Constant avogadro physics. Avogadro's constant

The simplest model oscillatory motion atoms in a diatomic molecule can serve as a system of two masses T/ and w?, connected by an elastic spring. The vibration of two atoms relative to the center of mass can be replaced by the vibration of one equivalent

mass relative to the initial zero point R= 0, where

R- distance between masses, R e- position of the equilibrium point.

In classical consideration, it is assumed that the spring is ideal - elastic force F is directly proportional to deformation - deviation from equilibrium x = R-R e, according to Hooke's law:

Where To- elasticity constant. Thus, the force is directed towards returning to the equilibrium position.

Using Hooke's and Newton's laws together (F-ta), can be written:

(denoting ). The solution to such an equation is known to be

serve harmonic functions

Where xo- amplitude, and

Using the reduced mass /l we get:

A measure of the potential energy of a system V serves work

IN quantum mechanics vibration analysis for a simple model harmonic oscillator quite complicated. It is based on solving the Schrödinger equation

(y/- vibrational wave function, E - total energy particles) and is beyond the scope of our presentation.

For a quantum oscillator it is only possible discrete series values ​​of energy E and frequencies in accordance with the formula E=hv. Besides, minimum value oscillator energy is not zero. This quantity is called zero energy, it corresponds to the lowest energy level of the oscillator and is equal to , its existence can be explained based on the Heisenberg uncertainty relation.

Thus, in accordance with quantum mechanics the energy of the harmonic oscillator is quantized:

Where v- oscillatory quantum number, which can take the value y=0, 1, 2, 3,....

When an oscillator interacts with quanta electromagnetic radiation three factors should be taken into account: 1) population of levels (probability of finding a molecule at a given energy level); 2) the frequency rule (Bohr), according to which the energy of a quantum must correspond to the difference in the energy of any two levels;

3) selection rule for quantum transitions: transition probability, i.e. the intensity of the lines in the absorption spectrum is determined by the quantity transition dipole moment (see theoretical introduction). In the case of the simplest harmonic oscillator, the selection rule is obtained from considering the wave functions. It states that transitions can only occur between adjacent levels (“one step”): the vibrational quantum number changes by one Av= 1. Since the distances between adjacent levels are the same, the absorption spectrum of a harmonic oscillator should contain only one line with a frequency

Since, in accordance with the Boltzmann distribution at room temperature and more low temperatures the lowest vibrational level is populated, then the most intense transition is from the very low level(d=0), and the frequency of this line coincides with the frequency of weaker transitions from higher levels to the adjacent, higher level.

Graphs of harmonic oscillator wave functions for different meanings energies are shown in Figure 2.3. They represent solutions of the Schrödinger equation for a harmonic oscillator

Where N, - normalizing factor, H 0- Hermite polynomials, x = R-R e- deviation from the equilibrium position.

Transition dipole moment for vibrational transitions, R0(or M„) equal to:

Where ju - dipole moment molecules; hesitation

body wave functions initial and final states, respectively. From the formula it is clear that the transition is allowed,

if at the equilibrium point - the dipole moment of the molecule

changes near the position of the equilibrium point, (curve ju=f(R) does not pass through the maximum at this point). The integral (the second factor in the formula) must also not be equal to zero. It can be shown that this condition is met if the transition occurs between adjacent levels, hence additional rule selection Ai = 1.

When diatomic molecules vibrational spectra can be observed only for heteronuclear molecules; homonuclear molecules have no dipole moment and do not change during vibrations. The vibrational spectra of CO2 exhibit vibrations (antisymmetric stretching and bending), in which the dipole moment changes, but symmetric vibrations, in which it remains unchanged, do not appear.

Bodies that, when moving, perform harmonic oscillations are called harmonic oscillators. Let's look at a number of examples of harmonic oscillators.

Example 1. A spring pendulum is a body of massm, capable of oscillating under the action of a weightless elastic force (m springs  m body ) springs (Fig. 4.2).

T

Fig.4.3. Physical pendulum.

Example 2. A physical pendulum is solid, oscillating under the influence of gravity around a moving horizontal axis that does not coincide with its center of gravity C (Fig. 4. 3). The axis passes through point O. If the pendulum is deflected from the equilibrium position by a small angle  and released, it will oscillate, following the basic equation of dynamics rotational movement solid
, Where J- moment of inertia pendulum relative to the axis, M is the moment of force restoring physical pendulum to a position of equilibrium. It is created by gravity, its moment is equal to
(l=OS). As a result we get
. This differential equation fluctuations for arbitrary angles deviations. At small angles, when
,
or, taking
, we obtain the differential equation of oscillation of a physical pendulum
. Its solutions have the form
or
. Thus, for small deviations from the equilibrium position, the physical pendulum performs harmonic oscillations with a cyclic frequency
and period
.

