Maxwellian velocity distribution of molecules. Maxwell distribution function

Lecture 5

As a result of numerous collisions of gas molecules with each other (~10 9 collisions per 1 second) and with the walls of the vessel, a certain statistical distribution molecules by speed. In this case, all directions of the molecular velocity vectors turn out to be equally probable, and the velocity modules and their projections on coordinate axes obey certain laws.

During collisions, the speeds of molecules change randomly. It may turn out that one of the molecules in a series of collisions will receive energy from other molecules and its energy will be significantly greater than the average energy value at a given temperature. The speed of such a molecule will be high, but still it will have final value, since the maximum possible speed– speed of light - 3·10 8 m/s. Consequently, the speed of a molecule can generally have values ​​from 0 to some υ max. It can be argued that very high speeds compared to average values ​​are rare, as are very small ones.

As theory and experiments show, the distribution of molecules by speed is not random, but quite definite. Let's determine how many molecules, or what part of the molecules have velocities that lie in a certain interval near a given speed.

Let a given mass of gas contain N molecules, while dN molecules have velocities ranging from υ before υ +dυ. Obviously this is the number of molecules dN proportional to the total number of molecules N and the value of the specified speed interval dυ

Where a- proportionality coefficient.

It is also obvious that dN depends on the speed υ , since in intervals of the same size, but at different absolute values ​​of speed, the number of molecules will be different (example: compare the number of people living at the age of 20 - 21 years and 99 - 100 years). This means that the coefficient a in formula (1) must be a function of speed.

Taking this into account, we rewrite (1) in the form

From (2) we get

Function f(υ ) is called the distribution function. Her physical meaning follows from formula (3)

Hence, f(υ ) is equal to the relative fraction of molecules whose velocities are contained in a unit velocity interval near the velocity υ . More precisely, the distribution function has the meaning of the probability of any gas molecule having a speed contained in unit interval near speed υ . That's why they call her probability density.

Integrating (2) over all speed values ​​from 0 to we obtain

From (5) it follows that

Equation (6) is called normalization condition functions. It determines the probability that a molecule has one of the speed values ​​from 0 to . The speed of a molecule has some meaning: this event is reliable and its probability is equal to one.

Function f(υ ) was found by Maxwell in 1859. She was named Maxwell distribution:

Where A– coefficient that does not depend on speed, m– molecular mass, T– gas temperature. Using the normalization condition (6) we can determine the coefficient A:

Taking this integral, we get A:

Taking into account the coefficient A The Maxwell distribution function has the form:

When increasing υ the factor in (8) changes faster than it grows υ 2. Therefore, the distribution function (8) starts at the origin, reaches a maximum at a certain speed value, then decreases, asymptotically approaching zero (Fig. 1).

Fig.1. Maxwellian distribution of molecules

by speed. T 2 > T 1

Using the Maxwell distribution curve, you can graphically find the relative number of molecules whose velocities lie in a given velocity range from υ before dυ(Fig. 1, area of ​​the shaded strip).

Obviously, the entire area under the curve gives total number molecules N. From equation (2), taking into account (8), we find the number of molecules whose velocities lie in the range from υ before dυ

From (8) it is also clear that specific type distribution function depends on the type of gas (molecule mass m) and on temperature and does not depend on the pressure and volume of the gas.

If isolated system remove it from a state of equilibrium and leave it to itself, then after a certain period of time it will return to a state of equilibrium. This period of time is called relaxation time. For various systems it is different. If the gas is in equilibrium state, then the distribution of molecules by speed does not change over time. The speeds of individual molecules are constantly changing, but the number of molecules dN, whose speeds lie in the range from υ before dυ remains constant all the time.

The Maxwellian velocity distribution of molecules is always established when the system reaches a state of equilibrium. The movement of gas molecules is chaotic. Precise definition The randomness of thermal movements is as follows: the movement of molecules is completely chaotic if the velocities of the molecules are distributed according to Maxwell. It follows that the temperature is determined by the average kinetic energy namely chaotic movements. No matter how high the speed strong wind, she won't make him "hot". Even the strongest wind can be both cold and warm, because the temperature of the gas is determined not by the directional speed of the wind, but by the speed of the chaotic movement of molecules.

From the graph of the distribution function (Fig. 1) it is clear that the number of molecules whose velocities lie in the same intervals d υ , but close different speeds υ , more if the speed υ approaches the speed that corresponds to the maximum of the function f(υ ). This speed υ n is called the most probable (most probable).

