MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF TATARSTAN
ALMETYEVSK STATE OIL INSTITUTE
Department of Physics
on the topic: "Debye's Law of Cubes"
Completed by a student of group 18-13B Gontar I.V. Teacher: Mukhetdinova Z.Z.
Almetyevsk 2010
1. Energy of the crystal lattice …………………………… 3
2. Einstein’s model…………………………………………….. 6
3. Debye model ………………………………………….. 7
4. Debye’s law of cubes……………………………………………… 8
5. Debye’s achievements…………………………………………… 9
6. References……………………………………………………….. 12
Energy of the crystal lattice
Peculiarity solid- the presence of long-range and short-range orders. In an ideal crystal, particles occupy certain positions and there is no need to take N into account! in statistical calculations.
The energy of the crystal lattice of a monatomic crystal consists of two main contributions: E = U o + E count. Atoms in the lattice vibrate. For polyatomic particles forming a crystal, it is necessary to take into account the internal degrees of freedom: vibrations and rotations. If we do not take into account the anharmonicity of atomic vibrations, which gives the dependence of U o on temperature (changes in the equilibrium positions of atoms), U o can be equated to the potential energy of the crystal and does not depend on T. At T = 0, the energy of the crystal lattice, i.e. the energy for removing crystal particles to an infinite distance will be equal to E cr = - E o = - (U o + E o,col).
Here E o,kol is the energy of zero-point oscillations. Typically this value is on the order of 10 kJ/mol and much less than U o. Consider Ecr = - Uo. (Method of largest term). Ecr in ionic and molecular crystals up to 1000 kJ/mol, in molecular and crystals with hydrogen bonds: up to 20 kJ/mol (CP 4 - 10, H 2 O - 50). The quantities are determined from experience or calculated on the basis of some model: ionic interaction according to the Coulomb, van der Waals forces according to the Sutherland potential.
Let us consider an ionic NaCl crystal having a face-centered cubic lattice: in the lattice each ion has 6 neighbors of the opposite sign at a distance R, in the next second layer there are 12 neighbors of the same sign at a distance of 2 1/2 R, 3rd layer: 8 ions at a distance of 3 1/2 R, 4th layer: 6 ions at a distance of 2R, etc.
The potential energy of a crystal of 2N ions will be U = Nu, where u is the energy of interaction of the ion with its neighbors. The interaction energy of ions consists of two terms: short-range repulsion due to valence forces (1st term) and attraction or repulsion of charges: 
+ sign for repulsion of like, - attraction of different ions. e - charge. Let us introduce the value of the reduced distance p ij = r ij / R, where r ij is the distance between the ions, R is the lattice parameter.
The energy of interaction of an ion with all its neighbors where
Madelung's constant = 6/1 - 12/2 1/2 + 8/3 1/2 - 6/2 + .... Here - for ions of the same charge sign, + for different ones. For NaCl a = 1.747558... A n = S 1/ p ij n in the first term. The distance R o (half the edge of the cube in this case) corresponds to the minimum potential energy at T = 0 and can be determined from crystallography data and knowing the repulsive potential. It's obvious that 
and then
From here we find A n and energy 
or 
.
n is the repulsive potential parameter and is usually ³ 10, i.e. The main contribution is made by the Coulomb interaction (we assume that R is not noticeably independent of T), and repulsion makes less than 10%.
For NaCl, the Coulomb interaction is 862, repulsion is 96 kJ/mol (n = 9). For molecular crystals, the potential can be calculated as 6-12 and the energy will be equal to 
z 1 is the number of atoms in the 1st coordination sphere, R 1 is the radius of the first coordination sphere, b is the potential parameter.
For nonionic crystals, the vibrational component of energy must be taken into account. There are no translational or rotational movements at absolute zero. The vibrational component of energy remains. There are 6 vibrations 3N, but translational and rotational vibrations apply to the crystal as a whole. Roughly it can be considered 3N, because N (large, the number of particles in the crystal). Then all 3N degrees of freedom of a crystal of N particles are vibrational. In principle, it is easy to calculate the sum over states and thermodynamic functions. But you need to know the frequency spectrum of crystal vibrations. The point is that the displacement of a particle causes the displacement of others and the oscillators are connected. The total sum over the states of oscillatory motion will be determined:
.
Because this is a crystal, then on N! no need to divide. The average energy is equal to the derivative of lnZ with respect to T at constant V, multiplied by kT 2. Hence the lattice energy is equal to the sum of the contributions of potential and vibrational energy, 
and entropy S = E/ T + k ln(Z).
Two main models are used for calculations.
Einstein's model
All frequencies are considered the same: a collection of one-dimensional harmonic oscillators. The sum over the states of a three-dimensional oscillator consists of 3 identical terms q = [ 2sh(hn/ 2kT)] -3. For N particles there will be 3N factors. Those. energy 
At high T, expanding the exponential into a series, the limit sh(hn/ 2kT) = hn/ 2kT and 
Entropy of vibrational motion 
Heat capacity of crystals: 
The OP has a mistake. Hence, at large T >> q E = hn/ k, the limit is C v ® 3Nk: Dulong-Ptied law for monatomic crystals. AND 
(The exponent quickly approaches 0).
In the classical approximation, Ecol without zero oscillations is equal to 3NkT and the contribution of oscillations to the heat capacity is 3Nk = 3R. Calculation according to Einstein: the lower curve, which deviates more noticeably from the experimental data.
Einstein's model gives the equation of state of a solid: (according to Melvin-Hughes)
u o = - q sublimation, m, n are experimental parameters, so for xenon m = 6, n = 11, a o is the interatomic distance at T = 0. That is pV/ RT = f(n, a o , n, m).
But near T = 0, Einstein’s assumption of equal frequencies does not work. Oscillators can differ in interaction strength and frequency. Experiments at low temperatures show a cubic dependence on temperature.
Debye model
Debye proposed a model for the existence of a continuous spectrum of frequencies (strictly for low frequencies, for thermal vibrations - phonons) up to a certain maximum. The frequency distribution function of harmonic oscillators has the form 
, where c l, c t- speed of propagation of longitudinal and transverse waves hesitation. At frequencies above the maximum g = 0.
The areas under the two curves must be the same. In reality, there is a certain spectrum of frequencies; the crystal is nonisotropic (this is usually neglected and the speeds of wave propagation in the directions are assumed to be the same). It may be that the maximum Debye frequency is higher than actually existing ones, which follows from the condition of equality of areas. The value of the maximum frequency is determined by the condition that the total number of oscillations is equal to 3N (we neglect the discreteness of energy) 
And 
, с is the speed of the wave. We assume that the speeds c l and c t are equal. Characteristic Debye temperature Q D = hn m/k.
Let's introduce x = hn/ kT. The average energy of oscillations then at maximum
The second term under the integral will give E zero-point vibrations E o = (9/8)NkQ D and then the vibrational energy of the crystal: 
Since U o and E o do not depend on T, the 2nd term in the expression for energy will contribute to the heat capacity.
