The number of degrees of freedom of a molecule. isoprocesses
The equation of state of a thermodynamic system. Clapeyron-Mendeleev equation. Ideal gas thermometer. Basic equation of molecular-kinetic theory. Uniform distribution of energy over the degrees of freedom of molecules. Internal energy of an ideal gas. Effective diameter and mean free path of gas molecules. Experimental confirmation of the molecular-kinetic theory.
The equation of state of a thermodynamic system describes the relationship between the parameters of the system . The state parameters are pressure, volume, temperature, amount of substance. In general, the equation of state is a functional dependence F (p, V, T) = 0.
For most gases, as experience shows, at room temperature and a pressure of about 10 5 Pa, the Mendeleev-Clapeyron equation :
p– pressure (Pa), V- occupied volume (m 3), R\u003d 8.31 J / molK - universal gas constant, T - temperature (K).
mole of substance
- the amount of a substance containing the number of atoms or molecules equal to Avogadro's number 
(so many atoms are contained in 12 g of the carbon isotope 12 C). Let m 0 is the mass of one molecule (atom), N is the number of molecules, then 
- mass of gas, 
is the molar mass of the substance. Therefore, the number of moles of a substance is:
.
A gas whose parameters satisfy the Clapeyron-Mendeleev equation is an ideal gas. Hydrogen and helium are closest in properties to the ideal.
Ideal gas thermometer.
A gas thermometer of constant volume consists of a thermometric body - a portion of an ideal gas enclosed in a vessel, which is connected to a pressure gauge by means of a tube.
With the help of a gas thermometer, it is possible to experimentally establish a relationship between the temperature of a gas and the pressure of a gas at a certain fixed volume. The constancy of the volume is achieved by the fact that by vertical movement of the left tube of the pressure gauge, the level in its right tube is brought to the reference mark and the difference in the heights of the liquid levels in the pressure gauge is measured. Taking into account various corrections (for example, thermal expansion of the glass parts of a thermometer, gas adsorption, etc.) makes it possible to achieve an accuracy of temperature measurement with a constant volume gas thermometer equal to 0.001 K.
Gas thermometers have the advantage that the temperature determined with their help at low densities gas does not depend on its nature, and the scale of such a thermometer coincides well with the absolute temperature scale determined using an ideal gas thermometer.
In this way, a certain temperature is related to the temperature in degrees Celsius by the relation: 
TO.
Normal gas conditions - a state in which the pressure is equal to normal atmospheric: R\u003d 101325 Pa10 5 Pa and temperature T \u003d 273.15 K.
From the Mendeleev-Clapeyron equation it follows that the volume of 1 mole of gas under normal conditions is equal to: 
m 3.
Fundamentals of ICT
The molecular kinetic theory (MKT) considers the thermodynamic properties of gases from the point of view of their molecular structure.
Molecules are in constant random thermal motion, constantly colliding with each other. In doing so, they exchange momentum and energy.
Gas pressure.
Consider a mechanical model of a gas in thermodynamic equilibrium with the vessel walls. Molecules elastically collide not only with each other, but also with the walls of the vessel in which the gas is located.
As an idealization of the model, we replace atoms in molecules with material points. The velocity of all molecules is assumed to be the same. We also assume that the material points do not interact with each other at a distance, so the potential energy of such an interaction is assumed to be zero.
P 
mouth
is the concentration of gas molecules, T is the gas temperature, u is the average speed of the translational motion of molecules. We choose a coordinate system so that the vessel wall lies in the XY plane, and the Z axis is directed perpendicular to the wall inside the vessel.
Consider the impact of molecules on the walls of a vessel. Because Since the impacts are elastic, after hitting the wall, the momentum of the molecule changes direction, but its magnitude does not change.
For a period of time t only those molecules that are at a distance from the wall at a distance of not more than L= ut. The total number of molecules in a cylinder with a base area S and height L, whose volume is V = LS = utS, equals N = nV = nutS.
At a given point in space, three different directions of molecular motion can be conventionally distinguished, for example, along the X, Y, Z axes. A molecule can move along each of the “forward” and “backward” directions.
Therefore, not all molecules in the selected volume will move towards the wall, but only a sixth of their total number. Therefore, the number of molecules that during the time t hit the wall, it will be equal to:
N 1 = N/6= nutS/6.
