In the statistical method, to determine the main characteristic (X is the set of coordinates and momenta of all particles of the system), one or another model of the structure of the body under consideration is used.

It turns out that it is possible to find general properties of general statistical patterns that do not depend on the structure of matter and are universal. The identification of such regularities is the main task of the thermodynamic method for describing thermal processes. All basic concepts and laws of thermodynamics can be revealed on the basis of statistical theory.

For an isolated (closed) system or a system in a constant external field, the state is called statistically equilibrium if the distribution function does not depend on time.

The specific form of the distribution function of the system under consideration depends both on the totality of external parameters and on the nature of the interaction with the surrounding bodies. Under the external parameters in this case we will understand the quantities determined by the position of the bodies not included in the system under consideration. These are, for example, the volume of the system V, the intensity of the force field, etc. Let's consider the two most important cases:

1) The system under consideration is energetically isolated. The total energy of particles E is constant. Wherein. E can be included in a, but highlighting it emphasizes the special role of E. The condition for the isolation of the system for given external parameters can be expressed by the equality:

2) The system is not closed - energy exchange is possible. In this case, it cannot be found; it will depend on the generalized coordinates and momenta of the particles of the surrounding bodies. This is possible if the interaction energy of the system under consideration with the surrounding bodies.

Under this condition, the distribution function of microstates depends on the average intensity of the thermal motion of the surrounding bodies, which is characterized by the temperature T of the surrounding bodies: .

Temperature also plays a special role. It has no (unlike a) analogue in mechanics: (does not depend on T).

In a state of statistical equilibrium does not depend on time, and all internal parameters are unchanged. In thermodynamics, this state is called the state of thermodynamic equilibrium. The concepts of statistical and thermodynamic equilibrium are equivalent.

Distribution function of a microscopic isolated system - Gibbs microcanonical distribution

The case of an energetically isolated system. Let us find the form of the distribution function for this case.

An essential role in finding the distribution function is played only by the integrals of motion - energy, - momentum of the system and - angular momentum. Only they are controlled.

The Hamiltonian plays a special role in mechanics, because it is the Hamiltonian function that determines the form of the particle motion equation. The conservation of the total momentum and angular momentum of the system in this case is a consequence of the equations of motion.

Therefore, it is precisely such solutions of the Liouville equation that are singled out when the dependence manifests itself only through the Hamiltonian:

Because, .

Of all possible values ​​of X (the set of coordinates and momenta of all particles of the system), those that are compatible with the condition are selected. The constant C can be found from the normalization condition:

where is the area of ​​the hypersurface in the phase space, distinguished by the condition of energy constancy.

Those. is the microcanonical Gibbs distribution.

In the quantum theory of the equilibrium state, there is also a microcanonical Gibbs distribution. Let's introduce the notation: - a complete set of quantum numbers characterizing the microstate of a system of particles, - the corresponding admissible energy values. They can be found by solving the stationary equation for the wave function of the system under consideration.

The distribution function of microstates in this case will be the probability for the system to be in a certain state: .

The quantum microcanonical Gibbs distribution can be written as:

where is the Kronecker symbol, - from the normalization: is the number of microstates with a given energy value (as well as). It's called statistical weight.

From the definition, all states that satisfy the condition have the same probability, equal. Thus, the quantum microcanonical Gibbs distribution is based on the principle of equal a priori probabilities.

The distribution function of the microstates of the system in the thermostat is the canonical Gibbs distribution.

Consider now a system exchanging energy with surrounding bodies. From a thermodynamic point of view, this approach corresponds to a system surrounded by a very large thermostat with temperature T. For a large system (our system + thermostat), the microcanonical distribution can be used, since such a system can be considered isolated. We will assume that the system under consideration is a small but macroscopic part of a larger system with temperature T and the number of particles in it. That is, equality (>>) is satisfied.

We will denote the variables of our system by X, and the thermostat variables by X1.

Then we write the microcanonical distribution for the entire system:

We will be interested in the probability of the state of a system of N particles for any possible states of the thermostat. This probability can be found by integrating this equation over the thermostat states

The Hamilton function of the system and thermostat can be represented as

We will neglect the energy of interaction between the system and the thermostat in comparison with both the energy of the system and the energy of the thermostat. This can be done because the interaction energy for a macrosystem is proportional to its surface area, while the energy of a system is proportional to its volume. However, neglecting the interaction energy compared to the energy of the system does not mean that it is equal to zero, otherwise the formulation of the problem loses its meaning.

Thus, the probability distribution for the system under consideration can be represented as

Let us turn to integration over the thermostat energy

Hence, using the -function property

In what follows, we will pass to the limiting case when the thermostat is very large. Let us consider a special case when the thermostat is an ideal gas with N1 particles with mass m each.