Let us introduce the Debye function 
At high T we obtain the obvious D(x) ® 1. Differentiating with respect to x, we obtain 
.
At high T the limit is C V = 3Nk, and at low T: 
.
At small T, the upper limit of integration tends to infinity, E - E o = 3Rp 4 T 4 /5Q D 3 and we obtain a formula for determining C v at T® 0: where
Received Debye's cube law.
Debye's law of cubes.
The characteristic Debye temperature depends on the density of the crystal and the speed of propagation of vibrations (sound) in the crystal. A strictly Debye integral must be solved on a computer.
Debye characteristic temperature (Phys. encyclopedia)
Na 150 Cu 315 Zn 234 Al 394 Ni 375 Ge 360 Si 625
A.U 157 342 316 423 427 378 647
Li 400 K 100 Be 1000 Mg 318 Ca 230 B 1250 Ga 240
As 285 Bi 120 Ar 85 In 129 Tl 96 W 310 Fe 420
Ag 215 Au 170 Cd 120 Hg 100 Gd 152 Pr 74 Pt 230
La 132 Cr 460 Mo 380 Sn(white) 170, (gray) 260 C(diamond) 1860
To estimate the characteristic Debye temperature, you can use Lindemann’s empirical formula: Q D =134.5[Tmel/ (AV 2/3)] 1/2, here A - atomic mass metal For the Einstein temperature it is similar, but the first factor is 100.
Debye's achievements
Debye is the author of fundamental works on the quantum theory of solids. In 1912, he introduced the concept of a crystal lattice as an isotropic elastic medium capable of oscillating in a finite frequency range (Debye's solid state model). Based on the spectrum of these vibrations, it was shown that at low temperatures the heat capacity of the lattice is proportional to the cube absolute temperature(Debye's law of heat capacity). Within the framework of his solid state model, he introduced the concept of characteristic temperature at which quantum effects become significant for each substance (Debye temperature). In 1913 one of the most famous works Debye, devoted to the theory of dielectric losses in polar liquids. Around the same time, his work on the theory of X-ray diffraction was published. The beginning of Debye's experimental activities is associated with the study of diffraction. Together with his assistant P. Scherrer, he obtained an x-ray diffraction pattern of finely ground LiF powder. The photographs clearly showed rings resulting from the intersection of X-rays, diffracted from randomly oriented crystals along the forming cones, with photographic film. The Debye–Scherrer method, or powder method, has long been used as the main one in X-ray diffraction analysis. In 1916, Debye, together with A. Sommerfeld, applied quantization conditions to explain the Zeeman effect and introduced the magnetic quantum number. In 1923 he explained the Compton effect. In 1923, Debye, in collaboration with his assistant E. Hückel, published two large articles on the theory of electrolyte solutions. The ideas presented in them served as the basis for the theory of strong electrolytes, called the Debye-Hückel theory. Since 1927 Debye's interests have focused on issues chemical physics, in particular on the study of molecular aspects of the dielectric behavior of gases and liquids. He also studied the diffraction of x-rays on isolated molecules, which made it possible to determine the structure of many of them.
Debye's main scientific interest during his time at Cornell University was polymer physics. He developed a method for determining the molecular weight of polymers and their shape in solution based on light scattering measurements. One of his last major works (1959) was devoted to an issue that is extremely relevant today - the study of critical phenomena. Among Debye's awards are the medals of H. Lorentz, M. Faraday, B. Rumford, B. Franklin, J. Gibbs (1949), M. Planck (1950), and others. Debye died in Ithaca (USA) on November 2, 1966.
Debye - an outstanding representative of Dutch science - received Nobel Prize in Chemistry in 1936. With exceptional versatility, he made major contributions not only to chemistry but also to physics. These achievements brought Debye great fame; he was awarded honorary degrees of Doctor of Science by more than 20 universities around the world (Brussels, Oxford, Brooklyn, Boston and others). He was awarded many medals and prizes, including Faraday and Lorentz. Plank. Since 1924 Debye has been a corresponding member. Academy of Sciences of the USSR.
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Indirect contributions and their contributions to the financial and government activities of enterprises
Thesis >> Financial SciencesType of application to the issues referred to in Article 5 Law, the delivery slip will have the entry “Without... 25”. deb itors' and creditors' debts - ... 3.0 euros per 1 cube. cm 2.4 euros per 1 cube. see Other cars from...
80. If we do not take into account the vibrational movements in the hydrogen molecule at a temperature of 200 TO, That kinetic energy V ( J) all molecules in 4 G hydrogen is equal to... Answer:
81. In physiotherapy, ultrasound is used with frequency and intensity. When such ultrasound acts on human soft tissues, the density amplitude of molecular vibrations will be equal to ...
(Assume the speed of ultrasonic waves in the human body to be equal. Express your answer in angstroms and round to the nearest whole number.) Answer: 2.
82. Two mutually perpendicular oscillations are added. Establish a correspondence between the number of the corresponding trajectory and the laws of point oscillations M along the coordinate axes 
Answer:
1 
2 
3 
4 
83. The figure shows the profile of a transverse traveling wave, which propagates at a speed of . The equation of this wave is the expression... 
Answer:
84. The law of conservation of angular momentum imposes restrictions on possible transitions electron in an atom from one level to another (selection rule). IN energy spectrum of the hydrogen atom (see figure) the transition is forbidden... 
Answer:
85. The energy of an electron in a hydrogen atom is determined by the value of the main quantum number. If , then equals... Answer: 3.
86. .
The angular momentum of an electron in an atom and its spatial orientation can be conventionally depicted by a vector diagram, in which the length of the vector is proportional to the modulus of the orbital angular momentum of the electron. The figure shows possible orientations of the vector. 
Answer: 3.
87. The stationary Schrödinger equation in the general case has the form 
. Here 
potential energy of a microparticle. The motion of a particle in a three-dimensional infinitely deep potential box is described by the equation... Answer: 
88. The figure schematically shows stationary orbits of an electron in a hydrogen atom according to the Bohr model, and also shows transitions of an electron from one stationary orbit to another, accompanied by the emission of an energy quantum. In the ultraviolet region of the spectrum these transitions give the Lyman series, in the visible - the Balmer series, in the infrared - the Paschen series. 
The highest quantum frequency in the Paschen series (for the transitions shown in the figure) corresponds to the transition... Answer: 
89. If a proton and a deuteron have passed through the same accelerating potential difference, then the ratio of their de Broglie wavelengths is ... Answer:
90. The figure shows the velocity vector of a moving electron: 
WITH directed... Answer: from us
91. A small electric boiler can be used to boil a glass of water for tea or coffee in the car. Battery voltage 12 IN. If he's over 5 min heats 200 ml water from 10 to 100° WITH, then the current strength (in A
J/kg. TO.)Answer: 21
92. Conducting flat circuit with an area of 100 cm 2 
Tl mV), is equal to... Answer: 0.12
93. Orientational polarization of dielectrics is characterized by... Answer: influence thermal movement molecules on the degree of dielectric polarization
94. The figures show graphs of the field strength for various charge distributions: 
R shown in the picture... Answer: 2.