Let us differentiate (8) and equate the derivative to zero:

then the last equality is satisfied when:

Equation (10) is satisfied when:

The first two roots correspond minimum values functions. Then we find the speed that corresponds to the maximum of the distribution function from the condition:

From the last equation:

Where R– universal gas constant, μ - molar mass.

Taking (11) into account, from (8) we can obtain maximum value distribution functions

From (11) and (12) it follows that with increasing T or when decreasing m curve maximum f(υ ) shifts to the right and becomes smaller, but the area under the curve remains constant (Fig. 1).

To solve many problems, it is convenient to use the Maxwell distribution in its reduced form. Let's introduce relative speed:

Where υ – given speed, υ n- the most probable speed. Taking this into account, equation (9) takes the form:

(13) is a universal equation. In this form, the distribution function does not depend on the type of gas or temperature.

Curve f(υ ) is asymmetrical. From the graph (Fig. 1) it is clear that most of molecules have speeds greater than υ n. The asymmetry of the curve means that the arithmetic mean speed of the molecules is not equal υ n. The arithmetic average speed is equal to the sum of the speeds of all molecules divided by their number:

Let us take into account that according to (2)

Substituting into (14) the value f(υ ) from (8) we obtain the arithmetic average speed:

The average square of the speed of molecules is obtained by calculating the ratio of the sum of the squares of the speeds of all molecules to their number:

After substitution f(υ ) from (8) we obtain:

From the last expression we find the root mean square speed:

Comparing (11), (15) and (16) we can conclude that and equally depend on temperature and differ only numerical values: (Fig. 2).

Fig.2. Maxwell distribution by absolute values speeds

The Maxwell distribution is valid for gases in a state of equilibrium; the number of molecules under consideration must be sufficiently large. For a small number of molecules, significant deviations from the Maxwell distribution (fluctuations) can be observed.

The first experimental determination of molecular velocities was carried out by Stern in 1920. Stern's device consisted of two cylinders of different radii mounted on the same axis. The air from the cylinders was pumped out to a deep vacuum. A platinum thread covered with thin layer silver When passed along a thread electric current she heated up to high temperature(~1200 o C), which led to the evaporation of silver atoms.

A narrow longitudinal slit was made in the wall of the inner cylinder, through which moving silver atoms passed. Deposited on the inner surface of the outer cylinder, they formed a clearly visible thin strip directly opposite the slot.

The cylinders began to rotate at a constant angular velocityω. Now the atoms that passed through the slit no longer settled directly opposite the slit, but were displaced by a certain distance, since during their flight the outer cylinder had time to rotate through a certain angle. When the cylinders rotate with constant speed, strip position, formed by atoms on the outer cylinder, shifted by some distance l.

Particles settle at point 1 when the installation is stationary; when the installation rotates, particles settle at point 2.

The obtained speed values ​​confirmed Maxwell's theory. However, this method provided approximate information about the nature of the velocity distribution of molecules.

The Maxwell distribution has been more accurately verified by experiments Lammert, Easterman, Eldridge and Costa. These experiments quite accurately confirmed Maxwell's theory.

Direct measurements of the velocity of mercury atoms in a beam were made in 1929 Lammert. A simplified diagram of this experiment is shown in Fig. 3.

Fig.3. Diagram of Lammert's experiment
1 - rapidly rotating disks, 2 - narrow slits, 3 - oven, 4 - collimator, 5 - trajectory of molecules, 6 - detector

Two disks 1, mounted on a common axis, had radial slots 2, shifted relative to each other at an angle φ . Opposite the slots there was furnace 3, in which fusible metal was heated to a high temperature. Heated metal atoms, in in this case mercury, flew out of the furnace and, using collimator 4, were directed in the required direction. The presence of two slits in the collimator ensured the movement of particles between the disks along straight path 5. Next, the atoms that passed through the slits in the disks were recorded using detector 6. The entire described installation was placed in deep vacuum.

When the disks rotated with a constant angular velocity ω, only atoms that had a certain speed passed through their slits freely υ . For atoms passing through both slits the equality must be satisfied:

where Δ t 1 - time of flight of molecules between disks, Δ t 2 - time to turn the disks at an angle φ . Then:

By changing the angular speed of rotation of the disks, it was possible to isolate molecules with a certain speed from the beam υ , and from the intensity recorded by the detector, judge their relative content in the beam.