The change in the momentum of the molecules upon impact is equal to the impulses of the force acting on the molecules from the side of the wall - with the same force, the molecules act on the wall:
P Z = P 2 Z – P 1 Z = Ft, or
N 1 m 0 u-( N 1 m 0 u)= Ft,
2N 1 m 0 u=Ft,
,
.
Where do we find the gas pressure on the wall: 
,
Where 
- kinetic energy of a material point (translational motion of a molecule). Therefore, the pressure of such a (mechanical) gas is proportional to the kinetic energy of the translational motion of the molecules:
.
This equation is called the basic equation of the MKT .
The law of uniform distribution of energy over degrees of freedom .
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).
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As shown, the average kinetic energy of the translational motion of a molecule is
Number of degrees of freedom called the smallest number of independent coordinates that must be entered to determine the position of the body in space. is the number of degrees of freedom.
Consider monatomic gas. The molecule of such a gas can be considered a material point, the position of the material point 
(Fig. 11.1) in space is determined by three coordinates.
A molecule can move in three directions (Fig. 11.2).
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Therefore, it has three translational degrees of freedom.
A molecule is a material point.
Energy of rotational motion 
, because the moment of inertia of a material point about the axis passing through the point is equal to zero 
For a monatomic gas molecule, the number of degrees of freedom 
.
Consider diatomic gas. In a diatomic molecule, each atom is taken as a material point and it is believed that the atoms are rigidly connected to each other, this is a dumbbell model of a diatomic molecule. Diatomic rigidly bound molecule(a set of two material points connected by a non-deformable bond), fig. 11.3.
The position of the center of mass of the molecule is given by three coordinates, (Fig. 11.4) these are three degrees of freedom, they determine translational movement of the molecule. But the molecule can also perform rotational movements around the axes 
And 
, these are two more degrees of freedom that determine rotation of the molecule. Rotation of a molecule around an axis 
impossible, because material points cannot rotate around an axis passing through these points.
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For a diatomic gas molecule, the number of degrees of freedom 
.
Consider triatomic gas. The model of a molecule is three atoms (material points) rigidly connected to each other (Fig. 11.5).
A triatomic molecule is a rigidly bound molecule.
For a triatomic gas molecule, the number of degrees of freedom 
.
For a polyatomic molecule, the number of degrees of freedom 
.
For real molecules that do not have rigid bonds between atoms, it is also necessary to take into account the degrees of freedom of vibrational motion, then the number of degrees of freedom of a real molecule is
i= i act + i rotate + i fluctuations (11.1)
The law of uniform distribution of energy over degrees of freedom (Boltzmann's law)
The law on the equipartition of energy over degrees of freedom states that if a system of particles is in a state of thermodynamic equilibrium, then the average kinetic energy of the chaotic movement of molecules per 1 degree of freedom translational and rotational movement is equal to 
Therefore, a molecule that has 
degrees of freedom, has energy
, (11.2)
Where 
is the Boltzmann constant; 
is the absolute temperature of the gas.
Internal energy 
ideal gas is the sum of the kinetic energies of all its molecules.
Finding internal energy 
one mole of an ideal gas. 
, Where 
is the average kinetic energy of one gas molecule, 
is the Avogadro number (the number of molecules in one mole). Boltzmann constant 
. Then
If the gas has mass 
, That 
is the number of moles, where 
is the mole mass, and the internal energy of the gas is expressed by the formula
. (11.3)
The internal energy of an ideal gas depends only on the temperature of the gas. The change in the internal energy of an ideal gas is determined by a change in temperature and does not depend on the process in which this change occurred.
Change in the internal energy of an ideal gas
, (11.4)
Where 
- temperature change.
The law of uniform distribution of energy applies to the oscillatory motion of atoms in a molecule. The vibrational degree of freedom accounts for not only kinetic energy, but also potential energy, and the average value of the kinetic energy per one degree of freedom is equal to the average value of the potential energy per one degree of freedom and is equal to 
Therefore, if a molecule has the number of degrees of freedom i= i act + i rotate + i vibrations, then the average total energy of the molecule: 
, and the internal energy of the mass gas 
:
. (11.5)
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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, i.e., we can talk about the amount of heat and work when the system transitions from the state of the first to the state of the second. 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 molecules of an ideal gas do not interact with each other and therefore have no potential energy. Therefore, the entire energy of ideal gas molecules consists only of the kinetic energy of translational and rotational motions. We determined the average kinetic energy of the translational motion of a molecule in the previous paragraph [formula (17)]. To take into account the average kinetic energy of the rotational motion of a molecule, it is necessary to introduce the concept of the number of degrees of freedom of a body into consideration.