95. Maxwell's equations are the basic laws of classical macroscopic electrodynamics, formulated on the basis of a generalization of the most important laws of electrostatics and electromagnetism. These equations in integral form have the form:
1). 
;
2). 
;
3). 
;
4). 0.
Maxwell's third equation is a generalization Answer: Ostrogradsky–Gauss theorems for electrostatic field in the environment
96. The dispersion curve in the region of one of the absorption bands has the form shown in the figure. Relationship between phase and group velocities for a section bc looks like... 
Answer:
1. 182 . An ideal heat engine operates according to the Carnot cycle (two isotherms 1-2, 3-4 and two adiabats 2-3, 4-1).
During the process of isothermal expansion 1-2, the entropy of the working fluid ... 2) does not change
2. 183. Change internal energy gas during an isochoric process is possible ... 2) without heat exchange with the external environment
3. 184. When the gun was fired, the projectile flew out of the barrel, located at an angle to the horizon, rotating around its longitudinal axis with an angular velocity. The moment of inertia of the projectile relative to this axis, the time of movement of the projectile in the barrel. A moment of force acts on the gun barrel during a shot... 1)
Electric motor rotor rotating at speed 
, after turning off it stopped after 10s. The angular acceleration of the rotor braking after turning off the electric motor remained constant. The dependence of rotation speed on braking time is shown in the graph. The number of revolutions that the rotor made before stopping is ... 3) 80
5. 186. An ideal gas has minimum internal energy in the state...
2) 1
6. 187. A ball of radius R and mass M rotates with angular velocity . The work required to double its rotation speed is... 4)
7. 189 . After a time interval equal to two half-lives, undecayed radioactive atoms will remain... 2)25%
8. 206 . A heat engine operating according to the Carnot cycle (see figure) performs work equal to...
4)
9. 207. If for polyatomic gas molecules at temperatures the contribution of nuclear vibration energy to the heat capacity of the gas is negligible, then of the ideal gases proposed below (hydrogen, nitrogen, helium, water vapor), one mole has an isochoric heat capacity (universal gas constant) ... 2) water vapor
10. 208.
An ideal gas is transferred from state 1 to state 3 in two ways: along the 1-3 and 1-2-3 paths. The ratio of work done by gas is... 3) 1,5
11. 210. When the pressure increases by 3 times and the volume decreases by 2 times, the internal energy ideal gas … 3) will increase by 1.5 times
12. 211.
13. A ball of radius rolls uniformly without slipping along two parallel rulers, the distance between which , and covers 120 cm in 2 s. The angular velocity of rotation of the ball is... 2)
14. 212 . A cord is wound around a drum with a radius, to the end of which a mass of mass is attached. The load descends with acceleration. Moment of inertia of the drum... 3)
15. 216. A rectangular wire frame is located in the same plane with a straight long conductor through which current I flows. The induction current in the frame will be directed clockwise when it ...
3) translational movement in the negative direction of the OX axis
16. 218. A frame with a current with a magnetic dipole moment, the direction of which is indicated in the figure, is in a uniform magnetic field:
The moment of forces acting on the magnetic dipole is directed... 2) perpendicular to the plane of the drawing to us
17. 219. The average kinetic energy of gas molecules at temperature depends on their configuration and structure, which is associated with the possibility various types movement of atoms in a molecule and the molecule itself. Provided that there is a progressive and rotational movement molecules as a whole, the average kinetic energy of a water vapor molecule () is ... 3)
18. 220. The eigenfunctions of an electron in a hydrogen atom contain three integer parameters: n, l and m. The parameter n is called the principal quantum number, the parameters l and m are called the orbital (azimuthal) and magnetic quantum numbers, respectively. Magnetic quantum number m determines... 1) projection of the orbital angular momentum of the electron to a certain direction
19. 221.
Stationary Schrödinger equation 
describes the motion of a free particle if the potential energy has the form... 2)
20. 222. The figure shows graphs reflecting the nature of the dependence of polarization P of the dielectric on the voltage of the external electric field E.
Non-polar dielectrics correspond to the curve ... 1) 4
21. 224. A horizontally flying bullet pierces a block lying on a smooth horizontal surface. In the “bullet-bar” system... 1) momentum is conserved, mechanical energy is not conserved
22. A hoop rolls down a slide 2.5 m high without slipping. The speed of the hoop (in m/s) at the base of the slide, provided that friction can be neglected, is ... 4) 5
23. 227. T The momentum of the body changed under the influence of a short-term impact and became equal, as shown in the figure:
At the moment of impact, the force acted in the direction... Answer:2
24. 228. The accelerator imparted speed to the radioactive nucleus (c is the speed of light in vacuum). At the moment of departure from the accelerator, the nucleus ejected a β-particle in the direction of its motion, the speed of which was relative to the accelerator. The speed of a beta particle relative to the nucleus is... 1) 0.5 s
25. 231. The average kinetic energy of gas molecules at temperature depends on their configuration and structure, which is associated with the possibility of various types of movement of atoms in the molecule and the molecule itself. Provided that there is translational, rotational motion of the molecule as a whole and vibrational motion of the atoms in the molecule, the ratio of the average kinetic energy of vibrational motion to the total kinetic energy of the nitrogen molecule () is equal to ... 3) 2/7
26. 232. The spin quantum number s determines... intrinsic mechanical torque of an electron in an atom
27. 233. If a hydrogen molecule, positron, proton and -particle have the same de Broglie wavelength, then the highest speed has ... 4) positron
28. A particle is located in a rectangular one-dimensional potential box with impenetrable walls 0.2 nm wide. If the energy of a particle at the second energy level is 37.8 eV, then at the fourth energy level it is equal to _____ eV. 2) 151,2
29. The stationary Schrödinger equation in the general case has the form 
. Here 
potential energy of a microparticle. An electron in a one-dimensional potential box with infinitely high walls corresponds to the equation... 1) 
30. Complete system Maxwell's equations for electrical magnetic field in integral form has the form:
,
,
The following system of equations:
valid for... 4) electromagnetic field in the absence of free charges
31. The figure shows sections of two straight long parallel conductors with oppositely directed currents, and . The magnetic field induction is zero in the area ...
4) d
32. Along parallel metal conductors located in a uniform magnetic field, a conducting jumper of length (see figure) moves with constant acceleration. If the resistance of the jumper and guides can be neglected, then the dependence of the induction current on time can be represented by a graph ...
33. The figures show the time dependence of the speed and acceleration of a material point oscillating according to a harmonic law.
The cyclic frequency of oscillations of a point is ______ Answer: 2
34. Two harmonic oscillations of the same direction with the same frequencies and amplitudes, equal to and , are added. Establish a correspondence between the phase difference of the added oscillations and the amplitude of the resulting oscillation.
