The number of degrees of freedom of molecules. The law of uniform distribution of energy over degrees of freedom The number of degrees of freedom of a gas molecule table

PHYSICAL FOUNDATIONS OF THERMODYNAMICS

1. First law of thermodynamics

§1. Internal energy

Any thermodynamic system in any state has an energy called total energy. The total energy of the system is the sum of the kinetic energy of the motion of the system as a whole, the potential energy of the system as a whole, and internal energy.

The internal energy of the system is the sum of all types of chaotic (thermal) motion of molecules: potential energy from intra-atomic and intra-nuclear motions. The internal energy is a function of the state of the gas. For a given state of the gas, the internal energy is uniquely determined, that is, it is a definite function.

During the transition from one state to another, the internal energy of the system changes. But at the same time, the internal energy in the new state does not depend on the process by which the system passed into this state.

§2. Warmth and work

There are two different ways of changing the internal energy of a thermodynamic system. The internal energy of a system can change as a result of doing work and as a result of transferring heat to the system. Work is a measure of the change in the mechanical energy of a system. When performing work, there is a movement of the system or individual macroscopic parts relative to each other. For example, by moving a piston into a cylinder containing gas, we compress the gas, as a result of which its temperature rises, i.e. the internal energy of the gas changes.

Internal energy can also change as a result of heat transfer, i.e. imparting some heat to the gasQ.

The difference between heat and work is that heat is transferred as a result of a number of microscopic processes in which the kinetic energy of the molecules of a hotter body during collisions is transferred to the molecules of a less heated body.

What is common between heat and work is that they are functions of the process, that is, we can talk about the amount of heat and work when the system transitions from the first state to the second state. Heat and the robot is not a state function, unlike internal energy. It is impossible to say what the work and heat of the gas in state 1 is equal to, but one can talk about the internal energy in state 1.

§3Ibeginning of thermodynamics

Let us assume that some system (a gas contained in a cylinder under a piston), having internal energy, has received a certain amount of heatQ, passing into a new state, characterized by internal energyU 2 , did the job A over the external environment, i.e. against external forces. The amount of heat is considered positive when it is supplied to the system, and negative when it is taken from the system. Work is positive when it is done by the gas against external forces, and negative when it is done on the gas.

Ibeginning of thermodynamics : Amount of heat (Δ Q ), the communicated system goes to increase the internal energy of the system and to perform work (A) by the system against external forces.

Recording Ithe beginning of thermodynamics in differential form

dU- an infinitesimal change in the internal energy of the system

elementary work,- an infinitesimal amount of heat.

If the system periodically returns to its original state, then the change in its internal energy is zero. Then

i.e. perpetual motion machineIkind, a periodically operating engine that would do more work than the energy communicated to it from the outside is impossible (one of their formulationsIthe beginning of thermodynamics).

§2 Number of degrees of freedom of a molecule. uniform law

distribution of energy over the degrees of freedom of the molecule

Number of degrees of freedom: a mechanical system is called the number of independent quantities, with the help of which the position of the system can be set. A monatomic gas has three translational degrees of freedomi = 3, since three coordinates (x, y, z ).

Hard connectionA bond is called a bond in which the distance between atoms does not change. Diatomic molecules with a rigid bond (N 2 , O 2 , H 2) have 3 translational degrees of freedom and 2 rotational degrees of freedom:i= ifast + ivr=3 + 2=5.

Translational degrees of freedom associated with the movement of the molecule as a whole in space, rotational - with the rotation of the molecule as a whole. Rotation of relative coordinate axesx And z on the corner will lead to a change in the position of molecules in space, during rotation about the axis at the molecule does not change its position, therefore, the coordinate φ ynot needed in this case. A triatomic molecule with a rigid bond has 6 degrees of freedom.

i= ifast + ivr=3 + 3=6

If the bond between the atoms is not rigid, then vibrational With degrees of freedom. For a nonlinear moleculei count . = 3 N - 6 , Where Nis the number of atoms in a molecule.