In this way, it was possible to experimentally verify Maxwell's law of molecular velocity distribution.

Since in a state of equilibrium the pressure in all parts of the system is the same, it is natural to assume that there are no directed movements of molecules in the gas, that is, the movements of the molecules are extremely disordered.

In terms of molecular speeds this means:

The speed of molecules and its projections are continuous quantities, since no speed value has an advantage over other values;

At thermal equilibrium in a gas, all directions of molecular velocities are equally probable. Otherwise, this would lead to the formation of directed macroscopic flows of molecules and the occurrence of pressure drops.

Since the speed and its projections are continuous quantities, the concept of the distribution density function f(v x), f(v y), f(v z) is introduced over the molecular velocity components (v x, v y, v z) and over the velocity modulus f(v)

Expressions for the probability density functions for the velocity components v x , v y , v z have the form

The graph of the function f(v x) is shown in Fig. 1.

The function has a maximum at v x = 0, is symmetrical with respect to it and exponentially tends to zero at v x ® ± ¥. Let us plot along the abscissa axis the elementary speed intervals dv x around the values ​​of v x equal to 0; ± v x ¢; ± v x ¢¢. The product f(v x) dv x is equal to the fraction of molecules whose velocity component v x lies in the range of about specified values. On the other hand, the product f(v x) dv x on the graph is equal to the shaded areas around the selected speeds.

From a comparison of the sizes of the shaded areas it follows:

The relative majority of molecules have a velocity projection along the v x axis close to zero;

Proportions of molecules having same values v x , but flying in opposite directions (different signs+v x and -v x) are the same;

Number of molecules having large values velocity components are small (small area about ± v x ¢¢).

A similar analysis can be carried out for f(v y), f(v z).

The graph of the function f(v) is shown in Fig. 2.

The function is equal to 0 at v = 0; tends to zero at v ® ¥, at v = v b has a maximum. The value of the speed v b at which the distribution density function reaches its maximum is called the most probable speed. Its value is found from the extremum condition.

The product f(v) dv gives the fraction of molecules whose velocities lie in the selected interval dv. On the graph, this product is equal to the shaded areas. As can be seen from the graph, the maximum area corresponds to the speed v b. With increasing speed, the proportion of molecules with high speeds decreases ( small area at v 3). Knowing analytical view f(v), can be found

The distribution of molecules by speed depends on temperature.

Maxwell's law of the distribution of gas molecules by speed describes the behavior of a very large number of particles, that is, it is a statistical law. The velocity distribution of molecules is established through their collisions. During collisions, the velocities of individual molecules change, but the law of velocity distribution does not change.

The characteristic parameters of the Maxwell distribution are the most probable speed υ in, corresponding to the maximum of the distribution curve, and the root mean square speed where is the average value of the square of the speed.

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§4 Maxwell's law on the distribution of velocities and energies

Molecular distribution law ideal gas by speed, theoretically obtained by Maxwell in 1860 determines what number dN homogeneous molecules (p= const) of a monatomic ideal gas from the totalNof its molecules per unit volume at a given temperature T speeds in the range fromv before v+ dv.

To derive the molecular velocity distribution functionf( v) equal to the ratio number of molecules dN, whose velocities lie in the intervalv÷ v+ dvto the total number of moleculesNand the size of the intervaldv

Maxwell used two sentences:

a) all directions in space are equal and therefore any direction of particle movement, i.e. any direction of speed is equally probable. This property is sometimes called the isotropy property of the distribution function.

b) movement along three mutually perpendicular axes are independent, i.e. x-components of velocitydoes not depend on the values ​​of its components or . And then the conclusion f ( v) done first for one component, and then generalized to all velocity coordinates.

It is also believed that the gas consists of a very large numberN identical molecules in a state of random thermal motion at the same temperature. Force fields Doesn't work on gas.

Functions f ( v) determines the relative number of moleculesdN( v)/ Nwhose speeds lie in the range fromv before v+ dv(for example: gas hasN= 10 6 molecules, whiledN= 100

molecules have speeds fromv=100 to v+ dv=101 m/s ( dv = 1 m) then .