35. Answer options:
36. If the frequency elastic wave increase by 2 times without changing its speed, then the intensity of the wave will increase by ___ times. Answer: 8
37. The equation of a plane wave propagating along the OX axis has the form 
. The wavelength (in m) is... 4) 3,14
38. A photon with an energy of 100 keV was deflected by an angle of 90° as a result of Compton scattering by an electron. The energy of a scattered photon is _____. Express your answer in keV and round to the nearest whole number. Please note that the rest energy of the electron is 511 keV Answer:84
39. The angle of refraction of a beam in a liquid is equal to If it is known that the reflected beam is completely polarized, then the refractive index of the liquid is equal to ... 3) 1,73
40. If the axis of rotation of a thin-walled circular cylinder is transferred from the center of mass to the generatrix (Fig.), then the moment of inertia relative to the new axis is _____ times.
1) will increase by 2
41. A disk rolls uniformly on a horizontal surface at speed without slipping. The velocity vector of point A, lying on the rim of the disk, is oriented in the direction ...
3) 2
42. A small washer begins to move without initial speed along a smooth ice slide from point A. Air resistance is negligible. The dependence of the potential energy of the puck on the x coordinate is shown on the graph:
The kinetic energy of the puck at point C is ______ than at point B. 4) 2 times more
43. Two small massive balls are attached to the ends of a weightless rod of length l. The rod can rotate in horizontal plane around a vertical axis passing through the middle of the rod. The rod was spun to angular velocity. Under the influence of friction, the rod stopped, and 4 J of heat were released.
44. If the rod is spun to angular velocity , then when the rod stops, an amount of heat (in J) will be released equal to ... Answer : 1
45. Light waves in a vacuum are... 3) transverse
46. The figures show the time dependence of the coordinates and speed of a material point oscillating according to a harmonic law:
47. The cyclic frequency of oscillations of a point (in) is equal to... Answer: 2
48. The energy flux density transferred by a wave in an elastic medium with density , increased 16 times at a constant speed and frequency of the wave. At the same time, the amplitude of the wave increased by _____ times. Answer: 4
49. The magnitude of the saturation photocurrent during the external photoelectric effect depends... 4) on the intensity of the incident light
50. The figure shows a diagram of the energy levels of the hydrogen atom, and also conventionally depicts the transitions of an electron from one level to another, accompanied by the emission of an energy quantum. In the ultraviolet region of the spectrum, these transitions give rise to the Lyman series, in the visible region – the Balmer series, in the infrared region – the Paschen series, etc.
The ratio of the minimum line frequency in the Balmer series to the maximum line frequency in the Lyman series of the spectrum of the hydrogen atom is ... 3)5/36
51. The ratio of the de Broglie wavelengths of a neutron and an alpha particle having the same speeds is ... 4) 2
52. The stationary Schrödinger equation has the form 
. This equation describes... 2) linear harmonic oscillator
53. The figure schematically shows the Carnot cycle in coordinates:
54. 
55. An increase in entropy takes place in the area ... 1) 1–2
56. Dependences of the pressure of an ideal gas in an external uniform field of gravity on height for two different temperatures are presented in the figure.
57. For the graphs of these functions, the statements that... 3) the dependence of the pressure of an ideal gas on height is determined not only by the temperature of the gas, but also by the mass of the molecules 4) temperature below temperature
1.
The stationary Schrödinger equation has the form 
.
This equation describes...an electron in a hydrogen-like atom
The figure schematically shows the Carnot cycle in coordinates: 
An increase in entropy occurs in areas 1–2
2. On ( P,V)-diagram shows 2 cyclic processes. 
The ratio of work performed in these cycles is equal to...Answer: 2.
3. Dependences of the pressure of an ideal gas in an external uniform field of gravity on height for two different temperatures are presented in the figure. 
For graphs of these functions unfaithful are statements that ... the temperature is below the temperature
the dependence of the pressure of an ideal gas on height is determined not only by the temperature of the gas, but also by the mass of the molecules
4. At room temperature, the ratio of molar heat capacities at constant pressure and constant volume is 5/3 for ... helium
5. The figure shows the trajectories of charged particles flying into a uniform magnetic field at the same speed, perpendicular to the plane drawing. At the same time, for charges and specific charges of particles, the statement is true... 
, , 
6. Unfaithful for ferromagnets is the statement...
The magnetic permeability of a ferromagnet is a constant value that characterizes its magnetic properties.
7. Maxwell's equations are the basic laws of classical macroscopic electrodynamics, formulated on the basis of a generalization of the most important laws of electrostatics and electromagnetism. These equations in integral form have the form:
1). 
;
2). 
;
3). 
;
4). 0.
Maxwell's fourth equation is a generalization...
Ostrogradsky–Gauss theorem for magnetic field
8. A bird sits on a power line wire whose resistance is 2.5 10 -5 Ohm for every meter of length. If a wire carries a current of 2 kA, and the distance between the bird’s paws is 5 cm, then the bird is energized...
9. Current strength in a conducting circular circuit with inductance 100 mH changes over time
according to the law (in SI units): 
Absolute value of self-induction emf at time 2 With equal to ____ ; in this case the induction current is directed...
0,12 IN; counterclockwise
10. An electrostatic field is created by a system of point charges. 
The field strength vector at point A is oriented in the direction ...
11. The angular momentum of an electron in an atom and its spatial orientation can be conventionally depicted by a vector diagram, in which the length of the vector is proportional to the modulus of the orbital angular momentum of the electron. The figure shows possible orientations of the vector. 
Minimum value of the principal quantum number n for the specified state is 3
12. The stationary Schrödinger equation in the general case has the form 
. Here 
potential energy of a microparticle. The motion of a particle in a three-dimensional infinitely deep potential box is described by the equation 
13. The figure schematically shows the stationary orbits of an electron in a hydrogen atom according to the Bohr model, and also shows transitions of an electron from one stationary orbit to another, accompanied by the emission of an energy quantum. In the ultraviolet region of the spectrum these transitions give the Lyman series, in the visible - the Balmer series, in the infrared - the Paschen series. 
The highest quantum frequency in the Paschen series (for the transitions shown in the figure) corresponds to the transition 
14. If a proton and a deuteron have passed through the same accelerating potential difference, then the ratio of their de Broglie wavelengths is
15. The figure shows the velocity vector of a moving electron: 
Vector of magnetic induction field created by an electron when moving, at a point WITH sent... from us
16. A small electric boiler can be used to boil a glass of water for tea or coffee in the car. Battery voltage 12 IN. If he's over 5 min heats 200 ml water from 10 to 100° WITH, then the current strength (in A) consumed from the battery is equal to...
(The heat capacity of water is 4200 J/kg. TO.) 21
17. Conducting flat circuit with an area of 100 cm 2 located in a magnetic field perpendicular to the lines of magnetic induction. If magnetic induction changes according to the law 
Tl, then the induced emf arising in the circuit at the moment of time (in mV), equal to 0.1
18. The orientational polarization of dielectrics is characterized by the influence of the thermal motion of molecules on the degree of polarization of the dielectric
19. The figures show graphs of the field strength for various charge distributions: 
Dependence graph for a charged metal sphere of radius R shown in the figure...Answer: 2.
