a sufficiently long period of time, or that it is the average velocity of all gas molecules in a given volume at some point in time.

There are various ways to determine the speed of movement of molecules. One of the simplest is the method carried out in 1920 in Stern's experiment.

Rice. 390. When the space under glass A is filled with hydrogen; then from the end of the funnel, closed by a porous vessel B, bubbles come out

To understand it, consider the following analogy. When shooting at a moving target, in order to hit it, you have to aim at a point in front of the target. If you take the sight on the target, then the bullets will hit behind the target. This deviation of the place of impact from the target will be the greater, the faster the target moves and the lower the speed of the bullets.

The experiment of Otto Stern (1888–1969) was devoted to experimental confirmation and visualization of the velocity distribution of gas molecules. This is another beautiful experience, which made it possible to “draw” a graph of this distribution on the experimental setup in the truest sense of the word. Stern's installation consisted of two rotating hollow cylinders with coinciding axes (see the figure on the right; the large cylinder is not fully drawn). In the inner cylinder, a silver thread 1 was stretched straight along its axis, through which a current was passed, which led to its heating, partial melting and subsequent evaporation of silver atoms from its surface. As a result, the inner cylinder, which initially had a vacuum, was gradually filled with gaseous silver of low concentration. In the inner cylinder, as shown in the figure, a thin slot 2 was made, so most of the silver atoms, reaching the cylinder, settled on it. A small part of the atoms passed through the gap and fell into the outer cylinder, in which the vacuum was maintained. Here, these atoms no longer collided with other atoms and therefore moved in the radial direction at a constant speed, reaching the outer cylinder after a time inversely proportional to this speed:

where are the radii of the inner and outer cylinders, and is the radial component of the particle velocity. As a result, over time, a layer of silver sputtering appeared on the outer cylinder 3. In the case of cylinders at rest, this layer had the form of a strip located exactly opposite the slot in the inner cylinder. But if the cylinders rotated with the same angular velocity, then by the time the molecule reached the outer cylinder, the latter had already shifted by a distance

compared to the point directly opposite the slot (i.e., the point on which the particles settled in the case of stationary cylinders).

57. Derivation of the equation of state of an ideal gas (Mendeleev-Claiperon equation)

Gases are often reactants and products in chemical reactions. It is not always possible to make them react with each other under normal conditions. Therefore, you need to learn how to determine the number of moles of gases under conditions other than normal.

To do this, use the ideal gas equation of state (it is also called the Clapeyron-Mendeleev equation): PV = nRT

where n is the number of moles of gas;

P is the gas pressure (for example, in atm;

V is the volume of gas (in liters);

T is the gas temperature (in kelvins);

R is the gas constant (0.0821 L atm/mol K).

I found the derivation of the equation, but it is very complicated. We still have to search.

58. Isothermal process.

An isothermal process is a change in the state of a gas in which its temperature remains constant. An example of such a process is the inflation of car tires with air. However, such a process can be considered isothermal if we compare the state of the air before it entered the pump with its state in the tire after the temperature of the tire and the surrounding air became equal. Any slow processes that occur with a small volume of gas surrounded by a large mass of gas, liquid or solid that has a constant temperature can be considered isothermal.

In an isothermal process, the product of the pressure of a given mass of gas and its volume is a constant value. This law, called the Boyle-Mariotte law, was discovered by the English scientist R. Boyle and the French physicist E. Mariotte and is written in the following form:

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59. Isobaric process.

An isobaric process is a change in the state of a gas that occurs at constant pressure.

In an isobaric process, the ratio of the volume of a given mass of gas to its temperature is constant. This conclusion, which is called the Gay-Lussac law in honor of the French scientist J. Gay-Lussac, can be written as:

One example of an isobaric process is the expansion of small bubbles of air and carbon dioxide contained in dough when it is placed in the oven. The air pressure inside and outside the oven is the same, and the temperature inside is approximately 50% higher than outside. According to Gay-Lussac's law, the volume of gas bubbles in the dough also grows by 50%, which makes the cake airy.