Let's find the value that represents the value

where is the volume of the phase space contained within the hypersurface. Then is the volume of the hyperspheric layer (compare with the expression for the three-dimensional space

For an ideal gas, the region of integration is given by the condition

As a result of integration within the specified boundaries, we obtain the volume of a 3N1-dimensional ball with a radius that will be equal to. Thus, we have

Where do we get

Thus, for the probability distribution we have

Let us now pass to the N1 limit, however, assuming that the ratio remains constant (the so-called thermodynamic limit). Then we get

Taking into account that

Then the distribution function of the system in the thermostat can be written as

where C is found from the normalization condition:

The function is called the classical statistical integral. Thus, the distribution function of the system in the thermostat can be represented as:

This is the canonical Gibbs distribution (1901).

In this distribution, T characterizes the average intensity of thermal motion - the absolute temperature of the particles of the environment.

Another form of writing the Gibbs distribution

When determining, microscopic states were considered different, differing only in the rearrangement of individual particles. This means that we are able to keep track of each particle. However, this assumption leads to a paradox.

The expression for the quantum canonical Gibbs distribution can be written by analogy with the classical one:

Statistical sum: .

It is a dimensionless analogue of the statistical integral. Then the free energy can be represented as:

Let us now consider a system located in a thermostat and capable of exchanging energy and particles with the environment. The derivation of the Gibbs distribution function for this case is in many ways similar to the derivation of the canonical distribution. For the quantum case, the distribution has the form:

This distribution is called the Gibbs grand canonical distribution. Here m is the chemical potential of the system, which characterizes the change in thermodynamic potentials when the number of particles in the system changes by one.

Z - from the normalization condition:

Here the summation goes not only over square numbers, but also over all possible values ​​of the number of particles.

Another form of writing: we introduce a function, but as previously obtained from thermodynamics, where is a large thermodynamic potential. As a result, we get

Here is the average value of the number of particles.

The classical distribution is similar.

Maxwell and Boltzmann distributions

The canonical Gibbs distribution establishes (for given) the explicit form of the distribution function for the values ​​of all coordinates and momenta of particles (6N-variables). But such a function is very complex. Often simpler functions are sufficient.

Maxwell distribution for an ideal monatomic gas. We can consider each gas molecule as a "system under consideration", belonging to a thermostat. Therefore, the probability of any molecule to have impulses in given intervals is given by the Gibbs canonical distribution: .

Replacing momenta with velocities and using the normalization conditions, we obtain

Maxwell's distribution function for velocity components. It is easy to get the distribution modulo as well.

In any system, the energy of which is equal to the sum of the energies of individual particles, there is an expression similar to Maxwell's. This is the Maxwell-Boltzmann distribution. Again, we will assume that the “system” is any one particle, while the rest play the role of a thermostat. Then the probability of the state of this chosen particle for any state of the others is given by the canonical distribution: , . For the rest of the quantities ... integrated

§4 Maxwell's law on the distribution of velocities and energies

The law of distribution of ideal gas molecules by velocities, theoretically obtained by Maxwell in 1860, determines what number dN molecules of homogeneous (p= const) of a monatomic ideal gas from the total numberNits molecules per unit volume has at a given temperature T speeds in the range fromv before v+ dv.

To derive the velocity distribution function of moleculesf( v) equal to the ratio of the number of molecules dN, whose velocities lie in the intervalv÷ v+ dvto the total number of moleculesNand the size of the intervaldv

Maxwell used two sentences:

a) all directions in space are equal and therefore any direction of motion of a particle, i.e. any direction of speed is equally likely. This property is sometimes called the property of isotropy of the distribution function.

b) movement along three mutually perpendicular axes are independent i.e. x-components of speeddoes not depend on the value of its components or . And then the conclusion f ( v) done first for one component, and then generalized to all velocity coordinates.

It is also believed that the gas consists of a very large numberN identical molecules in a state of random thermal motion at the same temperature. Force fields do not act on gas.

Functions f ( v) determines the relative number of moleculesdN( v)/ Nwhose speeds lie in the interval fromv before v+ dv(for example: gas hasN= 10 6 molecules, whiledN= 100

molecules have velocities fromv=100 to v+ dv=101 m/s ( dv = 1 m) then .

Using the methods of probability theory, Maxwell found the functionf ( v) - the law of distribution of molecules of an ideal gas in terms of velocities:

f ( v) depends on the type of gas (on the mass of the molecule) and on the state parameter (on the temperature T)

f( v) depends on the ratio of the kinetic energy of the molecule corresponding to the considered velocity to the value kTcharacterizing the average thermal energy of gas molecules.

At small v and function f( v) changes almost like a parabola. As v increases, the multiplier decreases faster than the multiplier increases, i.e. there is a max function f( v) . The velocity at which the velocity distribution function of ideal gas molecules is maximum is called most likely speed find from the condition

Therefore, with increasing temperature, the most probable speed increases, but the area S, bounded by the curve of the distribution function, remains unchanged, since from the normalization condition(since the probability of a certain event is 1), so as the temperature rises, the distribution curvef ( v) will expand and decrease.

