Fundamentals of molecular kinetic theory. Molecular-kinetic theory
DEFINITION
The equation underlying the molecular kinetic theory connects macroscopic quantities describing (for example, pressure) with the parameters of its molecules (and their velocities). This equation looks like:
Here, is the mass of a gas molecule, is the concentration of such particles per unit volume, and is the averaged square of the molecular velocity.
The basic equation of the MKT clearly explains how an ideal gas creates on the vessel walls surrounding it. Molecules all the time hit the wall, acting on it with a certain force F. Here it should be remembered: when a molecule hits an object, a force -F acts on it, as a result of which the molecule “bounces” from the wall. In this case, we consider the collisions of molecules with the wall to be absolutely elastic: the mechanical energy of the molecules and the wall is completely conserved without passing into . This means that only the molecules change during collisions, and the heating of the molecules and the wall does not occur.
Knowing that the collision with the wall was elastic, we can predict how the velocity of the molecule will change after the collision. The velocity modulus will remain the same as before the collision, and the direction of motion will change to the opposite with respect to the Ox axis (we assume that Ox is the axis that is perpendicular to the wall).
There are a lot of gas molecules, they move randomly and often hit the wall. Having found the geometric sum of forces with which each molecule acts on the wall, we find out the gas pressure force. To average the velocities of molecules, it is necessary to use statistical methods. That is why the basic MKT equation uses the averaged square of the molecular velocity , and not the square of the averaged velocity : the averaged velocity of randomly moving molecules is equal to zero, and in this case we would not get any pressure.
Now the physical meaning of the equation is clear: the more molecules are contained in the volume, the heavier they are and the faster they move, the more pressure they create on the walls of the vessel.
Basic MKT equation for the ideal gas model
It should be noted that the basic MKT equation was derived for the ideal gas model with the appropriate assumptions:
- Collisions of molecules with surrounding objects are absolutely elastic. For real gases, this is not entirely true; some of the molecules still pass into the internal energy of the molecules and the wall.
- The forces of interaction between molecules can be neglected. If the real gas is at high pressure and relatively low temperature, these forces become very significant.
- We consider molecules to be material points, neglecting their size. However, the dimensions of the molecules of real gases affect the distance between the molecules themselves and the wall.
- And, finally, the main equation of the MKT considers a homogeneous gas - and in reality we often deal with mixtures of gases. Such as, .
However, for rarefied gases, this equation gives very accurate results. In addition, many real gases at room temperature and at pressures close to atmospheric are very similar in properties to an ideal gas.
As is known from the laws, the kinetic energy of any body or particle. Replacing the product of the mass of each of the particles and the square of their speed in the equation we wrote down, we can represent it as:
Also, the kinetic energy of gas molecules is expressed by the formula , which is often used in problems. Here k is Boltzmann's constant, establishing the relationship between temperature and energy. k=1.38 10 -23 J/K.
The basic equation of the MKT underlies thermodynamics. It is also used in practice in astronautics, cryogenics and neutron physics.
Examples of problem solving
EXAMPLE 1
| Exercise | Determine the speed of movement of air particles under normal conditions. |
| Decision | We use the basic MKT equation, considering air as a homogeneous gas. Since air is actually a mixture of gases, the solution to the problem will not be absolutely accurate. Gas pressure:
We can notice that the product is a gas, since n is the concentration of air molecules (the reciprocal of volume), and m is the mass of the molecule. Then the previous equation becomes: Under normal conditions, the pressure is 10 5 Pa, the air density is 1.29 kg / m 3 - these data can be taken from the reference literature. From the previous expression we obtain air molecules:
|
| Answer | m/s |
EXAMPLE 2
| Exercise | Determine the concentration of homogeneous gas molecules at a temperature of 300 K and 1 MPa. Consider the gas to be ideal. |
| Decision | Let's start the solution of the problem with the basic equation of the MKT:  , as well as any material particles: . Then our calculation formula will take a slightly different form: |
Any substance is considered by physics as a collection of the smallest particles: atoms, molecules and ions. All these particles are in continuous chaotic motion and interact with each other through elastic collisions.