60. Isochoric process.

A process in which the state of a gas changes while its volume remains unchanged is called isochoric. From the Mendeleev-Clapeyron equation it follows that for a gas occupying a constant volume, the ratio of its pressure to temperature must also be constant:

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61. Evaporation and condensation.

Vapor is a gas formed from molecules that have enough kinetic energy to leave the liquid.

We are accustomed to the fact that water and its vapor can pass into each other. Puddles on the pavement dry up after rain, and the water vapor in the air in the morning often turns into tiny droplets of fog. All liquids have the ability to turn into vapor - go into a gaseous state. The process of changing liquid to vapor is called evaporation. The formation of a liquid from its vapor is called condensation.

The molecular kinetic theory explains the evaporation process as follows. It is known (see § 21) that an attractive force acts between the molecules of a liquid, which does not allow them to move away from each other, and the average kinetic energy of the molecules of the liquid is not enough to overcome the cohesive forces between them. However, at any given moment in time, different molecules of a liquid have different kinetic energies, and the energy of some molecules can be several times higher than its average value. These high-energy molecules have a much higher speed of movement and therefore can overcome the attractive forces of neighboring molecules and fly out of the liquid, thus forming vapor above its surface (see Fig. 26a).

The molecules that make up the vapor that have left the liquid move randomly, colliding with each other in the same way as gas molecules do during thermal motion. In this case, the chaotic movement of some vapor molecules can take them so far from the surface of the liquid that they never return there. Contributes to this, of course, and the wind. On the contrary, the random movement of other molecules can bring them back into the liquid, which explains the process of vapor condensation.

Only molecules with a kinetic energy much higher than the average can fly out of the liquid, which means that during evaporation, the average energy of the remaining liquid molecules decreases. And since the average kinetic energy of the molecules of a liquid, like that of a gas (see 23.6), is proportional to the temperature, the temperature of the liquid decreases during evaporation. Therefore, we become cold as soon as we leave the water, covered with a thin film of liquid, which immediately begins to evaporate and cool.

62. Saturated steam. Saturated steam pressure.

What happens if a vessel with a certain volume of liquid is closed with a lid (Fig. 26b)? Every second, the fastest molecules will still leave the surface of the liquid, its mass will decrease, and the concentration of vapor molecules will increase. At the same time, part of the vapor molecules will return to the liquid from the vapor, and the greater the vapor concentration, the more intense this condensation process will be. Finally, the vapor concentration over the liquid will become so high that the number of molecules returning to the liquid per unit time will become equal to the number of molecules leaving it. This state is called dynamic equilibrium, and the corresponding steam is called saturated steam. The concentration of vapor molecules above the liquid cannot be greater than their concentration in saturated vapor. If the concentration of vapor molecules is less than that of a saturated one, then such a vapor is called unsaturated.

Moving vapor molecules create pressure, the value of which, as for a gas, is proportional to the product of the concentration of these molecules and the temperature. Therefore, at a given temperature, the higher the concentration of steam, the greater the pressure it exerts. Saturated vapor pressure depends on the type of liquid and temperature. The harder it is to tear the molecules of a liquid apart, the lower will be the pressure of its saturated vapor. Thus, the pressure of saturated vapor of water at a temperature of 20 ° C is about 2 kPa, and the pressure of saturated vapor of mercury at 20 ° C is only 0.2 Pa.

The life of man, animals and plants depends on the concentration of water vapor (humidity) of the atmosphere, which varies widely depending on the place and season. As a rule, the water vapor around us is unsaturated. Relative humidity is the ratio of water vapor pressure to saturation vapor pressure at the same temperature, expressed as a percentage. One of the devices for measuring air humidity is a psychrometer, consisting of two identical thermometers, one of which is wrapped in a damp cloth.