Regardless of the total number of degrees of freedom of the molecules, the 3 degrees of freedom are always translational. None of the translational powers has an advantage over the others, so each of them has the same energy on average, equal to 1/3 of the value

Boltzmann established the law according to which for a statistical system (i.e., for a system in which the number of molecules is large), which is in a state of thermodynamic equilibrium, for each translational and rotational degree of freedom, there is an average kinematic energy equal to 1/2 kT , and for each vibrational degree of freedom - on average, the energy equal to kT . The vibrational degree of freedom "possesses" twice as much energy because it accounts not only for kinetic energy (as in the case of translational and rotational motion), but also for potential energy, andthus the average energy of a molecule

The digital resource can be used for teaching within the framework of the secondary school program (profile and advanced levels).

The computer model illustrates the features of the movement of molecules. One-atomic, diatomic and triatomic molecules are considered, the concept of "degrees of freedom" is introduced.

Brief theory

Working with the model

The model can be used in manual frame switching mode and in automatic demonstration mode (Movie).

This model can be used as an illustration in the lessons of studying new material, repetition in grade 10 on the topic "Basic Equation of Molecular Kinetic Theory".

The concept of "degree of freedom" is quite difficult for the perception of high school students. The model allows us to demonstrate the nature of the movement of various molecules.

Lesson planning example using the model

Topic "Basic Equation of Molecular Kinetic Theory"

The purpose of the lesson: to derive and analyze the basic equation of the MKT.

Examples of questions and tasks

Until now, we have used the concept of molecules as very small elastic balls, the average kinetic energy of which was assumed to be equal to the average kinetic energy of translational motion (see formula 6.7). This idea of ​​a molecule is valid only for monatomic gases. In the case of polyatomic gases, the contribution to the kinetic energy is also made by the rotational, and at high temperature, by the vibrational motion of molecules.

In order to estimate what fraction of the energy of a molecule falls on each of these motions, we introduce the concept degrees of freedom. The number of degrees of freedom of a body (in this case, a molecule) is understood as number of independent coordinates, which completely determine the position of the body in space. The number of degrees of freedom of the molecule will be denoted by the letter i.

If the molecule is monoatomic (inert gases He, Ne, Ar, etc.), then the molecule can be considered as a material point. Since the position of the material is determined by three coordinates x, y, z (Fig. 6.2, a), then a monatomic molecule has three degrees of freedom of translational motion (i = 3).

A diatomic gas molecule (H 2, N 2, O 2) can be represented as a set of two rigidly connected material points - atoms (Fig. 6.2, b). To determine the position of a diatomic molecule, linear coordinates x, y, z are not enough, since the molecule can rotate around the center of coordinates. It is obvious that such a molecule has five degrees of freedom (i=5): - three - translational motion and two - rotation around the coordinate axes (only two of the three angles  1 ,  2 ,  3 are independent).

If a molecule consists of three or more atoms that do not lie on one straight line (CO 2, NH 3), then it (Fig. 6.2, c) has six degrees of freedom (i = 6): three - translational motion and three - rotation around the coordinate axes.

It was shown above (see formula 6.7) that the average kinetic energy translational motion of an ideal gas molecule, taken as materialpoint, is equal to 3/2kT. Then, for one degree of freedom of translational motion, there is an energy equal to 1/2kT. This conclusion in statistical physics is generalized in the form of Boltzmann's law on the uniform distribution of the energy of molecules over degrees of freedom: statistically, on average, for any degree of freedom of molecules, there is the same energy, ε i , equal to:

Thus, the total average kinetic energy of the molecule

(6.12)

In reality, molecules can also perform oscillatory motions, and the energy of the vibrational degree of freedom is, on average, twice as large as that of the translational or rotational, i.e. kT. In addition, considering the model of an ideal gas, by definition, we did not take into account the potential energy of interaction of molecules.