In statistical physics, the average value of a quantity is defined as the integral from 0 to infinity of the product of the quantity and the probability density of this quantity (statistical weight)

< X >=

Then the arithmetic mean velocity of the molecules

And integrating by parts, we get

Velocities characterizing the state of the gas

§5 Experimental verification of Maxwell's distribution law - Stern's experiment

A platinum wire coated with a layer of silver is stretched along the axis of the inner cylinder for the purpose, which is heated by current. When heated, the silver evaporates, the silver atoms fly out through the slot and fall on the inner surface of the second cylinder. If both cylinders are stationary, then all atoms, regardless of their speed, fall into the same place B. When the cylinders rotate with an angular velocity ω, the silver atoms will fall into points B ', B '' and so on. In terms of ω, distance? and displacement X= BB', you can calculate the speed of the atoms that hit the point B'.

The slit image is blurry. By examining the thickness of the deposited layer, one can estimate the velocity distribution of molecules, which corresponds to the Maxwellian distribution.

§6 Barometric formula

Boltzmann distribution

Until now, the behavior of an ideal gas not subjected to external force fields has been considered. It is well known from experience that under the action of external forces, the uniform distribution of particles in space can be disturbed. So, under the influence of gravity, the molecules tend to sink to the bottom of the vessel. Intense thermal motion prevents settling, and the molecules spread out so that their concentration gradually decreases as the height increases.

Let's derive the law of change of pressure with height, assuming that the gravitational field is uniform, the temperature is constant and the mass of all molecules is the same. If atmospheric pressure is high h equals p , then at height h + dh it is equal to p + dp(at dh > 0, dp < 0, так как pdecreases with increaseh).

Pressure difference at heightsh and h+ dhwe can define as the weight of air molecules enclosed in a volume with a base area of ​​1 and a heightdh.

density at heighth, and since , then = const .

Then

From the Mendeleev-Clapeyron equation.

Then

Or

With height change fromh 1 before h 2 pressure changes fromp 1 before p 2

We propotentiate this expression (

The barometric formula shows how pressure changes with altitude.

Ticket

1) Kinematics of a material point. Reference system, radius - vector, displacement, path, speed, acceleration

Kinematics of a material point- a section of kinematics that studies the mathematical description of the movement of material points. The main task of kinematics is to describe the movement with the help of a mathematical apparatus without finding out the reasons that cause this movement.
Reference system- A set of bodies that are motionless relative to each other, in relation to which the movement is considered and the clock counting time.
Radius vector- A vector that specifies the position of a point in space (for example, Hilbert or vector) relative to some pre-fixed point
moving- change in the location of the physical body in space relative to the selected frame of reference.
Path is the length of the trajectory of the body.
moving is a segment connecting the initial and final position of the body.
Speed- The speed of movement of the body and the direction in which the particle moves at each moment of time.
Acceleration is a vector quantity characterizing the rate of change in the speed of a moving body in magnitude and direction.

2) Waves. General characteristics of wave processes. Plane wave equation. Phase and group wave velocities

Waves– There are two types of waves: Longitudinal and transverse. If the oscillatory process is perpendicular to the direction of wave propagation - transverse. If the oscillation is along - longitudinal.

Longitudinal waves- fluctuations of the medium occur along the direction of wave propagation, while there are areas of compression and rarefaction of the medium.
transverse waves- fluctuations of the medium occur perpendicular to the direction of their propagation, while there is a shift in the layers of the medium.

Plane wave equation -
Wave phase velocity- the speed of movement of a point with a constant phase of oscillatory motion in space
along the given direction.
Group velocity - determines the speed and direction of energy transfer by waves

Ticket

1) Rectilinear and curvilinear motion. Tangential and normal acceleration

Rectilinear motion- mechanical movement, in which the displacement vector ∆r does not change in direction, its module is equal to the length of the path traveled by the body
Curvilinear motion- this is a movement whose trajectory is a curved line (for example, a circle, an ellipse, a hyperbola, a parabola). An example of a curvilinear movement is the movement of the planets, the end of the clock hand on the dial, etc. In the general case, the speed during curvilinear motion changes in magnitude and in direction.
Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.
Normal acceleration- a vector quantity characterizing the rate of change in the speed of a moving body in magnitude and direction.

2) Principles of relativity of Galileo, transformations of Galileo.

Galileo's principle of relativity- says that all physical processes in inertial reference frames proceed in the same way, regardless of whether the system is stationary or it is in a state of uniform and rectilinear motion.
Galilean transformations- Galileo's transformations are based on Galileo's principle of relativity, which implies the same time in all frames of reference ("absolute time")

Ticket

1) Kinematics of rotary motion

If in the process of motion of an absolutely rigid body its points A and B remain motionless, then any point C of the body located on the line AB must also remain motionless. Otherwise, the distances AC and BC would have to change, which would contradict the assumption of absolute rigidity of the body. Therefore, the motion of a rigid body, in which its two points A and B remain motionless, is called the rotation of the body around a fixed axis, and the fixed line AB is called the axis of rotation.