20. Maxwell's equations are the basic laws of classical macroscopic electrodynamics, formulated on the basis of a generalization of the most important laws of electrostatics and electromagnetism. These equations in integral form have the form:
1). 
;
2). 
;
3). 
;
4). 0.
Maxwell's third equation is a generalization of the Ostrogradsky–Gauss theorem for the electrostatic field in a medium
21. The dispersion curve in the region of one of the absorption bands has the form shown in the figure. Relationship between phase and group velocities for a section bc looks like... 
22. Sunlight falls on a mirror surface along the normal to it. If the solar radiation intensity is 1.37 kW/m 2, then the light pressure on the surface is _____. (Express your answer in µPa and round to the nearest whole number). Answer: 9.
23. The phenomenon of external photoelectric effect is observed. In this case, as the wavelength of the incident light decreases, the magnitude of the retarding potential difference increases
24. A plane light wave with wavelength is incident on a diffraction grating along the normal to its surface. If the grating constant is , then total number the main maxima observed in the focal plane of the collecting lens are equal to ...Answer: 9.
25. A particle moves in a two-dimensional field, and its potential energy is given by the function. The work of field forces to move a particle (in J) from point C (1, 1, 1) to point B (2, 2, 2) is equal to ...
(The function and coordinates of the points are given in SI units.) Answer: 6.
26. A skater rotates around a vertical axis with a certain frequency. If he presses his hands to his chest, thereby reducing his moment of inertia relative to the axis of rotation by 2 times, then the speed of the skater’s rotation and his kinetic energy of rotation will increase by 2 times
27. On board spaceship emblem in the form geometric figure:
If the ship moves in the direction indicated by the arrow in the figure at a speed comparable to the speed of light, then in a stationary frame of reference the emblem will take the shape shown in the figure
28. Three bodies are considered: a disk, a thin-walled pipe and a ring; and the masses m and radii R their bases are the same. 
For the moments of inertia of the bodies under consideration relative to the indicated axes, the following relation is correct: 
29. The disk rotates uniformly around a vertical axis in the direction indicated by the white arrow in the figure. At some point in time, a tangential force was applied to the disk rim. 
At the same time, it correctly depicts the direction angular acceleration disk vector 4
30. The figure shows a graph of body speed versus time t. 
If body weight is 2 kg, then the force (in N), acting on the body, is equal to...Answer: 1.
31. Establish a correspondence between the types of fundamental interactions and radii (in m) their actions.
1.Gravitational
2.Weak
3. Strong
32. -decay is a nuclear transformation that occurs according to the scheme 
33. The charge in electron charge units is +1; the mass in electron mass units is 1836.2; spin in units is 1/2. These are the main characteristics of the proton
34. The law of conservation of lepton charge prohibits the process described by the equation 
35. In accordance with the law of uniform distribution of energy over degrees of freedom, the average kinetic energy of an ideal gas molecule at temperature T equal to: . Here , where , and are the number of degrees of freedom of translational, rotational and vibrational motions of the molecule, respectively. For hydrogen () number i equals 7
36. A diagram of the cyclic process of an ideal monatomic gas is shown in the figure. The ratio of the work during heating to the work of gas for the entire cycle in modulus is equal to ...
37. The figure shows graphs of the distribution functions of ideal gas molecules in an external uniform field of gravity versus height for two different gases, where are the masses of gas molecules (Boltzmann distribution). 
For these functions it is true that...
mass greater than mass
the concentration of gas molecules with a lower mass at the “zero level” is less
38. When heat enters a non-isolated thermodynamic system during a reversible process for the increment of entropy, the following relation will be correct:
39. The traveling wave equation has the form: , where is expressed in millimeters, – in seconds, – in meters. The ratio of the amplitude value of the velocity of particles of the medium to the velocity of wave propagation is 0.028
40. The amplitude of damped oscillations decreased by a factor of ( – base natural logarithm) for . The attenuation coefficient (in) is equal to...Answer: 20.
41. Two harmonic oscillations of the same direction with the same frequencies and equal amplitudes are added. Establish a correspondence between the amplitude of the resulting oscillation and the phase difference of the added oscillations.
1. 2. 3. Answer: 2 3 1 0
42. The figure shows the orientation of the electric () and magnetic () field strength vectors in an electromagnetic wave. The energy flux density vector of the electromagnetic field is oriented in the direction... 
43. Two conductors are charged to potential 34 IN and –16 IN. Charge 100 nCl need to be transferred from the second conductor to the first. In this case, it is necessary to perform work (in µJ), equal to...Answer: 5.
44. The figure shows bodies of the same mass and size rotating around a vertical axis with the same frequency. Kinetic energy of the first body J. If kg, cm, then the angular momentum (in mJ s) of the second body is equal to ... 
The main task of theories chemical kinetics- propose a method for calculating the rate constant of an elementary reaction and its dependence on temperature, using different ideas about the structure of reagents and reaction paths. We will consider the two simplest theories of kinetics - the theory of active collisions (TAC) and the theory activated complex(SO).
Active collision theory is based on counting the number of collisions between reacting particles, which are represented as hard spheres. It is assumed that a collision will lead to a reaction if two conditions are met: 1) the translational energy of the particles exceeds the activation energy E A; 2) the particles are correctly oriented in space relative to each other. The first condition introduces the factor exp(- E A/RT), which is equal proportion of active collisions in the total number of collisions. The second condition gives the so-called steric factor P- a constant characteristic of a given reaction.
In TAS, two main expressions for the rate constant of a bimolecular reaction are obtained. For the reaction between different molecules(A + B products) rate constant is
Here N A- Avogadro's constant, r- radii of molecules, M - molar masses substances. The multiplier in large parentheses is the average speed relative motion particles A and B.
The rate constant of a bimolecular reaction between identical molecules (2A products) is equal to:
(9.2)
From (9.1) and (9.2) it follows that the temperature dependence of the rate constant has the form:
.
According to TAS, the pre-exponential factor depends weakly on temperature. Experienced Activation Energy E op, determined by equation (4.4), is related to the Arrhenius, or true activation energy E A ratio:
E op = E A - RT/2.
Monomolecular reactions within the framework of TAS are described using the Lindemann scheme (see problem 6.4), in which the activation rate constant k 1 is calculated using formulas (9.1) and (9.2).
IN activated complex theory an elementary reaction is represented as a monomolecular decomposition of an activated complex according to the scheme:
It is assumed that a quasi-equilibrium exists between the reactants and the activated complex. The rate constant of monomolecular decomposition is calculated using statistical thermodynamics methods, representing decomposition as a one-dimensional translational motion of the complex along the reaction coordinate.
The basic equation of the activated complex theory is:
, (9.3)
Where kB= 1.38. 10 -23 J/K - Boltzmann constant, h= 6.63. 10 -34 J. s - Planck's constant, - the equilibrium constant of the formation of an activated complex, expressed in terms of molar concentrations (in mol/l). Depending on how the equilibrium constant is estimated, the statistical and thermodynamic aspects of TAC are distinguished.