Atomic theory - the basis of molecular kinetic theory
Democritus
Molecular kinetic theory originated in ancient Greece about 2500 years ago. Its foundation is considered atomic hypothesis , sponsored by ancient Greek philosopher Leucippus and his student Ancient Greek scholar Democritus from the city of Abdera.
Leucippus
Leucippus and Democritus assumed that all material things consist of indivisible smallest particles, which are called atoms (from Greekἄτομος - indivisible). And the space between the atoms is filled with emptiness. All atoms have a size and shape, and are able to move. The proponents of this theory in the Middle Ages were Giordano Bruno, Galileo, Isaac Beckman and other scientists. The foundations of the molecular kinetic theory were laid in the work "Hydrodynamics", published in 1738. Its author was a Swiss physicist, mechanic and mathematician Daniel Bernoulli.
Basic Provisions of Molecular Kinetic Theory
Mikhail Vasilievich Lomonosov
The closest thing to modern physics was the theory of the atomic structure of matter, which was developed in the 18th century by the great Russian scientist Mikhail Vasilievich Lomonosov. He argued that all substances are made up of molecules which he called corpuscles . And corpuscles, in turn, consist of atoms . Lomonosov's theory was called corpuscular .
But as it turned out, the atom is divided. It consists of a positively charged nucleus and negative electrons. In general, it is electrically neutral.
Modern science calls atom the smallest part of a chemical element, which is the carrier of its basic properties. Connected by interatomic bonds, atoms form molecules. A molecule can contain one or more atoms of the same or different chemical elements.
All bodies are made up of a huge number of particles: atoms, molecules and ions. These particles are constantly and randomly moving. Their movement does not have any definite direction and is called thermal motion . During their motion, the particles interact with each other by absolutely elastic collisions.
We cannot observe molecules and atoms with the naked eye. But we can see the result of their actions.
Confirmation of the main provisions of the molecular kinetic theory are: diffusion , Brownian motion and change aggregate states of substances .
Diffusion
Diffusion in liquid
One of the proofs of the constant movement of molecules is the phenomenon diffusion .
In the process of movement, the molecules and atoms of one substance penetrate between the molecules and atoms of another substance in contact with it. Molecules and atoms of the second substance behave in exactly the same way. relation to the first. And after a while, the molecules of both substances are evenly distributed throughout the volume.
The process of penetration of molecules of one substance between the molecules of another is called diffusion . We encounter the phenomenon of diffusion at home every day when we drop a tea bag into a glass of boiling water. We observe how colorless boiling water changes its color. Throwing a few crystals of manganese into a test tube with water, you can see that the water turns pink. This is also diffusion.
The number of particles per unit volume is called concentration substances. During diffusion, molecules move from those parts of the substance where the concentration is higher to those parts where it is less. The movement of molecules is called diffusion flow . As a result of diffusion, the concentrations in different parts of substances are aligned.
Diffusion can be observed in gases, liquids and solids. In gases, it occurs at a faster rate than in liquids. We know how quickly smells spread in the air. The liquid in the test tube stains much more slowly if ink is dropped into it. And if we put salt crystals on the bottom of a container with water and do not mix it, then more than one day will pass before the solution becomes homogeneous.
Diffusion also occurs at the boundary of the contacting metals. But its speed in this case is very small. If you cover copper with gold, then at room temperature and atmospheric pressure, gold will penetrate copper by only a few microns in a few thousand years.
Lead from an ingot placed under a load on a gold ingot will penetrate into it only to a depth of 1 cm in 5 years.
Diffusion in metals
Diffusion rate
The diffusion rate depends on the cross-sectional area of the flow, the difference in the concentrations of substances, the difference in their temperatures or charges. Through a rod with a diameter of 2 cm, heat spreads 4 times faster than through a rod with a diameter of 1 cm. The higher the temperature difference of the substances, the higher the diffusion rate. During thermal diffusion, its rate depends on thermal conductivity material, and in the case of a flow of electric charges - from electrical conductivity .
Fick's Law
Adolf Fick
In 1855, the German physiologist Adolf Eugene Fick made the first quantitative description of diffusion processes:
where J - density diffusion flow of matter,
D - diffusion coefficient,
C - substance concentration.
Diffusion flux density of matterJ [cm -2 s -1 ] is proportional to the diffusion coefficientD [cm -2 s -1 ] and the concentration gradient taken with the opposite sign.