Consider an arbitrary point M of the body that does not lie on the axis of rotation AB. When a rigid body rotates, the distances M A and MB and the distance ρ the M points up to the axis of rotation must remain unchanged. Thus, all points of a body rotating around a fixed axis describe circles whose centers lie on the axis of rotation, and the planes are perpendicular to this axis. The movement of an absolutely rigid body, fixed at one fixed point, is called the rotation of the body around a fixed point - the center of rotation. Such a motion of an absolutely rigid body at each moment of time can be considered as a rotation around some axis passing through the center of rotation and called the instantaneous axis of rotation of the body. The position of the instantaneous axis relative to the fixed frame of reference and the body itself can change over time.

2) Michelson's experiment. SRT postulates. Lorentz transformations, consequences from Lorentz transformations

Michelson's experience- a physical experiment set by Albert Michelson on his interferometer in 1881, in order to measure the dependence of the speed of light on the motion of the Earth relative to the ether. The ether was then understood as a medium similar to volumetrically distributed matter, in which light propagates like sound vibrations. The result of the experiment, according to Michelson, was negative - the displacement of the bands did not coincide in phase with the theoretical ones, but the fluctuations of these displacements were only slightly less than the theoretical ones. The existence of the ether has been refuted.
1) all natural phenomena proceed in exactly the same way in all inertial frames of reference.
2) C is a constant value and does not depend on the speed of the light source and receiver
3) from position 2 of the postulate it is easy to prove that events are simultaneous in one frame of reference are non-simultaneous in another frame of reference

Ticket

1) The concept of mass, force, momentum.

Pulse The product of a body's mass and its speed.
Weight is a property of a body that characterizes its inertia. With the same impact from the surrounding bodies, one body can quickly change its speed, and the other, under the same conditions, much more slowly.
Strength is a quantitative measure of the interaction of bodies. Force is the cause of a change in the speed of a body. In Newtonian mechanics, forces can have various physical causes: friction force, gravity force, elastic force, etc. Force is a vector quantity.

2) Addition of speeds. Space-time interval

When considering a complex movement (that is, when a point or body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in 2 frames of reference.
In classical mechanics, the absolute velocity of a point is equal to the vector sum of its relative and translational velocities:
This equality is the content of the statement of the theorem on the addition of velocities.
The speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of this body relative to the moving frame of reference and the speed (relative to the fixed frame) of that point of the moving frame of reference where the body is currently located.

Ticket

1) Newton's laws. Inertial and non-inertial frames of reference. Forces of inertia.

Newton's laws- three laws that underlie classical mechanics and allow writing the equations of motion for any mechanical system, if the force interactions for its constituent bodies are known.

1) If no external force acts on the body, then the body is at rest or uniform rectilinear motion.
2) F=ma Acceleration of a body is directly proportional to the resultant force and inversely proportional to its mass
3) The force of action is equal to the force of reaction F1 = - F2

Inertial Reference System (ISO)- a frame of reference in which the law of inertia is valid: all free bodies (that is, those on which external forces do not act or the action of these forces is compensated) move in them rectilinearly and uniformly or rest in them. Only in these systems Newton's laws are fulfilled.

Non-inertial system reference - an arbitrary frame of reference, which is not inertial. Any frame of reference moving with acceleration relative to inertial is non-inertial.

inertia force, a vector quantity numerically equal to the product of the mass t material point on its acceleration w and directed opposite to the acceleration. At curvilinear movement of S. and. can be decomposed into a tangent or tangential component J t directed opposite to the tangential acceleration w t , and on the normal or centrifugal component J n, directed along the main normal to the trajectory from the center of curvature; numerically J t = nw t , J n =mv2 / r, where v- speed of the point, r - radius of curvature of the trajectory. When studying motion in relation to the inertial frame of reference, S. and. introduced in order to have a formal opportunity to compose the equations of dynamics in the form of simpler equations

2) Impulse. Law of motion in relativistic dynamics. Energy, interrelations of mass and energy. Conservation laws in SRT.

The relativistic law of adding the velocities of a body and the speed of a moving system in one

where u" - the speed of the body in a moving frame of reference; v is the speed of the moving system K"relative to the fixed system K;
u is the speed of the body relative to the fixed frame of reference K(Fig. 1).

Relativistic time dilation Time t 0 , counted by a clock resting relative to a given body, is called own time. It is always less than the time measured by the moving clock: t 0 < t.

Relativistic contraction of length The transverse dimensions of the moving rod do not change. Linear size of the rod l 0 in the reference frame where it rests is called its own length. This length is maximum: l 0 > l.

