First law of thermodynamics. The number of degrees of freedom of a molecule. The law of uniform distribution of energy over the degrees of freedom of a molecule. Heat capacity. The number of degrees of freedom of the molecule The number of degrees of freedom of neon
Let us now turn to a detailed consideration of the concept of the internal energy of an ideal gas and the relationship of this energy with the number of degrees of freedom of molecules. Previously, in the ideal gas model, we took into account only the energy of the translational motion of molecules. This approach describes well a monatomic gas. According to classical mechanics, the number degrees of freedom of a monatomic molecule is equal to the number of coordinates needed to set their position in space. In our three-dimensional space, the number of coordinates and the number of degrees of freedom of a monatomic gas is three. In accordance with (9.6), the average kinetic energy of the translational motion of molecules, determined through the average square of the velocity v? b, proportional to gas temperature
At the same time, from the isotropy of space (equality of all directions), the mean squares of the velocity components are equal to v * KB = Vy KB == Vz KB , which makes it possible to compare each of the coordinates and each degree of freedom with a third of the average kinetic energy of the translational motion of molecules. Thus, we can assume that for each degree of freedom there is an average energy
If the gas molecule is not monatomic, but consists of N atoms, then to set their position in space it is necessary 3N coordinates. Thus, a molecule N atoms has 3N degrees of freedom. Since a polyatomic molecule is a single whole, it is convenient to consider the motion of its center of mass with three translational degrees of freedom. In this case, the remaining degrees of freedom fall on the rotational and vibrational motions of the molecule. Theoretical mechanics states that a nonlinear molecule consisting of three or more atoms is capable of participating in three independent rotational movements about three coordinate axes. Any other rotation can be thought of as a combination of them. Therefore, the number of rotational degrees of freedom of a nonlinear molecule is three. For a linear molecule of two or more atoms (lined up along one line), taking into account the rotation around the axis connecting the atoms, which are considered to be material points, does not contribute to the energy. Therefore, the number of rotational degrees of freedom of a linear molecule is two. The remaining degrees of freedom fall on the oscillatory motion. It is easy to calculate that the number of vibrational degrees of freedom for a nonlinear molecule is equal to 3N-6, and for a linear molecule - 3N-5.
In the case of a polyatomic gas (as well as for a monatomic one), the law on the uniform distribution of energy over degrees of freedom: the average kinetic energy per one degree of freedom of a molecule at thermal equilibrium is ~kT.
The energy of vibrational degrees of freedom should be taken into account in particular. At normal and low temperatures, the vibrational motion of molecules is usually described by the laws of quantum mechanics. These laws justify the rigidity of molecules and the absence of vibrational energy - in this case, it is believed that the vibrational degrees of freedom frozen out(missing). At high temperatures, on the vibrational degree of freedom, in addition to the kinetic energy - kT have the same potential energy, so that in total we get kt.(It follows from the model of a harmonic oscillator that the average potential energy of oscillatory motion is equal to the average kinetic energy.)
Thus, in the general case, the average internal energy of a molecule is equal to
and the internal energy of a mole of an ideal gas is
Where i is the effective number of degrees of freedom of the molecule.
As follows from the above reasoning, for a monatomic molecule /=3, for a linear molecule at normal and low temperatures /=5, for a nonlinear molecule at normal and low temperatures /=6. At high temperatures of the order of 10 3 K for a linear molecule i=6N-5, for a nonlinear molecule i=6 N-6 .
We note that at very low temperatures (of the order of 10 K) the rotational degrees of freedom also freeze out. This is due to the fact that the laws of classical statistical mechanics, on which the law of the uniform distribution of energy over degrees of freedom is based, cease to work, and the application of quantum mechanical laws is necessary.
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.
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, 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 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 .