This equation is called Fick's first equation .
Diffusion, as a result of which the concentrations of substances are equalized, is called non-stationary diffusion . With such diffusion, the concentration gradient changes with time. And in case stationary diffusion this gradient remains constant.
Brownian motion
Robert Brown
This phenomenon was discovered by the Scottish botanist Robert Brown in 1827. Studying under a microscope cytoplasmic grains suspended in water isolated from pollen cells of a North American plantClarkia pulchella, he drew attention to the smallest solid grains. They trembled and moved slowly for no apparent reason. If the temperature of the liquid increased, the speed of the particles increased. The same thing happened when the particle size decreased. And if their size increased, the temperature of the liquid decreased or its viscosity increased, the movement of the particles slowed down. And these amazing "dances" of particles could be observed indefinitely. Deciding that the reason for this movement is that the particles are alive, Brown replaced the grains with small particles of coal. The result was the same.
Brownian motion
To repeat Brown's experiments, it is enough to have the most ordinary microscope. The molecular size is too small. And it is impossible to consider them with such a device. But if we color the water in a test tube with watercolor paint and then look at it through a microscope, we see tiny colored particles that move randomly. These are not molecules, but paint particles suspended in water. And they are forced to move by water molecules that hit them from all sides.
This is the behavior of all particles visible in a microscope that are suspended in liquids or gases. Their random movement, caused by the thermal motion of molecules or atoms, is called brownian motion . A Brownian particle is continuously subjected to impacts from the molecules and atoms that make up liquids and gases. And this movement does not stop.
But particles up to 5 microns (micrometers) in size can participate in Brownian motion. If their size is larger, they are immobile. The smaller the size of a Brownian particle, the faster it moves. Particles smaller than 3 microns move progressively along all complex trajectories or rotate.
Brown himself could not explain the phenomenon he discovered. And only in the 19th century, scientists found the answer to this question: the movement of Brownian particles is caused by the influence of the thermal movement of molecules and atoms on them.
Three states of matter
The molecules and atoms that make up matter are not only in motion, but also interact with each other, mutually attracting or repelling.
If the distance between the molecules is comparable to their size, then they experience attraction. If it becomes smaller, then the repulsive force begins to prevail. This explains the resistance of physical bodies to deformation (compression or tension).
If the body is compressed, then the distance between the molecules decreases, and the repulsive forces will try to return the molecules to their original state. When stretched, the deformation of the body will interfere with the forces of attraction between the molecules.
Molecules interact not only within one body. Dip a piece of cloth into the liquid. We will see that it gets wet. This is due to the fact that the molecules of a liquid are attracted to the molecules of solids more strongly than to each other.
Each physical substance, depending on temperatures and pressures, can be in three states: solid, liquid or gaseous . They're called aggregate .
In gases the distance between molecules is large. Therefore, the forces of attraction between them are so weak that they perform a chaotic and almost free movement in space. They change the direction of their movement by hitting each other or the walls of blood vessels.
in liquids molecules are closer together than in a gas. There is more attraction between them. The molecules in them no longer move freely, but oscillate randomly near the equilibrium position. But they are able to jump in the direction of the external force, changing places with each other. The result of this is fluid flow.
In solids the forces of interaction between molecules are very large due to the close distance between them. They cannot overcome the attraction of neighboring molecules, therefore they are only capable of performing oscillatory movements around the equilibrium position.
Solid bodies retain volume and shape. The liquid has no form, it always takes the form of the vessel in which it is located at the moment. But its volume remains the same. Gaseous bodies behave differently. They easily change both shape and volume, taking the form of the vessel in which they were placed, and occupying the entire volume provided to them.
However, there are also such bodies that have the structure of a liquid, have a slight fluidity, but at the same time are able to retain their shape. Such bodies are called amorphous .
Modern physics singles out the fourth aggregate state of matter - plasma .
§ 2. Molecular physics. Thermodynamics
Main provisions of molecular kinetic theory(MKT) are as follows.1. Substances are made up of atoms and molecules.