The momentum of the moving body ( relativistic momentum):

Total energy of a body or system of bodies:

6 ticket
1) The law of conservation of momentum. Center of Mass. The movement of the center of mass.

Law of conservation of momentum- In a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other. This fundamental law of nature is called the law of conservation.
impulse. It is a consequence of Newton's second and third laws.

P is the momentum of the system; F is the resultant of all forces acting on the particles of the system

Center of mass- a geometric point that characterizes the motion of a body or a system of particles as a whole.
Theorem on the motion of the center of mass(center of inertia) of the system - a general problem of dynamics. that the acceleration of the center of mass of a mechanical system does not depend on the internal forces acting on the bodies of the system, and relates this acceleration to the external forces acting on the system. The center of mass moves in the same way as a material point, the mass of which is equal to the mass of the system, would move under the action of a force equal to the sum of all external forces acting on the system. ma=(sum F)

2) Thermodynamic parameters. Ideal and real gases. Equation of state of ideal and real gases.

Thermodynamic quantities name the physical quantities used in the description of states and processes in thermodynamic systems.

1) Temperature- a physical quantity that approximately characterizes the average kinetic energy of particles of a macroscopic system per one degree of freedom, which is in a state of thermodynamic equilibrium.
2) Pressure- this is normal to the surface (perpendicular) force acting per unit area: p \u003d F / A.
3) Volume- a quantitative characteristic of the space occupied by a body or substance. The volume of a body or the capacity of a vessel is determined by its shape and linear dimensions.
4) Entropy is the degree of system disorder. Spontaneously in nature, all processes go in one direction: towards the growth of entropy. St.-va (either grows or does not change; this is a function of the state; the entropy of a system of bodies is the sum of the entropy of the bodies included in the system; internal entropy \u003d free energy + bound energy)

Ideal gas is a gas in which the mutual potential energy of the molecules and the intrinsic volume of the molecules can be neglected.
In real gases, the density is so high that the mutual potential energy cannot be neglected. The intrinsic volume of molecules also plays a role. As an experiment, you can do the following: take a balloon, put an ideal gas in it, compress it very slowly. In this case, the temperature must be constant due to heat exchange with the environment.
The relationship between pressure and volume obeys the Boyle-Mariot law. Pressure is inversely proportional to volume.
If you increase the concentration, then the mutual attraction will increase. Potential energy cannot be neglected
(real gas). There is no inverse relationship between pressure and volume.

Ticket

1) Moment of inertia, moment of force and moment of impulse. Steiner's theorem

moment of inertia system relative to the axis of rotation is called a physical quantity equal to the sum of the product of the masses of n material points of the system by the squares of their distances to the considered axis.
If the moment of inertia of the body about the axis passing through its center of mass is known, the moment of inertia about any other axis parallel to the given one is determined using the Steiner theorem: the moment of inertia of the body I about the parallel axis of rotation is equal to the moment of inertia I c about the parallel axis passing through the center of mass From the body, folded with the product of the mass m of the body and the square of the distance a between the axes

Moment of force relative to a fixed point O is called a pseudovector quantity equal to the vector product of the radius vector drawn from point O to the point of application of the force, by the force

Modulus of moment of force:

The angular momentum of a rigid body relative to a fixed axis Z is a scalar value equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point O of this axis. The value of the angular momentum does not depend on the position of the point O on the Z axis.

The moment of momentum of a rigid body about the axis is the sum of the moments of momentum of individual particles.

Moment of impulse - characterizes the amount of rotational motion. The angular momentum of a material point relative to some reference point is determined by the vector product of its radius vector and momentum:

L=r×p,
where r is the radius-vector of the particle with respect to the selected reference point fixed in the given reference frame, p is the momentum of the particle.

2) Internal energy of ideal and real gases.

Based on the definition of an ideal gas, there is no potential component of internal energy in it (there are no forces of interaction of molecules, except for shock). In this way , the internal energy of an ideal gas represents only the kinetic energy of the movement of its molecules.

Ticket

1) The basic equation of the dynamics of rotational motion. Law of conservation of angular momentum.

2) Degrees of freedom of molecules. The theorem of equipartition of energy over degrees of freedom.

degrees of freedom of molecules- the number of independent coordinates that must be set in order to uniquely determine the position of this object relative to the reference frame under consideration.

a - monoatomic (3), b - diatomic (5), c - triatomic (6).

Average kinetic energy of motion molecules of an ideal gas can be determined by the formula: i is the number of independent quantities determined by the position of the body in space.

Any body in translational motion has three degrees of freedom. Each degree of freedom of a statistical system has the same energy equal to . ΣƩ

This is the essence of the theorem on the equipartition of thermal energy over degrees of freedom.

For monatomic

For diatomic - 2 degrees of freedom. The oscillations of the degrees of freedom occur with a significant increase in temperature, because interatomic bonds are weakened and oscillations inside molecules are intensified.