IN statistical approach, the equilibrium constant is expressed through sums over states:
, (9.4)
where is the total sum over the states of the activated complex, Q react is the product of the total sums over the states of the reactants, is the activation energy at absolute zero, T = 0.
Complete sums by state are usually decomposed into factors corresponding to individual types of molecular motion: translational, electronic, rotational and vibrational:
Q = Q fast. Q email . Q vr. . Q count
Translational sum over states for a particle with mass m is equal to:
Q post = .
This progressive sum has dimension (volume) -1, because concentrations of substances are expressed through it.
The electron sum over states at ordinary temperatures is, as a rule, constant and equal to the degeneracy of the ground electronic state: Q el = g 0 .
The rotational sum over states for a diatomic molecule is equal to:
Q vr = ,
where m = m 1 m 2 / (m 1 +m 2) - reduced mass of the molecule, r- internuclear distance, s = 1 for asymmetric molecules AB and s = 2 for symmetric molecules A 2. For linear polyatomic molecules, the rotational sum over states is proportional T, and for nonlinear molecules - T 3/2. At ordinary temperatures, rotational sums over states are of the order of 10 1 -10 2 .
The vibrational sum over the states of a molecule is written as a product of factors, each of which corresponds to a specific vibration:
Q count = 
,
Where n- number of vibrations (for a linear molecule consisting of N atoms, n = 3N-5, for a nonlinear molecule n = 3N-6), c= 3 . 10 10 cm/s - speed of light, n i- vibration frequencies, expressed in cm -1. At ordinary temperatures, the vibrational sums over states are very close to 1 and differ noticeably from it only under the condition: T>n. At very high temperatures, the vibrational sum for each vibration is directly proportional to the temperature:
Q i 
.
The difference between an activated complex and ordinary molecules is that it has one less vibrational degree of freedom, namely: the vibration that leads to the decomposition of the complex is not taken into account in the vibrational sum over states.
IN thermodynamic approach, the equilibrium constant is expressed through the difference between the thermodynamic functions of the activated complex and the starting substances. To do this, the equilibrium constant expressed in terms of concentrations is converted into a constant expressed in terms of pressures. The last constant, as is known, is associated with a change in the Gibbs energy in the reaction of formation of an activated complex:
.
For a monomolecular reaction in which the formation of an activated complex occurs without changing the number of particles, = and the rate constant is expressed as follows:
Entropy factor exp ( S /R) is sometimes interpreted as a steric factor P from the theory of active collisions.
For a bimolecular reaction occurring in the gas phase, the factor is added to this formula RT / P 0 (where P 0 = 1 atm = 101.3 kPa), which is needed to transition from to:
For a bimolecular reaction in solution, the equilibrium constant is expressed in terms of the Helmholtz energy of formation of the activated complex:
Example 9-1. Bimolecular reaction rate constant
2NO 2 2NO + O 2
at 627 K it is equal to 1.81. 10 3 cm 3 /(mol. s). Calculate the true activation energy and the fraction of active molecules if the diameter of the NO 2 molecule can be taken to be 3.55 A and the steric factor for this reaction is 0.019.
Solution. When calculating, we will rely on the theory of active collisions (formula (9.2)):
.
This number represents the fraction of active molecules.
When calculating rate constants using various theories of chemical kinetics, it is necessary to handle dimensions very carefully. Please note that the radius of the molecule and average speed expressed in cm in order to obtain a constant in cm 3 /(mol. s). The coefficient 100 is used to convert m/s to cm/s.
The true activation energy can be easily calculated through the fraction of active molecules:
J/mol = 166.3 kJ/mol.
Example 9-2. Using activated complex theory, determine the temperature dependence of the rate constant for the trimolecular reaction 2NO + Cl 2 = 2NOCl at temperatures close to room temperature. Find the connection between the experienced and true activation energies.
Solution. According to the statistical version of SO, the rate constant is equal to (formula (9.4)):
.
In the sums for the states of the activated complex and reagents, we will not take into account the vibrational and electronic degrees of freedom, because at low temperatures, the vibrational sums over the states are close to unity, and the electronic sums are constant.
The temperature dependences of the sums by state, taking into account translational and rotational motions, have the form:
The activated complex (NO) 2 Cl 2 is a nonlinear molecule, therefore its rotational sum over states is proportional T 3/2 .
Substituting these dependencies into the expression for the rate constant, we find:
We see that trimolecular reactions are characterized by a rather unusual dependence of the rate constant on temperature. Under certain conditions, the rate constant can even decrease with increasing temperature due to the pre-exponential factor!
The experimental activation energy for this reaction is:
.
Example 9-3. Using the statistical version of the activated complex theory, obtain an expression for the rate constant of a monomolecular reaction.
Solution. For a monomolecular reaction
AN products
the rate constant, according to (9.4), has the form:
.
The activated complex in a monomolecular reaction is an excited reagent molecule. The translational amounts of reactant A and complex AN are the same (the mass is the same). If we assume that the reaction occurs without electronic excitation, then the electronic sums for the states are the same. If we assume that upon excitation the structure of the reagent molecule does not change very much, then the rotational and vibrational sums for the states of the reagent and the complex are almost the same, with one exception: the activated complex has one less vibration than the reagent. Consequently, the vibration leading to bond breaking is taken into account in the sum over the states of the reagent and is not taken into account in the sum over the states of the activated complex.
By reducing identical sums across states, we find the rate constant of a monomolecular reaction:
where n is the frequency of vibration that leads to the reaction. Speed of light c is a multiplier that is used if the vibration frequency is expressed in cm -1. At low temperatures, the vibrational sum over states is equal to 1:
.
At high temperatures, the exponential in the vibrational sum over states can be expanded into the series: exp(- x) ~ 1 - x:
.
This case corresponds to the situation where, at high temperatures, each vibration leads to a reaction.
Example 9-4. Determine the temperature dependence of the rate constant for the reaction of molecular hydrogen with atomic oxygen:
H2+O. HO. +H. (linear activated complex)
at low and high temperatures.
Solution. According to the activated complex theory, the rate constant for this reaction is:
We will assume that the electron multipliers do not depend on temperature. All progressive sums across states are proportional T 3/2, rotational sums over states for linear molecules are proportional T, the vibrational sums over states at low temperatures are equal to 1, and at high temperatures they are proportional to the temperature to a degree equal to the number of vibrational degrees of freedom (3 N- 5 = 1 for molecule H 2 and 3 N- 6 = 3 for a linear activated complex). Taking all this into account, we find that at low temperatures
and at high temperatures
Example 9-5. The acid-base reaction in a buffer solution proceeds according to the mechanism: A - + H + P. The dependence of the rate constant on temperature is given by the expression
k = 2.05. 10 13.e -8681/ T(l. mol -1. s -1).
Find the experimental activation energy and activation entropy at 30 o C.