2. Atoms and molecules are in continuous chaotic motion.
3. Atoms and molecules interact with each other with forces of attraction and repulsion
The nature of the movement and interaction of molecules can be different, in this regard, it is customary to distinguish 3 states of aggregation of matter: solid, liquid and gaseous. The interaction between molecules is strongest in solids. In them, the molecules are located in the so-called nodes of the crystal lattice, i.e. in positions where the forces of attraction and repulsion between molecules are equal. The motion of molecules in solids is reduced to oscillatory motion around these equilibrium positions. In liquids, the situation differs in that, having fluctuated around some equilibrium positions, the molecules often change them. In gases, the molecules are far from each other, so the interaction forces between them are very small and the molecules move forward, occasionally colliding with each other and with the walls of the vessel in which they are located.
Relative molecular weight M r call the ratio of the mass m o of a molecule to 1/12 of the mass of a carbon atom moc:
The amount of a substance in molecular physics is usually measured in moles.
Molem ν called the amount of a substance that contains the same number of atoms or molecules (structural units) as they are contained in 12 g of carbon. This number of atoms in 12 g of carbon is called Avogadro's number:
Molar mass M = M r 10 −3 kg/mol is the mass of one mole of a substance. The number of moles in a substance can be calculated using the formula
The basic equation of the molecular kinetic theory of an ideal gas is:
where m0 is the mass of the molecule; n- concentration of molecules; Ṽ
is the root mean square velocity of the molecules.
2.1. Gas laws
The equation of state of an ideal gas is the Mendeleev-Clapeyron equation:Isothermal process(Boyle-Mariotte law):
For a given mass of gas at a constant temperature, the product of pressure and its volume is a constant value:
In coordinates p − V isotherm is a hyperbola, and in coordinates V − T and p − T- straight (see fig. 4) 
Isochoric process(Charles law):
For a given mass of gas with a constant volume, the ratio of pressure to temperature in degrees Kelvin is a constant value (see Fig. 5). 
isobaric process(Gay-Lussac's law):
For a given mass of gas at constant pressure, the ratio of gas volume to temperature in degrees Kelvin is a constant value (see Fig. 6). 
Dalton's law:
If a vessel contains a mixture of several gases, then the pressure of the mixture is equal to the sum of the partial pressures, i.e. the pressures that each gas would create in the absence of the others.
2.2. Elements of thermodynamics
Internal energy of the body is equal to the sum of the kinetic energies of the random motion of all molecules relative to the center of mass of the body and the potential energies of the interaction of all molecules with each other.Internal energy of an ideal gas is the sum of the kinetic energies of the random movement of its molecules; Since the molecules of an ideal gas do not interact with each other, their potential energy vanishes.
For an ideal monatomic gas, the internal energy
The amount of heat Q called a quantitative measure of the change in internal energy during heat transfer without doing work.
Specific heat is the amount of heat that 1 kg of a substance receives or gives off when its temperature changes by 1 K
Work in thermodynamics:
work during isobaric expansion of a gas is equal to the product of the gas pressure and the change in its volume:
The law of conservation of energy in thermal processes (the first law of thermodynamics):
the change in the internal energy of the system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system:
Applying the first law of thermodynamics to isoprocesses:
a) isothermal process T = const ⇒ ∆T = 0.
In this case, the change in the internal energy of an ideal gas
Hence: Q=A.
All the heat transferred to the gas is spent on doing work against external forces;
b) isochoric process V = const ⇒ ∆V = 0.
In this case, the work of the gas
Hence, ∆U = Q.
All the heat transferred to the gas is spent on increasing its internal energy;
in) isobaric process p = const ⇒ ∆p = 0.
In this case:
adiabatic A process that occurs without heat exchange with the environment is called:
In this case A = −∆U, i.e. the change in the internal energy of the gas occurs due to the work of the gas on external bodies.
As the gas expands, it does positive work. The work A performed by external bodies on the gas differs from the work of the gas only in sign:
The amount of heat required to heat up a body in a solid or liquid state within one state of aggregation, calculated by the formula
where c is the specific heat of the body, m is the mass of the body, t 1 is the initial temperature, t 2 is the final temperature.
The amount of heat required to melt the body at the melting point, calculated by the formula
where λ is the specific heat of fusion, m is the mass of the body.