For the largest increase in temperature

Ticket

1) Work of constant and variable force. The kinetic energy of a body involved in translational and rotational motion.

The work of a constant force. To characterize the effectiveness of the force impact on the body, a quantity called mechanical work is used. Let under the action of a constant force F the particle moved arbitrarily from position 1 to position 2. The work of the force F on the move ∆r called a scalar quantity, defined by the following relationship: The work of a constant force is equal to the scalar product of force and displacement.

The unit of measure for work is Joule. 1 J = 1 Nm.
Variable force work

Variable force work. In the case of movement under the action of a variable force, the amount of work is calculated as follows. The entire trajectory is mentally divided into separate sections of such a small length |d r| that the force acting on them can be considered constant (see Fig. 7.2). The projection of the force on the direction of the elementary displacement vector d r is its tangential component. Therefore, the elementary work on the displacement d r can be calculated using the ratio.

2) The first law of thermodynamics and its application to isoprocesses. adiabatic process

isoprocesses- processes occurring at a constant value of one of the parameters.

Isothermal process (T = const, hence ΔU = 0).
According to the first law of thermodynamics: Q = A".
The gas does work A "due to the input heat Q (A"> 0, Q> 0).
The performance of work by external forces A (gas compression) requires the removal of heat Q from the gas in order to maintain its temperature (A>0, Q<0).

Isochoric process (V = const, hence A = 0).
According to the first law of thermodynamics: ΔU = Q.
Heating a gas in a closed vessel leads to an increase in its internal energy U (temperature) (Q>0, ΔU>0).
Cooling a gas in a closed vessel leads to a decrease in its internal energy U (temperature) (Q<0, ΔU<0).

Isobaric process (p = const).
According to the first law of thermodynamics: Q = ΔU + A".
The heat Q supplied to the gas is partly used to increase the internal energy U, and partly to do work with the gas A" (Q>0, ΔU>0, A">0).
The work of external forces A during isobaric compression of the gas requires the removal of heat Q from the gas, while its internal energy U (Q<0, ΔU<0, A>0).

An adiabatic process is a process that proceeds without heat exchange with the environment (Q = 0).
According to the first law of thermodynamics: ΔU = A.
All the work of external forces A goes only to increase the internal energy of the gas (A>0, ΔU>0).
The work of gas A "is performed only due to the loss of internal energy of the gas (A"> 0, ΔU<0).

Ticket

1) Potential energy. Potential energy of a compressed spring, a body in a gravitational field.

Potential energy- a scalar physical quantity, which is a part of the total mechanical energy of the system in the field of conservative forces. Depends on the position of the material points that make up the system, and characterizes the work done by the field when they move. Another definition: potential energy is a function of coordinates, which is a term in the Lagrangian of the system, and describes the interaction of the elements of the system] . The term "potential energy" was introduced in the 19th century by the Scottish engineer and physicist William Rankine.

The unit of energy in the International System of Units (SI) is the joule.

A correct definition of potential energy can only be given in the field of forces, the work of which depends only on the initial and final position of the body, but not on the trajectory of its movement.

Potential energy of the body in the Earth's gravitational field near the surface is approximately expressed by the formula:

where is the mass of the body, is the acceleration of free fall, is the height of the position of the center of mass of the body above an arbitrarily chosen zero level.

2) The work of gravitational forces, the connection between force and potential energy. Work of gas in isoprocesses.

The space in which conservative forces act is called the potential field. Each point of the potential field corresponds to a certain value of the force F acting on the body, and a certain value of the potential energy U. This means that there must be a connection between the force F and U, on the other hand, dA = -dU, therefore Fdr = -dU, hence:

Projections of the force vector on the coordinate axes:

The force vector can be written in terms of projections: , F = –grad U, where .

AT isochoric process (V= const) the gas does no work, A = 0.

AT isobaric process (p= const) the work done by the gas is expressed by the ratio.

We have established a function that describes the distribution of molecules by velocities (Maxwell distribution) and a dependence that characterizes the distribution of molecules by potential energy values ​​(Boltzmann distribution). Both dependencies can be combined into one generalized distribution.

Consider an infinitesimal volume dV gas located at a point with a radius vector in a large system representing an ideal gas at a constant temperature in external force fields. The number of molecules in the selected volume is n( ) d 3 r. Since the volume is small, within its limits the particle density can be considered constant. This means that the condition for the validity of the Maxwell distribution is satisfied. Then for the number of molecules dN, having speeds from v before v+dv and located in the volume d 3 r, as a result of combining dependencies (3.11) and (3.27), we obtain the following formula:

But the concentration of molecules n(r) depends on the location of this volume in external force fields:

where n 0 - concentration of molecules at the point where E p = 0. Then

Since the expression

is the total energy of the particle in the external potential force field, we arrive at the generalized Maxwell-Boltzmann energy distribution molecules:

where N- the total number of particles in the system, a dN - number of particles with coordinates between r and r + dr and (simultaneously) with speeds between v and v+dv.