Solution. Since the bimolecular reaction occurs in solution, we use expression (9.7) to calculate the thermodynamic functions. The experimental activation energy must be introduced into this expression. Since the pre-exponential factor in (9.7) linearly depends on T, That E op = + RT. Replacing in (9.7) with E oops, we get:
.
It follows that the experimental activation energy is equal to E op = 8681. R= 72140 J/mol. The activation entropy can be found from the pre-exponential factor:
,
whence = 1.49 J/(mol. K).
9-1. The diameter of the methyl radical is 3.8 A. What is the maximum rate constant (in l/(mol. s)) of the recombination reaction of methyl radicals at 27 o C? (answer)
9-2. Calculate the value of the steric factor in the ethylene dimerization reaction
2C 2 H 4 C 4 H 8
at 300 K, if the experimental activation energy is 146.4 kJ/mol, the effective diameter of ethylene is 0.49 nm, and the experimental rate constant at this temperature is 1.08. 10 -14 cm 3 /(mol. s).
9-7. Determine the temperature dependence of the rate constant for the reaction H. + Br 2 HBr + Br. (nonlinear activated complex) at low and high temperatures. (answer)
9-8. For the reaction CO + O 2 = CO 2 + O, the dependence of the rate constant on temperature at low temperatures has the form:
k( T) ~ T-3/2. exp(- E 0 /RT)
(answer)
9-9. For the reaction 2NO = (NO) 2, the dependence of the rate constant on temperature at low temperatures has the form:
k( T) ~ T-1 exp(- E 0/R T)
What configuration - linear or nonlinear - does the activated complex have? (answer)
9-10. Using active complex theory, calculate the true activation energy E 0 for reaction
CH3. + C 2 H 6 CH 4 + C 2 H 5 .
at T= 300 K, if the experimental activation energy at this temperature is 8.3 kcal/mol. (answer)
9-11. Derive the relationship between the experimental and true activation energies for the reaction
9-12. Determine the activation energy of a monomolecular reaction at 1000 K if the frequency of vibrations along the broken bond is n = 2.4. 10 13 s -1 , and the rate constant is k= 510 min -1 .(answer)
9-13. The first-order reaction rate constant for the decomposition of bromoethane at 500 o C is 7.3. 10 10 s -1 . Estimate the activation entropy of this reaction if the activation energy is 55 kJ/mol. (answer)
9-14. Decomposition of di-peroxide rubs-butyl in the gas phase is a first-order reaction, the rate constant of which (in s -1) depends on temperature as follows:
Using the activated complex theory, calculate the enthalpy and entropy of activation at a temperature of 200 o C. (answer)
9-15. The isomerization of diisopropyl ether to allyl acetone in the gas phase is a first order reaction, the rate constant of which (in s -1) depends on temperature as follows:
Using the activated complex theory, calculate the enthalpy and entropy of activation at a temperature of 400 o C. (answer)
9-16. Dependence of the rate constant for the decomposition of vinyl ethyl ether
C 2 H 5 -O-CH=CH 2 C 2 H 4 + CH 3 CHO
depending on temperature has the form
k = 2.7. 10 11. e -10200/ T(s -1).
Calculate the entropy of activation at 530 o C. (answer)
9-17. In the gas phase, substance A is monomolecularly converted into substance B. The reaction rate constants at temperatures of 120 and 140 o C are equal to 1.806, respectively. 10 -4 and 9.14. 10 -4 s -1 . Calculate the average entropy and heat of activation in this temperature range.
The figure shows a graph of the distribution function of oxygen molecules over speeds (Maxwell distribution) for temperature T=273 K, at speed the function reaches its maximum. Here is the probability density or the fraction of molecules whose velocities lie in the speed interval from to per unit of this interval. For the Maxwell distribution, the following statements are true: ...
Specify at least two answer options
The area of the shaded strip is equal to the fraction of molecules with velocities in the range from to or the probability that the molecule's speed has a value in that speed range
As temperature increases, the most probable speed of molecules will increase
Exercise
Kinetic energy of rotational motion of all molecules in 2 g of hydrogen at a temperature of 100 K is ...
The efficiency of the Carnot cycle is 40%. If you increase the heater temperature by 20% and reduce the coolant temperature by 20%, the efficiency (in%) will reach the value ...
The diagram shows two cyclic processes The ratio of work done in these cycles is….
To melt a certain mass of copper, it is required more heat than for melting the same mass of zinc, since specific heat copper melting is 1.5 times greater than zinc (J/kg, J/kg). The melting point of copper is approximately 2 times higher than the melting point of zinc (,). The destruction of the metal crystal lattice during melting leads to an increase in entropy. If the entropy of zinc increased by , then the change in the entropy of copper will be ...
Answer: ¾ DS
Dependence of ideal gas pressure in an external homogeneous the gravity field from height for two different temperatures () is presented in the figure ...
From the ideal gases given below, select those for which the ratio of molar heat capacities is equal (neglect vibrations of atoms inside the molecule).
Oxygen
The diagram shows the Carnot cycle for an ideal gas.
For the magnitude of the work of adiabatic expansion of a gas and adiabatic compression, the relation is valid...
The figure shows a graph of the distribution function molecules of an ideal gas by speed (Maxwell distribution), where is the fraction of molecules whose speeds are contained in the speed interval from to per unit of this interval.
For this function it is true that...
When temperature changes, the area under the curve does not change
The figure shows the Carnot cycle in coordinates (T, S), where S– entropy. Adiabatic expansion occurs at the stage...
An ideal gas is transferred from the first state to the second in two methods ( and ), as shown in the figure. The heat received by the gas, the change in internal energy and the work of the gas during its transition from one state to another are related by the relations ...
Diagram of the cyclic process of an ideal monatomic gas presented in the figure. The work done by a gas in kilojoules in a cyclic process is...
Boltzmann's formula characterizes the distribution particles in a state of chaotic thermal motion in a potential force field, in particular the distribution of molecules by height in an isothermal atmosphere. Match the pictures with the corresponding statements.
1. Distribution of molecules in a force field at very high temperature, when the energy of chaotic thermal motion significantly exceeds the potential energy of molecules.
2. The distribution of molecules is not Boltzmann and is described by the function.
3. Distribution of air molecules in the Earth's atmosphere.
4. Distribution of molecules in a force field at temperature.
To a monatomic ideal gas as a result of isobaric the amount of heat supplied to the process. To increase the internal energy of gas
a portion of heat is consumed equal to (in percent) ...
Adiabatic expansion of gas (pressure, volume, temperature, entropy) corresponds to the diagram...
Molar heat capacity of an ideal gas at constant pressure equal to where is the universal gas constant. The number of rotational degrees of freedom of a molecule is...
Dependence of the concentration of ideal gas molecules in the external uniform field of gravity versus height for two different temperatures () is shown in the figure...
If we do not take into account vibrational movements in a linear molecule carbon dioxide (see figure), then the ratio of the kinetic energy of rotational motion to the total kinetic energy of the molecule is equal to ...