The number of degrees of freedom of a body is the number of independent coordinates that determine the position of bodies in space.
Let us explain this definition. If the body moves in space completely arbitrarily, then this movement can always be composed of six simultaneous independent movements: three translational (along three axes of a rectangular coordinate system) and three rotational (around three mutually perpendicular axes passing through the center of gravity of the body) (Fig. 75 ). In other words, the position of the body in space is determined in this case by six independent coordinates: three linear and three angular. Therefore, according to the definition, the number of degrees of freedom of a body arbitrarily moving in space is six (three translational and three rotational degrees of freedom). If the body's freedom of movement is limited, then its number of degrees of freedom is less than six. For example, a body moves only along a plane, while having the possibility of arbitrary rotation (a rolling ball). Then the number of its degrees of freedom is five (two translational and three rotational). The railroad car has one degree of freedom (translational) since it only moves along the line. The wagon wheel has two degrees of freedom: one translational (together with the hagon) and one rotational (around the horizontal axis).
Let us now return to the question of the kinetic energy of a gas molecule. In view of the complete randomness of the movement of molecules, all types of their movements (both translational and rotational) are equally possible (equiprobable). Therefore, for each degree of freedom of a molecule, there is on average the same amount of energy (Boltzmann's theorem on the uniform distribution of energy over degrees of freedom).
Since the molecules move completely randomly, they would have to have six degrees of freedom. However, the following circumstance must be taken into account here.
A monatomic gas molecule (for example, He) can be represented as a material point whose rotation around its own axes does not change its position in space. This means that to determine the position of a monatomic molecule, it is sufficient to specify only its linear coordinates. Therefore, a monatomic molecule should be assigned a number of degrees of freedom equal to three (translational). From a physical point of view, this circumstance can be explained as follows. The kinetic energy of the rotational motion of the body (see § 23) is equal to
where is the angular velocity of rotation, I is the moment of inertia of the body. For material point
where is the mass of a material point, its distance from the axis of rotation. If a material point rotates around its axis, then But then and Consequently, a monatomic molecule has an infinitely small energy for rotational motion (per rotational degrees of freedom), which can be neglected. A rigorous proof of this proposition is possible only on the basis of quantum mechanics.
A diatomic gas molecule (for example,) can be represented as a set of two material points - atoms, rigidly interconnected by chemical bonds (Fig. 76, a). The rotation of such a molecule around an axis passing through both atoms does not change the position of the molecule in space. From a physical point of view, the energy pertaining to the rotation of a molecule around an axis passing through the atoms is close to zero. Therefore, a diatomic molecule should be assigned five degrees of freedom (three translational and two rotational).
As for the triatomic molecule (Fig. 76, b), it obviously has all six degrees of freedom (three translational and three rotational). Other polyatomic molecules (four-atom, five-atom, etc.) have the same number of degrees of freedom.
To calculate the average kinetic energy per one degree of freedom of a molecule, we use formula (17):
Since this energy was obtained for a monatomic molecule (as a material point) that has three degrees of freedom, then one degree of freedom of the molecule accounts for the energy
Then, according to the mentioned Boltzmann theorem, a molecule having degrees of freedom will have a total kinetic energy
Therefore, the total kinetic energy of a gas molecule is proportional to its absolute temperature and depends only on it.
From formula (19) follows the physical meaning of the absolute zero of temperature: at will, i.e., at absolute zero, the movement of gas molecules stops.
According to formula (19), a monatomic molecule has a total energy
a diatomic molecule has a total energy
triatomic and polyatomic molecules have a total energy
Then the internal energy of a certain mass of gas is equal to the product of the number of molecules contained in this mass and the total kinetic energy of one molecule:
Since for a mole of gas, then for the internal energy of a mole we obtain (considering that