Average energy of a quantum oscillator. The Maxwell-Boltzmann distribution was obtained in classical physics, but it turned out to be valid in quantum mechanics, where many seemingly unshakable provisions were revised. As an example, consider the problem of a load with a mass t, fixed at the end of a spring with stiffness k. The equation of motion is well known, and its solution is the harmonic vibrations of the body with a circular frequency

The classical energy of a system simulating vibrations of atoms in a molecule is given by formula (3.62) and can take any value depending on the vibration amplitude. As we know from quantum mechanics, the vibrational energy quantized, that is, it takes a discrete series of values ​​determined by the formula:

In accordance with the general principles of statistical physics, the probability P n find an oscillator in a state characterized by a certain value n vibrational quantum number, is determined by the formula

where BUT - normalization constant. To determine it, it is necessary to use the probability normalization condition

To do this, in the well-known formula for a geometric progression

substitute the value

We get then instead of (2)

whence follows the expression for the constant BUT. Using it in expression (1), we arrive at the probability

It can be seen that the larger the value of the quantum number n, the less likely it is to find an oscillator in this state. The higher the temperature, the higher the values n become practically significant for the system. At

all probabilities go to zero P n With n > 1, and only

In other words, at zero temperature there are no thermal excitations, and the oscillator performs "zero oscillations" - it is in mostly state of lowest energy

The energy distribution of oscillators depending on the temperature of the system is shown in Fig. . 3.9

Rice. 3.9. Approximate distribution of N = 30 quantum oscillators over energy levels depending on temperature. Only the ground and the first five excited energy levels are shown. At T = 0, all oscillators are in the ground state. As the temperature rises, higher and higher energies become available, and the distribution of oscillators over levels becomes more and more uniform.

For clarity, we took the system from a small ( N=30) the number of oscillators (strictly speaking, statistical laws apply to systems with a much larger number of particles).

The question arises: what average meaning vibrational quantum number n at some temperature T? To answer, we must calculate the sum:

To do this, we differentiate with respect to q both parts of equality (3.67) for a geometric progression:

where we get

Using (7) for

we obtain from (6) an expression for the desired average

Now it is easy to get the average energy of the oscillator

where is the function cth - hyperbolic cotangent defined by the relation

On fig. 3.10 the solid line shows the average energy of the quantum oscillator, measured in units ħω ,

depending on "dimensionless temperature"

Rice. 3.10. Average energy of a quantum oscillator as a function of temperature

Dotted line

corresponds to the result of classical physics. Indeed, the energy

per degree of freedom is the average of both the kinetic and potential energies of a classical oscillator, so that the average of the total energy is just

It can be seen that quantum corrections are important at low temperatures: at q< 0,3 the average energy of the oscillator is close to the energy of the ground state ħω/2. In this case, we say that the vibrational degrees of freedom are "frozen", that is, there is not enough thermal energy to excite vibrations. But already at q = 2 both energies practically coincide, that is, the quantum corrections are small. Meaning q = 1 can be taken as a conditional boundary between the quantum and classical fields. Its meaning is clear:

thermal energy is equal to the minimum excitation energy of the oscillator, that is, the difference between the energy

first excited state and energy

ground state of the oscillator.

What temperatures can be considered low for an oscillator simulating a real system, for example, a hydrogen molecule H 2? The characteristic frequencies of molecular vibrations are usually located in the infrared region and are of the order n = 10 14 Hz. This corresponds to the energy

and temperature

Average energy of a quantum rotator. Thus, the room temperatures familiar to us turn out to be sufficiently low from the point of view of excitation of molecular vibrations. Let's see what happens to molecules at temperatures T< Т К0Л. Since there are no vibrations, a diatomic molecule can be represented as a "dumbbell" - two atoms rigidly connected to each other. Such a system is called rotator and, as we saw earlier, it has five degrees of freedom - three translational (movement of the center of mass) and two rotational. The energy of the rotational motion of a classical rotator has the form (3.61). Given the connection

between angular frequency of rotation ω , moment of inertia I and angular momentum L we write the classical rotational energy of the molecule as

In quantum mechanics, the square of angular momentum is quantized,

Here J- rotational quantum number, therefore, the energy of the rotational motion of the molecule is also quantized

Using this ratio and the Maxwell-Boltzmann distribution, one can obtain an expression for the average energy of a quantum rotator. However, in this case, the formulas are rather complicated, and we restrict ourselves to qualitative results. At high temperatures, the average energy tends to the classical value k B T, corresponding to two degrees of freedom (rotation around two orthogonal axes). At low temperatures, the rotator will be in the ground state corresponding to the value J= 0 (no rotation). The "transition" between these two limiting cases is obviously carried out at such a temperature T BP when thermal motion can excite rotational degrees of freedom. The minimum (non-zero) rotational energy is

as follows from the formula for E BP at j = 1. That's why

For the moment of inertia of the molecule, we can take the estimate

where m p = 1.67· 10 -27 kg(proton mass), and a B =5 10 -11 m is the Bohr radius. We get then