63. The dependence of saturated steam pressure on temperature.

Steam is a gas formed by evaporated liquid molecules, and therefore equation (23.7) is valid for it, relating the vapor pressure, p, the concentration of molecules in it, n, and the absolute temperature, T:

From (27.1) it follows that the saturated vapor pressure must increase linearly with increasing temperature, as is the case for ideal gases in isochoric processes (see §25). However, measurements have shown that the pressure of saturated vapor increases with temperature much faster than the pressure of an ideal gas (see Fig. 27a). This happens due to the fact that with increasing temperature, and hence the average kinetic energy, more and more molecules of the liquid leave it, increasing the concentration, n of the vapor above it. And since according to (27.1), the pressure is proportional to n, then this increase in vapor concentration explains the faster increase in saturated vapor pressure with temperature, compared to an ideal gas. The increase in saturated vapor pressure with temperature explains the well-known fact - when heated, liquids evaporate faster. Note that as soon as the increase in temperature leads to the complete evaporation of the liquid, the vapor will become unsaturated.

When the liquid in each of the bubbles is heated, the evaporation process is accelerated, and the saturated vapor pressure increases. The bubbles expand and, under the action of the buoyant force of Archimedes, break away from the bottom, float up and burst on the surface. In this case, the vapor that filled the bubbles is carried away into the atmosphere.

The lower the atmospheric pressure, the lower the temperature at which this liquid boils (see Fig. 27c). So, on the top of Mount Elbrus, where the air pressure is half normal, ordinary water boils not at 100 o C, but at 82 o C. On the contrary, if it is necessary to increase the boiling point of the liquid, then it is heated at elevated pressure. This, for example, is the basis for the work of pressure cookers, where food containing water can be cooked at a temperature of more than 100 ° C without boiling.

64. Boiling.

Boiling is an intense evaporation process that occurs throughout the volume of a liquid and on its surface. A liquid begins to boil when its saturated vapor pressure approaches the pressure inside the liquid.

Boiling is the formation of a large number of vapor bubbles that pop up and burst on the surface of a liquid when it is heated. In fact, these bubbles are always present in the liquid, but their size grows, and they become noticeable only when boiling. One reason why liquids always contain microbubbles is as follows. The liquid, when it is poured into a vessel, displaces air from there, but it cannot do this completely, and its small bubbles remain in microcracks and irregularities on the inner surface of the vessel. In addition, liquids usually contain micro-bubbles of vapor and air that adhere to the smallest dust particles.

When the liquid in each of the bubbles is heated, the evaporation process is accelerated, and the saturated vapor pressure increases. The bubbles expand and, under the action of the buoyant force of Archimedes, break away from the bottom, float up and burst on the surface. In this case, the vapor that filled the bubbles is carried away into the atmosphere. Therefore, boiling is called evaporation, which occurs in the entire volume of the liquid. Boiling begins at the temperature when the gas bubbles have the opportunity to expand, and this occurs if the saturation vapor pressure exceeds atmospheric pressure. Thus, the boiling point is the temperature at which the saturation vapor pressure of a given liquid is equal to atmospheric pressure. As long as a liquid boils, its temperature remains constant.

The boiling process is impossible without the participation of the Archimedean buoyant force. Therefore, there is no boiling at space stations under weightless conditions, and heating water only leads to an increase in the size of vapor bubbles and their combination into one large vapor bubble inside a vessel with water.

65. Critical temperature.

There is also such a thing as a critical temperature, if the gas is at a temperature above the critical temperature (individual for each gas, for example, for carbon dioxide about 304 K), then it can no longer be turned into a liquid, no matter what pressure is applied to it. This phenomenon occurs due to the fact that at the critical temperature of the force surface tension fluids are zero.

Table 23. Critical temperature and critical pressure of some substances

What does the existence of a critical temperature indicate? What happens at even higher temperatures?

Experience shows that at temperatures higher than critical, a substance can only exist in a gaseous state.

The existence of a critical temperature was first pointed out in 1860 by Dmitri Ivanovich Mendeleev.

After the discovery of the critical temperature, it became clear why for a long time it was not possible to turn gases such as oxygen or hydrogen into liquid. Their critical temperature is very low (Table 23). To turn these gases into a liquid, they must be cooled below a critical temperature. Without this, all attempts to liquefy them are doomed to failure.