Using methods of probability theory, Maxwell found the functionf ( v) - law of distribution of ideal gas molecules by speed:

f ( v) depends on the type of gas (on the mass of the molecule) and on the state parameter (on temperature T)

f( v) depends on attitude kinetic energy molecule corresponding to the considered speed to the size kTcharacterizing the average thermal energy gas molecules.

At small v and function f( v) changes almost parabolically. As v increases, the multiplier decreases faster than the multiplier increases, i.e. there are max functions f( v) . The speed at which the speed distribution function of ideal gas molecules is maximum is called most likely speed we find from the condition

Therefore, as the temperature increases, the most probable speed increases, but the area S, limited by the distribution function curve remains unchanged, since from the normalization condition(since the probability of a reliable event is 1), therefore, with increasing temperature, the distribution curvef ( v) will stretch and shrink.

IN statistical physics the average value of a quantity is defined as the integral from 0 to infinity of the product of the quantity by the probability density of this quantity (statistical weight)

< X >=

Then the arithmetic mean speed of molecules

And integrating by parts we got

Velocities characterizing the state of gas

§5 Experimental verification of Maxwell's distribution law - Stern's experiment

A platinum wire covered with a layer of silver, which is heated by current, is stretched along the axis of the inner cylinder for the purpose. When heated, silver evaporates, silver atoms fly out through the slit and land on inner surface second cylinder. If both cylinders are motionless, then all the atoms, regardless of their speed, end up in the same place B. When the cylinders rotate with an angular velocity ω, the silver atoms fall into points B’, B '' and so on. By magnitude ω, distance? and displacement X= BB', you can calculate the speed of the atoms hitting point B'.

The image of the slit appears blurry. By examining the thickness of the deposited layer, it is possible to estimate the velocity distribution of molecules, which corresponds to the Maxwellian distribution.

§6 Barometric formula

Boltzmann distribution

So far we have considered the behavior of an ideal gas not affected by external forces new fields. It is well known from experience that under the action of external forces the uniform distribution of particles in space can be disrupted. So, under the influence of gravity, the molecules tend to sink to the bottom of the vessel. Intense thermal motion prevents sedimentation, and the molecules spread out so that their concentration gradually decreases as altitude increases.

Let us derive the law of pressure change with height, assuming that the gravitational field is uniform, the temperature is constant and the mass of all molecules is the same. If Atmosphere pressure on high h is equal to p, then at the height h + dh it is equal p + dp(at dh > 0, dp < 0, так как pdecreases with increasingh).

Pressure difference at altitudesh And h+ dhwe can define it as the weight of air molecules enclosed in a volume with a base area of ​​1 and heightdh.

density at altitudeh, and since , then = const .

Then

From the Mendeleev-Clapeyron equation.

Then

Or

With a change in height fromh 1 before h 2 pressure varies fromp 1 before p 2

Let's potentiate this expression (

The barometric formula shows how pressure changes with altitude

At

Then

Because

That

n h,

n 0 concentration of molecules at heighth =0.

Because

That

potential energy molecules in a gravitational field

Boltzmann distribution in an external potential field. It follows from this that whenT= const The gas density is greater where the potential energy of the molecules is less.

§7 Experienced determination Avogadro's constant

J. Perrin (French scientist) in 1909 studied the behavior of Brownian particles in a gum gum emulsion (tree sap) with dimensions examined using a microscope that had a field depth of 1 μm. By moving the microscope in the vertical direction, it was possible to study the height distribution of Brownian particles.

Applying the Boltzmann distribution to them, we can write

n = - where m is the mass of the particle

m - mass of displaced liquid:

If n 1 and n 2 concentration of particles at levels h 1 and h 2, and k = R / N A, then

N A =

The value agrees well with the reference value, which confirms the Boltzmann particle distribution

The movement of gas molecules obeys the laws of statistical physics. On average, the speeds and energies of all molecules are the same. However, at any given time, the energy and speed of individual molecules can differ significantly from the average value.

By using probability theory Maxwell managed to derive a formula for relative frequency, with which molecules occur in a gas at a given temperature with velocities in a certain range of values.

Maxwell's distribution law determines the relative number of molecules dN/N, whose velocities lie in the interval ( u, u + du).

It looks like:

Where N- total number of gas molecules; - the number of molecules whose velocities are within a certain range; u is the lower limit of the speed interval; d u is the value of the speed interval; T- gas temperature; e= 2.718… - the base of natural logarithms;

k= 1.38×10 -23 J/K - Boltzmann constant; m 0 is the mass of the molecule.