The estimates obtained are confirmed by measurements of the molar heat capacity at constant volume with nV , which we have already discussed in the previous chapter. At temperatures below 100 K only the translational degrees of freedom of the molecule are involved in thermal motion. The average energy of a molecule is 3kW/2, and the energy of one mole - 3N A k B T/2=3RT/2, whence follows the expression for the heat capacity with nV = 3R/2. In the temperature range from 100 K before 200 K molar heat capacity increases to the value with nV = 5R/2, which indicates the "unfreezing" of two additional (rotational) degrees of freedom (that is, the addition k B T energy per molecule). In the temperature range from 4 000 K before 5 000 K the molar heat capacity increases again, this time to a value with nV = 7R/2. This "unfrozen" vibrational degree of freedom, which brought additional energy k B T on the molecule.

The rate of chemical reactions. Chemists have a rule of thumb that when the temperature rises by 10 °С the reaction rate doubles. This is just a rough generalization, there are many exceptions to it, but on the whole it is more or less true. An explanation can also be given here on the basis of the Maxwell-Boltzmann distribution.

For many chemical reactions to occur, it is necessary that the energy of the particles involved in them exceed a certain threshold value, which we will denote E 0 . T 2 \u003d 310 K this ratio is E 0 /k B T 2 = 14.0. The number of particles participating in the reaction is determined by the relations

Indeed, an increase in temperature of only 10 degrees led to an increase in 60 % the number of particles whose energy exceeds the threshold value.

For an ideal gas, the Hamilton function can be simply replaced by energy, and then, according to formula (6.2), the probability of finding a system with energy in an element of the phase space will be:

For a system of non-interacting particles, the energy can be represented as the sum of the energies of individual particles. Then the probability (6.28) can be divided into factors

Integrating the variable of all particles, except for the 1st one, we obtain the probability expression for the particle:

Here considered as a function of 6 variables Distribution (6.30) can be

considered in -dimensional phase space of one molecule, which is called -space (from the word molecule).

The energy of an individual particle can be represented by the sum of the kinetic and potential energies, depending on the momentum and coordinates of the particle, respectively:

Substituting this expression into (6.30), we get:

This is the Maxwell-Boltzmann distribution.

The fact that the kinetic and potential energies depend on different variables makes it possible to consider one distribution (6.32) as two independent distributions in the three-dimensional momentum space and in the three-dimensional space of coordinates:

Here are the constants determined from the distribution normalization condition.

The momentum distribution (6.33) coincides with the Maxwellian distribution (3.22) for an ideal gas. But it should be noted that the momentum distribution obtained here does not depend on the nature of the interaction of the particles of the system, since the interaction energy can always be introduced into the potential energy of the particle. In other words, the Maxwellian velocity distribution is suitable for particles of any classical systems: gases, liquids, and solids.

If we consider the molecules or atoms that make up the molecules as the smallest particles, then the Maxwellian distribution is also valid for them. However, already for electrons in an atom or in a metal, or for other quantum

systems, the Maxwellian distribution will not be valid, since it is a consequence of classical statistics.

The distribution function over particle coordinates (6.34) in a potential field represents the so-called Boltzmann distribution (1877).

For the case when the potential energy depends only on one variable, for example, one can integrate (6.34) over two other variables and obtain (taking into account the normalization) the expression:

For an ideal gas in a uniform gravity field, the well-known barometric formula is derived from (6.35). Indeed, in this case, the height distribution function of particles also takes the form:

Due to the proportionality of the number of particles to the distribution function (6.36), we obtain the following distribution of the number of particles per unit volume in height (Fig. 30):

Since there will be particles in a unit volume, then for the distribution of particles in height we get:

If we take into account that the pressure in a gas is proportional to the density, then from (6.37) we get the barometric formula

Rice. 30. Change in the number of particles per unit volume with a change in height according to the Boltzmann distribution

Experimental studies have shown that at high altitudes in the atmosphere deviations of the number of particles from the distribution described by formula (6.37) are observed, associated with the inhomogeneous composition of the atmosphere, with the difference in temperatures at different altitudes and with the fact that the atmosphere is not in a state of equilibrium.

In the atmospheres of planets, the phenomenon of scattering of the atmosphere into outer space occurs. It is explained by the fact that any particle that has a speed greater than the second space velocity for a given planet can leave the atmosphere of the planet. In a gas, as follows from the Mackwellian distribution, there is always a certain fraction of molecules with very high velocities, the departure of which determines the gradual scattering of the upper layers of the atmosphere. The scattering of the atmosphere of planets occurs the faster, the smaller the mass of the planet and the higher its temperature. For the Earth, this effect turns out to be negligible, and the planet Mercury and the Moon have already lost their atmospheres in this way.