66. Partial pressure. relative humidity. 67. Instruments for measuring the relative humidity of the air.

The life of man, animals and plants depends on the concentration of water vapor (humidity) of the atmosphere, which varies widely depending on the place and season. As a rule, the water vapor around us is unsaturated. Relative humidity is the ratio of water vapor pressure to saturation vapor pressure at the same temperature, expressed as a percentage. One of the devices for measuring air humidity is a psychrometer, consisting of two identical thermometers, one of which is wrapped in a damp cloth. When the air humidity is less than 100%, the water from the cloth will evaporate, and thermometer B will cool, showing a lower temperature than A. And the lower the air humidity, the greater the difference, Dt, between the readings of thermometers A and B. Using a special psychrometric table, this temperature difference can be used to determine the humidity of the air.

Partial pressure is the pressure of a certain gas that is part of the gas mixture, which this gas would exert on the walls of the container containing it, if it alone occupied the entire volume of the mixture at the temperature of the mixture.

Partial pressure is not measured directly, but is estimated from the total pressure and composition of the mixture.

Gases dissolved in water or body tissues also exert pressure because the dissolved gas molecules are in random motion and have kinetic energy. If a gas dissolved in a liquid hits a surface, such as a cell membrane, it exerts a partial pressure in the same way as a gas in a gas mixture.

P. D. cannot be measured directly; it is calculated based on the total pressure and composition of the mixture.

Factors Determining the Value of the Partial Pressure of a Gas Dissolved in a Liquid. The partial pressure of a gas in a solution is determined not only by its concentration, but also by its solubility coefficient, i.e. some types of molecules, such as carbon dioxide, are physically or chemically attached to water molecules, while others are repelled. This relationship is called Henry's law and is expressed by the following formula: Partial pressure = Dissolved gas concentration / Solubility coefficient.

68. Surface tension.

The most interesting feature of liquids is the presence of a free surface. Liquid, unlike gases, does not fill the entire volume of the vessel into which it is poured. An interface is formed between the liquid and the gas (or vapor), which is in special conditions compared to the rest of the mass of the liquid. The molecules in the boundary layer of a liquid, in contrast to the molecules in its depth, are not surrounded by other molecules of the same liquid from all sides. The forces of intermolecular interaction acting on one of the molecules inside the liquid from the neighboring molecules are, on average, mutually compensated. Any molecule in the boundary layer is attracted by molecules inside the liquid (the forces acting on a given molecule of the liquid from the gas (or vapor) molecules can be neglected). As a result, some resultant force appears, directed deep into the liquid. Surface molecules are drawn into the liquid by the forces of intermolecular attraction. But all molecules, including those of the boundary layer, must be in a state of equilibrium. This equilibrium is achieved due to some decrease in the distance between the molecules of the surface layer and their nearest neighbors inside the liquid. As can be seen from fig. 3.1.2, when the distance between molecules decreases, repulsive forces arise. If the average distance between molecules inside the liquid is equal to r0, then the molecules of the surface layer are packed somewhat more densely, and therefore they have an additional reserve of potential energy compared to the internal molecules (see Fig. 3.1.2). It should be borne in mind that, due to the extremely low compressibility, the presence of a more densely packed surface layer does not lead to any noticeable change in the volume of the liquid. If the molecule moves from the surface into the liquid, the forces of intermolecular interaction will do positive work. On the contrary, in order to pull a certain number of molecules from the depth of the liquid to the surface (i.e., increase the surface area of ​​the liquid), external forces must perform a positive work ΔAext, proportional to the change ΔS of the surface area: ΔAext = σΔS.

The coefficient σ is called the coefficient of surface tension (σ > 0). Thus, the coefficient of surface tension is equal to the work required to increase the surface area of ​​a liquid at a constant temperature by one unit.

In SI, the surface tension coefficient is measured in joules per square meter (J/m2) or in newtons per meter (1 N/m = 1 J/m2).

It is known from mechanics that the equilibrium states of a system correspond to the minimum value of its potential energy. It follows that the free surface of the liquid tends to reduce its area. For this reason, a free drop of liquid takes on a spherical shape. The fluid behaves as if forces are acting tangentially to its surface, reducing (contracting) this surface. These forces are called surface tension forces.