In obtaining this formula, Maxwell was based on the following assumptions:

1. Gas consists of a large number N identical molecules.

2. The gas temperature is constant.

3. Gas molecules undergo thermal chaotic motion.

4. Gas is not affected by force fields.

Note, that under the exponential sign in formula (8.29) is the ratio of the kinetic energy of the molecule to the quantity kT, which characterizes the average (over molecules) value of this energy.

The Maxwell distribution shows what fraction dN/N of the total number of molecules of a given gas has a speed in the range from u to u + du.

Graph of distribution functions (Fig. 8.5) asymmetrical. The maximum position characterizes the most frequently occurring speed, which is called the most probable speed u m. Speeds exceeding u m, are more common than lower speeds. With increasing temperature, the maximum of the distribution shifts towards higher velocities.

At the same time, the curve becomes flatter (the area under the curve cannot change, since the number of molecules N remains constant).

Rice. 8.5

To determine the most probable speed, you need to examine the Maxwell distribution function to its maximum (equate the first derivative to zero and solve for u). As a result we get:

We have omitted factors that do not depend on u. Having carried out differentiation, we arrive at the equation:

The first factor (exponent) vanishes at u = ¥, and the third factor (u) at u = 0. However, from the graph (Fig. 8.5) it is clear that the values ​​of u = 0 and u = ¥ correspond to the minima of function (8.29). Therefore, the value u, corresponding to the maximum, is obtained from the equality to zero of the second bracket: . From here

Let us introduce notation for the velocity distribution function of molecules (8.29):

It is known that the average value of some physical quantity j( x) can be calculated using the formula:

From (8.32) we obtain expressions for the average value of the velocity modulus u and the average value of the square u:

Thus, average speed molecules (also called the arithmetic mean speed) has the meaning:

The square root of expression (8.34) gives the root mean square speed of molecules:

Note, that it coincides with formula (8.24). In Fig. Figure 8.5 shows a graph of the Maxwell distribution function. Vertical lines three characteristic speeds are noted.

Statistical distributions

During thermal motion, the positions of particles, the magnitude and direction of their velocities change randomly. Due to the gigantic number of particles, the random nature of their movement is manifested in the existence of certain statistical patterns in the distribution of particles of the system according to coordinates, velocity values, etc. Such distributions are characterized relevant functions distributions. The distribution function (probability density) characterizes the distribution of particles according to the corresponding variable (coordinates, velocities, etc.). At the core classical statistics the following provisions lie:

All particles classical system distinguishable (i.e. they can be numbered and each particle can be tracked);

All dynamic variables characterizing the state of the particle change continuously;

An unlimited number of particles can exist in a given state.

Able thermal equilibrium no matter how the speeds of molecules change during collisions, the root mean square speed of molecules in a gas, at T=const, remains constant and equal

This is explained by the fact that a certain stationary statistical distribution of molecules over velocity values, called the Maxwell distribution, is established in the gas. The Maxwell distribution is described by some function f(u), called the molecular velocity distribution function.

where N is the total number of molecules, dN(u) is the number of molecules whose velocities belong to the velocity range from u to u + du.

Thus, the Maxwell function f(u) is equal to the probability that the velocity value of a randomly selected molecule belongs to a unit velocity interval near the value u. Or it is equal to the fraction of molecules whose velocities belong to a unit velocity interval near the value of u.

The explicit form of the function f(u) was obtained theoretically by Maxwell:

The distribution function graph is shown in Fig. 12. From the graph it follows that the distribution function tends to zero at u®0 and u®¥ and passes through a maximum at a certain speed u B, called most likely speed. Has this speed and close to it greatest number molecules. The curve is asymmetrical with respect to u B. The value of the most probable speed can be found using the condition for the maximum of the function f(u).

In Fig. Figure 13 shows the displacement of uB with a change in temperature, while the area under the graph remains constant and equal to 1, which follows from normalization conditions Maxwell functions

The normalization condition follows from the meaning of this integral - it determines the probability that the speed of a molecule falls in the speed range from 0 to ¥. This reliable event, its probability, by definition, is taken equal to 1.

Knowing the velocity distribution function of gas molecules allows one to calculate the average values ​​of any velocity functions, in particular arithmetic average speed .