Mean number of collisions and mean free path of molecules

The process of collision of molecules is conveniently characterized by the value of the effective diameter of molecules d, which is understood as the minimum distance at which the centers of two molecules can approach each other.

The average distance traveled by a molecule between two successive collisions is called mean free path molecules .

Due to the randomness of the thermal motion, the trajectory of the molecule is a broken line, the break points of which correspond to the points of its collision with other molecules (Fig. 6.3). In one second, a molecule travels a path equal to the arithmetic mean speed . If is the average number of collisions in 1 second, then the mean free path of a molecule between two successive collisions

=/(6.13)

For determining Let us represent the molecule as a ball with a diameter d (other molecules will be assumed to be immobile). The length of the path traveled by the molecule in 1 s will be equal to . A molecule on this path will collide only with those molecules whose centers lie inside a broken cylinder with radius d (Fig. 6.3). These are molecules A, B, C.

The average number of collisions in 1 s will be equal to the number of molecules in this cylinder:

=n 0 V,

where n 0 is the concentration of molecules;

V is the volume of the cylinder, equal to:

V = πd 2

So the average number of collisions

= n 0 π d2

When taking into account the motion of other molecules, more accurately

=
πd 2 n 0 (6.14)

Then the mean free path according to (6.13) is equal to:

(6.15)

Thus, the mean free path depends only on the effective molecular diameter d and their concentration n 0 . For example, let's evaluate And . Let d ~ 10 -10 m, ~ 500 m / s, n 0 \u003d 3 10 25 m -3, then 3 10 9 s –1 and 7 10 - 8 m at a pressure of ~10 5 Pa. With decreasing pressure (see formula 6.8) increases and reaches a value of several tens of meters.

Basic concepts of thermodynamics.

Unlike MKT, thermodynamics studies the macroscopic properties of bodies and natural phenomena without being interested in their microscopic picture. Without introducing atoms and molecules into consideration, without entering into a microscopic consideration of processes, thermodynamics makes it possible to draw a number of conclusions regarding their course.

Thermodynamics is based on several fundamental laws (called the principles of thermodynamics), established on the basis of a generalization of a large set of experimental facts.

Approaching the consideration of changes in the state of matter from different points of view, thermodynamics and MKT mutually complement each other, forming essentially one whole.

Thermodynamics- a branch of physics that studies the general properties of macroscopic systems in a state of thermodynamic equilibrium and the processes of transition between these states.

Thermodynamic method is based on the introduction of the concept of energy and considers processes from an energy point of view, that is, based on the law of conservation of energy and its transformation from one form to another.

Thermodynamic system- a set of bodies that can exchange energy with each other and with the environment.

To describe a thermodynamic system, physical quantities are introduced, which are called thermodynamic parameters or system state parameters: p, V, T.

Physical quantities characterizing the state of a thermodynamic system are called thermodynamic parameters.

By pressure called a physical quantity numerically equal to the force acting per unit area of ​​the surface of the body in the direction of the normal to this surface:, .

Normal atmospheric pressure 1atm=10 5 Pa.

Absolute temperature is a measure of the average kinetic energy of molecules.

.

The states in which the thermodynamic system is located can be different.

If one of the parameters at different points of the system is not the same and changes over time, then this state of the system is called nonequilibrium.

If all thermodynamic parameters remain constant at all points of the system for an arbitrarily long time, then such a state is called equilibrium, or a state of thermodynamic equilibrium.

Any closed system after a certain time spontaneously passes into an equilibrium state.

Any change in the state of the system associated with a change in at least one of its parameters is called thermodynamic process. A process in which each subsequent state differs infinitely little from the previous one, i.e. is a sequence of equilibrium states, is called equilibrium.

Obviously, all equilibrium processes proceed infinitely slowly.