The presence of surface tension forces makes the liquid surface look like an elastic stretched film, with the only difference that the elastic forces in the film depend on its surface area (i.e., on how the film is deformed), and the surface tension forces do not depend on the surface area liquids.

Some liquids, such as soapy water, have the ability to form thin films. All well-known soap bubbles have the correct spherical shape - this also manifests the action of surface tension forces. If a wire frame is lowered into the soap solution, one of the sides of which is movable, then the whole of it will be covered with a film of liquid.

69. Wetting.

Everyone knows that if you place a drop of liquid on a flat surface, it will either spread over it or take on a rounded shape. Moreover, the size and convexity (the value of the so-called contact angle) of a sessile drop is determined by how well it wets the given surface. The wetting phenomenon can be explained as follows. If the molecules of a liquid are attracted to each other more strongly than to the molecules of a solid body, the liquid tends to collect into a droplet.

An acute contact angle occurs on a wetted (lyophilic) surface, while an obtuse one occurs on a non-wettable (lyophobic) surface.

This is how mercury behaves on glass, water on paraffin or on a “greasy” surface. If, on the contrary, the molecules of a liquid are attracted to each other weaker than to the molecules of a solid body, the liquid is “pressed” to the surface and spreads over it. This happens with a drop of mercury on a zinc plate, or with a drop of water on clean glass. In the first case, it is said that the liquid does not wet the surface (the contact angle is greater than 90°), and in the second case, it wets it (the contact angle is less than 90°).

It is water-repellent lubricant that helps many animals escape from excessive wetting. For example, studies of marine animals and birds - fur seals, seals, penguins, loons - have shown that their downy hair and feathers have hydrophobic properties, while the guard hairs of animals and the upper part of the contour feathers of birds are well wetted with water. As a result, an air layer is created between the animal's body and water, which plays a significant role in thermoregulation and thermal insulation.

But lubrication isn't everything. The structure of the surface also plays a significant role in the wetting phenomenon. Rough, bumpy, or porous terrain can improve wetting. Recall, for example, sponges and terry towels that perfectly absorb water. But if the surface is initially “afraid” of water, then the developed relief will only aggravate the situation: water droplets will collect on ledges and roll off.

70. Capillary phenomena.

Capillary phenomena are called the rise or fall of liquid in tubes of small diameter - capillaries. Wetting liquids rise through the capillaries, non-wetting liquids descend.

On fig. 3.5.6 shows a capillary tube of some radius r, lowered by its lower end into a wetting liquid of density ρ. The upper end of the capillary is open. The rise of the liquid in the capillary continues until the force of gravity acting on the column of liquid in the capillary becomes equal in modulus to the resulting Fn of the surface tension forces acting along the boundary of contact between the liquid and the capillary surface: Ft = Fn, where Ft = mg = ρhπr2g, Fн = σ2πr cos θ.

This implies:

Figure 3.5.6.

Rise of the wetting liquid in the capillary.

With complete wetting θ = 0, cos θ = 1. In this case

With complete nonwetting, θ = 180°, cos θ = –1 and, therefore, h< 0. Уровень несмачивающей жидкости в капилляре опускается ниже уровня жидкости в сосуде, в которую опущен капилляр.

Water almost completely wets clean surface glass. Conversely, mercury does not completely wet the glass surface. Therefore, the level of mercury in the glass capillary falls below the level in the vessel.

71. Crystalline bodies and their properties.

Unlike liquids, a solid body retains not only its volume, but also its shape and has considerable strength.

The various solids encountered can be divided into two groups that differ significantly in their properties: crystalline and amorphous.

Basic properties of crystalline bodies

1. Crystalline bodies have a certain melting point tmelt, which does not change during melting at constant pressure (Fig. 1, curve 1).

2. Crystalline bodies are characterized by the presence of a spatial crystal lattice, which is an ordered arrangement of molecules, atoms or ions, repeating throughout the entire volume of the body (long-range order). For any crystal lattice, the existence of such an element of its structure is characteristic, by repeated repetition of which in space one can obtain the entire crystal. This is a single crystal. A polycrystal consists of many very small, intergrown single crystals, which are randomly oriented in space.