Example 3. A mathematical pendulum is a material point with massm(a heavy ball of small size), suspended on a weightless (compared tomball), elastic, inextensible thread longl. If you remove the ball from its equilibrium position by deflecting it from the vertical by a small angle , and then release it, it will oscillate. If we consider this system as a physical pendulum with moment of inertia of a material point J = ml 2, then from the formulas for a physical pendulum we obtain expressions for the cyclic frequency and period of oscillation of a mathematical pendulum

,
.

4. 4. Damped oscillations. @

In the considered examples of harmonic oscillations, the only force acting on material point(body), was quasi-elastic force F and did not take into account the resistance forces that are present in any real system. Therefore, the considered oscillations can be called ideal undamped harmonic oscillations.

Availability in real oscillatory system the resistance force of the environment leads to a decrease in the energy of the system. If the loss of energy is not replenished through the work of external forces, the oscillations will die out. Damped oscillations are those whose amplitude decreases with time.

Let us consider free damped oscillations. At low speeds, the drag force F C is proportional to the speed v and inversely proportional to it in direction
, where r - drag coefficient environment. Using Newton's second law, we obtain the differential equation damped oscillations
,
,
. Let's denote
,
. Then the differential equation takes the form:

Fig.4.4. Dependence of the displacement and amplitude of damped oscillations on time.

This is a differential equation of damped oscillations. Here  0 is the natural frequency of oscillations of the system, i.e. the frequency of free oscillations at r=0,  - the damping coefficient determines the rate of decrease in the amplitude. The solutions to this equation under the condition  0 are

or
.

The graph of the last function is shown in Fig. 4.4. The upper dotted line gives the graph of the function
, And 0 is the amplitude at the initial moment of time. The amplitude decreases over time according to an exponential law,  - the attenuation coefficient is inverse in magnitude relaxation time , i.e. time during which the amplitude decreases by e times, since

,
, = 1, . Frequency and period of damped oscillations
,
; at very low resistance of the medium ( 2  0 2), the oscillation period is almost equal to
. As  increases, the period of oscillation increases and at > 0, the solution of the differential equation shows that oscillations do not occur, but that the system moves monotonously towards the equilibrium position. This kind of motion is called aperiodic.

To characterize the rate of attenuation of oscillations, two more parameters are used: the damping decrement D and logarithmic decrement . The damping decrement shows how many times the oscillation amplitude decreases during one period T.

N

Fig.4.5. Type of resonance curves.

Because , That
, where N is the number of oscillations per time.

Harmonic oscillator

Harmonic oscillator(in classical mechanics) - a system that, when displaced from an equilibrium position, experiences a restoring force F, proportional to the displacement x(according to Hooke's law):

Where k- system rigidity coefficient.

If F is the only force acting on the system, then the system is called simple or conservative harmonic oscillator. Free vibrations of such a system are periodic motion near the equilibrium position ( harmonic vibrations). The frequency and amplitude are constant, and the frequency does not depend on the amplitude.

Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small angles of deflection), a torsion pendulum, and acoustic systems. Among other analogues of a harmonic oscillator, it is worth highlighting the electric harmonic oscillator (see LC circuit).

Free vibrations

Conservative harmonic oscillator

As a model of a conservative harmonic oscillator, we take a mass load m, fixed to the spring by rigidity k .

Let x- displacement of the load relative to the equilibrium position. Then, according to Hooke's law, a restoring force will act on it:

Then total energy has a constant value

Simple harmonic motion - this is the movement of a simple harmonic oscillator, periodic motion that is neither forced nor damped. A body in simple harmonic motion is exposed to a single variable force, which is directly proportional to the displacement in magnitude x from the equilibrium position and is directed in the opposite direction.

This movement is periodic: the body oscillates around the equilibrium position according to a sinusoidal law. Each subsequent oscillation is the same as the previous one, and the period, frequency and amplitude of the oscillations remain constant. If we assume that the equilibrium position is at a point with coordinate, equal to zero, then the offset x body from the equilibrium position at any time is given by the formula:

Where A- amplitude of oscillations, f- frequency, φ - initial phase.

The frequency of movement is determined characteristic properties system (for example, the mass of a moving body), while the amplitude and initial phase are determined by the initial conditions - the displacement and speed of the body at the moment the oscillations begin. The kinetic and potential energies of the system also depend on these properties and conditions.

Simple harmonic motion can be mathematical models various types movements such as the oscillation of a spring. Other cases that can be roughly considered as simple harmonic motion are the motion of a pendulum and the vibration of molecules.

Simple harmonic motion is the basis of some ways of analyzing more complex types of motion. One of these methods is the method based on the Fourier transform, the essence of which boils down to the expansion of more complex type movements into a series of simple harmonic movements.