The equilibrium process can be carried out in the opposite direction, and the system will pass through the same states as in the forward course, but in reverse order. Therefore, equilibrium processes are called reversible.

The process by which a system returns to its original state after a series of changes is called circular process or cycle.

All quantitative conclusions of thermodynamics are strictly applicable only to equilibrium states and reversible processes.

The number of degrees of freedom of a molecule. The law of uniform distribution of energy over degrees of freedom.

Number of degrees of freedom is the number of independent coordinates that completely determine the position of the system in space. A monatomic gas molecule can be considered as a material point with three degrees of freedom of translational motion.

A diatomic gas molecule is a set of two material points (atoms) rigidly connected by a non-deformable bond; in addition to three degrees of freedom of translational motion, it has two more degrees of freedom of rotational motion (Fig. 1).

Three- and polyatomic molecules have 3+3=6 degrees of freedom (Fig. 1).

Naturally, there is no rigid bond between atoms. Therefore, for real molecules, one should also take into account the degrees of freedom of vibrational motion (except for monatomic ones).

Let us write the expression for pressure and the equation of state for an ideal gas side by side:

;

,

average kinetic energy of the translational motion of molecules:

.

Conclusion: the absolute temperature is a quantity proportional to the mean energy progressive molecular movements.

This expression is remarkable in that the average energy turns out to depend only on temperature and does not depend on the mass of the molecule.

However, along with progressive rotation of the molecule and vibrations of the atoms that make up the molecule are also possible by motion. Both of these types of movement rotation and oscillation) are associated with a certain energy reserve, which can be determined position on the equipartition of energy over the degrees of freedom of a molecule.

The number of degrees of freedom of a mechanical system is the number of independent quantities that can be used to set the position of the system.

For example: 1. A material point has 3 degrees of freedom, since its position in space is completely determined by setting the values ​​of its three coordinates.

2. An absolutely rigid body has 6 degrees of freedom, since its position can be determined by setting the coordinates of its center of mass ( x, y, z) and angles ,  and . The measurement of the coordinates of the center of mass at constant angles ,  and  is determined by the translational motion of a rigid body, therefore, the corresponding degrees of freedom are called translational. The degrees of freedom associated with the rotation of a rigid body are called rotational.

3. System of N material points has 3 N degrees of freedom. Any rigid connection that establishes an invariable mutual arrangement of two points reduces the number of degrees of freedom by one. So, if there are two points, then the number of degrees of freedom is 5: 3 translational and 2 rotational (around the axes

).

If the connection is not rigid, but elastic, then the number of degrees of freedom is 6 - three translational, two rotational and one vibrational degree of freedom.

From experiments on measuring the heat capacity of gases, it follows that when determining the number of degrees of freedom of a molecule, atoms should be considered as material points. A monatomic molecule is assigned 3 translational degrees of freedom; diatomic molecule with a rigid bond - 3 translational and 2 rotational degrees of freedom; a diatomic molecule with an elastic bond - 3 translational, 2 rotational and 1 vibrational degrees of freedom; a triatomic molecule is assigned 3 translational and 3 rotational degrees of freedom.

Boltzmann's law on the equipartition of energy over degrees of freedom: no matter how many degrees of freedom a molecule has, three of them are translational. Since none of the translational degrees of freedom has advantages over the others, any of them should have on average the same energy equal to 1/3 of the value
, i.e. .

So, the distribution law: for each degree of freedom, there is on average the same kinetic energy equal to (translational and rotational), and the vibrational degree of freedom - the energy equal to KT. According to the equipartition law, the average value of the energy of one molecule
the more complex the molecule, the more degrees of freedom it has.

The vibrational degree of freedom must have twice the energy capacity than the translational or rotational degree of freedom, because it accounts for not only kinetic, but also potential energy (the average value of potential and kinetic energy for a harmonic oscillator turns out to be the same); thus, the average energy of a molecule must be equal to
, Where.

Table 11.1