The refrigerator will double, then the efficiency of the heat engine...
will decrease by
Average kinetic energy of gas molecules at temperature depends on their configuration and structure, which is associated with the possibility of various types of movement of atoms in the molecule and the molecule itself. Provided that only translational and rotational motion of the molecule as a whole takes place, the average kinetic energy of nitrogen molecules is equal to ...
If the amount of heat given off by the working fluid refrigerator will double, then the efficiency of the heat engine
The actual circuit consists of an inductor and a capacitor. A real coil cannot be considered only as an inductance that stores magnetic energy. Firstly, the wire has finite conductivity, and secondly, between the turns it accumulates electrical energy, i.e. there is interturn capacitance. The same can be said about capacity. Real capacitance, in addition to the capacitance itself, will include lead inductance and loss resistance.
To simplify the problem, consider a model of a real oscillatory circuit with an inductor consisting of only two turns.
The equivalent circuit will look like the one shown in Fig. 4. (and - inductance and resistance of one turn, - interturn capacitance).
However, as the experience of a radio engineer shows, in most cases there is no need for this complex circuit.
The equation for the electrical circuit shown in Fig. We obtain 5 based on Kirchhoff’s law. We use the second rule: the sum of the voltage drops on the circuit elements is equal to the algebraic sum of the external emfs included in this circuit. In our case, the EMF is zero, and we get:
Divide the terms by and denote
The equation for an ideal contour will take the form:
Having models of two dynamic systems, we can already draw some conclusions.
A simple comparison of equations (B.6) and (B.9) shows that a pendulum at small deviations and an ideal circuit are described by the same equation, known as the harmonic oscillator equation, which in standard form is:
Consequently, both the pendulum and the circuit as oscillatory systems have the same properties. This is a manifestation of the unity of oscillatory systems.
Having these models, the equations that describe them, and generalizing the results obtained, we will give a classification of dynamic systems according to the type differential equation. Systems can be linear or nonlinear.
Linear systems are described linear equations(see (B.11) and (B.15)). Nonlinear systems are described nonlinear equations(for example, the equation of a mathematical pendulum (B.9)).
Another classification feature is number of degrees of freedom. The formal sign is the order of the differential equation describing the motion in the system. A system with one degree of freedom is described by a 2nd order equation (or two first order equations); a system with N degrees of freedom is described by an equation or system of equations of order 2N.
Depending on how the energy of vibrational motion in the system changes, all systems are divided into two classes: conservative systems - those in which the energy remains unchanged, and non-conservative systems - those in which the energy changes over time. In a system with losses, energy decreases, but there may be cases when energy increases. Such systems are called active.
A dynamic system may or may not be subject to external influences. Depending on this, four types of movement are distinguished.
1.Natural or free vibrations systems. In this case, the system receives a finite supply of energy from an external source and the source is turned off. The motion of the system with a finite initial supply of energy represents its own oscillations.
2.Forced vibrations. The system is under the influence of an external periodic source. The source has a “force” effect, i.e. the nature of the source is the same as that of dynamic system(V mechanical system– source of force, in electrical – EMF, etc.). Oscillations caused by an external source are called forced. When turned off they disappear.
3.Parametric oscillations are observed in systems in which some parameter changes periodically over time, for example, the capacitance in the circuit or the length of the pendulum. The nature of the external source that changes the parameter may differ from the nature of the system itself. For example, the capacity can be changed mechanically.
It should be noted that a strict separation of forced and parametric oscillations is possible only for linear systems.
4.A special type of movement is self-oscillation. The term was first introduced by Academician Andronov. Self-oscillation is a periodic oscillation, the period, shape and amplitude of which depend on the internal state of the system and do not depend on the initial conditions. From an energy point of view, self-oscillating systems are converters of the energy of some source into the energy of periodic oscillations.
Chapter 1. NATURAL VIBRATIONS IN A LINEAR CONSERVATIVE SYSTEM WITH ONE DEGREE OF FREEDOM (HARMONIC OSCILLATOR)
The equation of such a system is:
(examples include a mathematical pendulum at small angles of deflection and an ideal oscillatory circuit). Let us solve equation (1.1) in detail using classical method Euler. We are looking for a particular solution in the form:
where and are constants, as yet unknown constants. Let's substitute (1.2) into equation (1.1)
Let's divide both sides of the equation by and get an algebraic, so-called characteristic, equation:
The roots of this equation
where is the imaginary unit. The roots are imaginary and complex conjugate.
As is known, the general solution is the sum of the partial ones, i.e.
We believe that there is a real value. For this to work, the constants and must be complex conjugate, i.e.
Two constants are determined from two initial conditions:
The solution in the form (1.8) is mainly used in theory; for applied tasks it is not convenient, since it is not measured. Let's move on to the form of the solution that is most commonly used in practice. Let us represent complex constants in polar form:
Let's substitute them into (1.8) and use Euler's formula
where is the oscillation amplitude and is the initial phase.
And they are determined from the initial conditions. Note that the initial phase depends on the origin of time. Indeed, the constant can be represented as:
If the origin of time coincides with , the initial phase is zero. For a harmonic oscillation, the phase shift and time shift are equivalent.
Let us decompose the cosine in (1.13) into cosine and sinusoidal components. Let's get another idea:
If they are known, then it is not difficult to find the amplitude and phase of the oscillation using the following relations:
All three notations (1.8, 1.12, 1.15) are equivalent. The use of a specific form is determined by the convenience of considering a specific task.
Analyzing the solution, we can say that the natural oscillations of a harmonic oscillator are a harmonic oscillation, the frequency of which depends on the parameters of the system and does not depend on the initial conditions; The amplitude and initial phase depend on the initial conditions.
Independence of the initial conditions of the frequency (period) of natural oscillations is called isochoric.
Let's consider the energy of a harmonic oscillator using an oscillatory circuit as an example. Equation of motion in a circuit
Let's multiply the terms of this equation by:
After transformation it can be represented as:
Let's find the law of energy change in a capacitor. The current in the capacitive branch can be found using the following expression
Substituting (1.28) into the formula for finding electrical energy, we obtain the law of change in electrical energy on the capacitor
Thus, the energy in each element of the circuit oscillates at twice the frequency. The graph of these fluctuations is shown in Fig. 6.
At the initial moment of time, all the energy is concentrated in the container, the magnetic energy is equal to zero. As the capacitance discharges through the inductance, electrical energy from the capacitance is converted into magnetic energy from the inductance. After a quarter of the period, all the energy is concentrated in the inductance, i.e. The container is completely discharged. This process is then repeated periodically.
Thus, an oscillation in an ideal circuit is a transition of electrical energy into magnetic energy and vice versa, periodically repeating in time.
This conclusion is valid for any electromagnetic oscillatory systems, in particular for volumetric resonators, where magnetic and electrical energy are not spatially separated.
Summarizing this result, we can say that oscillatory process in a linear conservative system, this is a periodic transition of energy of one type to another. Thus, when a pendulum oscillates, kinetic energy transforms into potential energy and vice versa.