The amount of heat required for evaporation, is calculated by the formula
where r is the specific heat of vaporization, m is the mass of the body.
In order to convert part of this energy into mechanical energy, heat engines are most often used. Heat engine efficiency The ratio of the work A done by the engine to the amount of heat received from the heater is called:
The French engineer S. Carnot came up with an ideal heat engine with an ideal gas as a working fluid. The efficiency of such a machine 
Air, which is a mixture of gases, contains water vapor along with other gases. Their content is usually characterized by the term "humidity". Distinguish between absolute and relative humidity.
absolute humidity called the density of water vapor in the air ρ ([ρ] = g/m 3). You can characterize absolute humidity by the partial pressure of water vapor - p([p] = mm Hg; Pa).
Relative humidity (ϕ)- the ratio of the density of water vapor present in the air to the density of the water vapor that would have to be contained in the air at that temperature in order for the vapor to be saturated. You can measure relative humidity as the ratio of the partial pressure of water vapor (p) to that partial pressure (p 0) that saturated steam has at this temperature: 
DEFINITION
Atom - the smallest particle of a given chemical element. All atoms that exist in nature are represented in Mendeleev's periodic system of elements.
Atoms are connected into a molecule due to chemical bonds based on electrical interaction. The number of atoms in a molecule can be different. A molecule can consist of one, two, three, or even several hundred atoms.
DEFINITION
Molecule- the smallest particle of a given substance that has its chemical properties.
Molecular Kinetic Theory- the doctrine of the structure and properties of matter based on the concept of the existence of atoms and molecules.
The founder of the molecular kinetic theory is M.V. Lomonosov (1711-1765), who formulated its main provisions and applied them to explain various thermal phenomena.
Basic Provisions of Molecular Kinetic Theory
The main provisions of the ICT:
- all bodies in nature consist of the smallest particles (atoms and molecules);
- particles are in continuous chaotic motion, which is called thermal;
- particles interact with each other: forces of attraction and repulsion act between the particles, which depend on the distance between the particles.
The molecular kinetic theory is confirmed by many phenomena.
The mixing of various liquids, the dissolution of solids in liquids, is explained by the mixing of molecules of various kinds. In this case, the volume of the mixture may differ from the total volume of its constituent components. which indicates different sizes of molecular compounds.
DEFINITION
Diffusion- the phenomenon of penetration of two or more adjoining substances into each other.
Diffusion proceeds most intensively in gases. The spread of odors is due to diffusion. Diffusion indicates that the molecules are in constant chaotic motion. Also, the phenomenon of diffusion indicates that there are gaps between the molecules, i.e. matter is discrete.
DEFINITION
Brownian motion- thermal motion of the smallest microscopic particles suspended in a liquid or gas.
This phenomenon was first observed by the English botanist R. Brown in 1827. While observing flower pollen suspended in water through a microscope, he saw that each pollen particle makes rapid random movements, moving over a certain distance. As a result of individual movements, each pollen particle moved along a zigzag trajectory (Fig. 1a).
Fig.1. Brownian motion: a) trajectories of motion of individual particles suspended in a liquid; b) transfer of momentum by liquid molecules to a suspended particle.
Further studies of Brownian motion in various liquids and with various solid particles showed that this motion becomes more intense, the smaller the particle size and the higher the temperature of the experiment. This movement never stops and does not depend on any external causes.
R. Brown could not explain the observed phenomenon. The theory of Brownian motion was built by A. Einstein in 1905 and received experimental confirmation in the experiments of the French physicist J. Perrin (1900-1911).
Liquid molecules that are in constant chaotic motion, when colliding with a suspended particle, transfer some impulse to it (Fig. 1, b). In the case of a large particle, the number of molecules incident on it from all sides is large, their impacts are compensated at each moment of time, and the particle remains practically motionless. If the particle size is very small, then the impacts of the molecules are not compensated - on the one hand, a larger number of molecules can hit it than on the other, as a result of which the particle will begin to move. It is precisely such a movement under the influence of random impacts of molecules that Brownian particles perform. Although Brownian particles are billions of times larger than the mass of individual molecules and move at very low speeds (compared to the speeds of molecules), their movement can still be observed under a microscope.
Examples of problem solving
EXAMPLE 1
EXAMPLE 2