Any system in which simple harmonic motion occurs has two key properties:

1. When a system is thrown out of equilibrium, there must be a restoring force that tends to return the system to equilibrium.
2. The restoring force must be exactly or approximately proportional to the displacement.

The load-spring system satisfies both of these conditions.

Once a displaced load is subjected to a restoring force, it accelerates and tends to return to its original position. starting point, that is, to the equilibrium position. As the load approaches the equilibrium position, the restoring force decreases and tends to zero. However, in the situation x = 0 the load has a certain amount of motion (impulse), acquired due to the action of the restoring force. Therefore, the load overshoots the equilibrium position, beginning to deform the spring again (but already in opposite direction). The restoring force will tend to slow it down until the speed becomes zero; and the force will again strive to return the load to its equilibrium position.

As long as there is no energy loss in the system, the load will oscillate as described above; such a movement is called periodic.

Further analysis will show that in the case of a load-spring system, the motion is simple harmonic.

Dynamics of simple harmonic motion

For vibrations in one-dimensional space, taking into account Newton's Second Law ( F= m  d² x/d t² ) and Hooke's law ( F = −kx, as described above), we have a second-order linear differential equation:

The solution to this differential equation is sinusoidal; one solution is:

Where A, ω and φ - constants, and the equilibrium position is taken as the initial one. Each of these constants represents an important physical property movements: A is the amplitude, ω = 2π f- circular frequency, and φ - initial phase.

Universal circular motion

Simple harmonic motion can in some cases be considered as a one-dimensional projection of universal circular motion. If an object moves with a constant angular velocity ω along a circle of radius r, the center of which is the origin of the plane x−y, then such a movement along each of coordinate axes is simple harmonic with amplitude r and circular frequency ω.

A weight like a simple pendulum

In the approximation of small angles, the motion of a simple pendulum is close to simple harmonic. The period of oscillation of such a pendulum attached to a rod of length ℓ with acceleration free fall g is given by the formula

This shows that the period of oscillation does not depend on the amplitude and mass of the pendulum, but depends on the acceleration of gravity g, therefore, with the same length of the pendulum, on the Moon it will swing more slowly, since gravity is weaker there and less value free fall acceleration.

This approximation is correct only for small deflection angles, since the expression for angular acceleration is proportional to the sine of the coordinate:

what is he doing angular acceleration directly proportional to the angle θ, and this satisfies the definition of simple harmonic motion.

Damped harmonic oscillator

Taking the same model as a basis, we will add the force of viscous friction to it. The force of viscous friction is directed against the speed of movement of the load relative to the medium and is proportional to this speed. Then full strength, acting on the load, is written as follows:

Carrying out similar actions, we obtain a differential equation describing damped oscillator:

Here the designation is introduced: . The coefficient is called the attenuation constant. It also has the dimension of frequency.

The solution breaks down into three cases.

Critical damping is noteworthy in that it is at critical damping that the oscillator most quickly tends to the equilibrium position. If the friction is less than critical, it will reach the equilibrium position faster, but will “overshoot” it due to inertia and will oscillate. If the friction is greater than critical, then the oscillator will exponentially tend to the equilibrium position, but the more slowly, the greater the friction.

Therefore, in dial indicators (for example, in ammeters), they usually try to introduce critical attenuation so that its readings can be read as quickly as possible.

The damping of an oscillator is also often characterized by a dimensionless parameter called the quality factor. Quality factor is usually denoted by the letter . By definition, the quality factor is equal to:

The higher the quality factor, the slower the oscillator oscillations decay.

An oscillator with critical damping has a quality factor of 0.5. Accordingly, the quality factor indicates the behavior of the oscillator. If the quality factor is greater than 0.5, then the free movement of the oscillator represents oscillations; Over time, it will cross the equilibrium position an unlimited number of times. A quality factor less than or equal to 0.5 corresponds to non-oscillatory motion of the oscillator; V free movement it will cross the equilibrium position at most once.

The quality factor is sometimes called the gain factor of the oscillator, since with some methods of excitation, when the excitation frequency coincides with the resonant one, the amplitude of oscillations turns out to be approximately times greater than when excited at a low frequency.

Also, the quality factor is approximately equal to the number of oscillatory cycles during which the oscillation amplitude decreases by a factor, multiplied by .

In the case of oscillatory motion, damping is also characterized by such parameters as:

• Lifetime vibrations (aka decay time, it's the same relaxation time) τ - time during which the amplitude of oscillations will decrease in e once.

Forced vibrations

Oscillator oscillations are called forced when some additional external influence is applied to it. This effect can be produced by various means and by various laws. For example, force excitation is the effect on a load of a force that depends only on time according to a certain law. Kinematic excitation is the effect on the oscillator by the movement of the spring attachment point along given law. It is also possible to be affected by friction, when, for example, the medium with which the load experiences friction moves according to a given law.