Chaotic movement in an external field. Gas in a potential field. Barometric formula. Boltzmann's law for the distribution of particles in an external potential field. Gas work during expansion

Let an ideal gas be in a field of conservative forces under conditions of thermal equilibrium. In this case, the gas concentration will be different at points with different potential energy, which is necessary to comply with the conditions of mechanical equilibrium. So, the number of molecules in a unit volume n decreases with distance from the Earth's surface, and the pressure, due to the relation P = nkT, falls.

If the number of molecules in a unit volume is known, then the pressure is also known, and vice versa. Pressure and density are proportional to each other, since the temperature in our case is constant. The pressure must increase as the height decreases, because the bottom layer has to support the weight of all the atoms located on top.

Based on the basic molecular equation kinetic theory: P = nkT, replace P And P0 in the barometric formula (2.4.1) at n And n 0 and we get Boltzmann distribution for molar mass of gas:

Since a , then (2.5.1) can be represented in the form

Figure 2.11 shows the dependence of the concentration of various gases on altitude. It can be seen that the number of heavier molecules decreases faster with height than light ones.

Boltzmann proved that relation (2.5.3) is valid not only in the potential field of gravitational forces, but also in any potential field, for a collection of any identical particles in a state of chaotic thermal motion.

Boltzmann's law for the distribution of particles in an external potential field

MOLECULAR PHYSICS AND THERMODYNAMICS

BOLZMANN (Boltzmann) Ludwig(1844-1906), Austrian physicist, one of the founders of statistical physics and physical kinetics, foreign corresponding member of the St. Petersburg Academy of Sciences (1899). He derived the distribution function named after him and the basic kinetic equation of gases. Gave (1872) a statistical substantiation of the second law of thermodynamics. He derived one of the laws of thermal radiation (Stefan-Boltzmann law).

Due to chaotic motion, changes in the position of each particle (molecule, atom, etc.) physical system(macroscopic body) are of the nature random process. Therefore, we can talk about the probability of detecting a particle in a particular region of space.

From kinematics it is known that the position of a particle in space is characterized by its radius vector or coordinates.

Let's consider the probability dW() of detecting a particle in a region of space defined by a small interval of values of the radius vector if the physical system is in a state of thermodynamic equilibrium.

We will measure the vector interval by volume dV=dxdydz.

Probability density (probability function of the distribution of radius vector values)

Particle in at the moment time is actually located somewhere in the specified space, which means the normalization condition must be satisfied:

Let us find the particle distribution probability function f() for a classical ideal gas. The gas occupies the entire volume V and is in a state of thermodynamic equilibrium with temperature T.

In the absence of an external force field, all positions of each particle are equally probable, i.e. gas occupies the entire volume with the same density. Therefore f() = c onst.

Using the normalization condition we find that

If the number of gas particles is N, then the concentration n = N/V.

Therefore, f(r) =n/N.

Conclusion: in the absence of an external force field, the probability dW() of detecting an ideal gas particle in a volume dV does not depend on the position of this volume in space, i.e. .

Let us place an ideal gas in an external force field.

As a result of the spatial redistribution of gas particles, the probability density f() ¹ c onst.

The concentration of gas particles n and its pressure P will be different, i.e. in the limit where D N is the average number of particles in the volume D V and pressure in the limit where D F is the absolute value of the average force acting normally on the area D S.

If the external field forces are potential and act in one direction (for example, the Earth’s gravity is directed along the z axis), then the pressure forces acting on the upper dS 2 and lower dS 1 bases of the volume dV will not be equal to each other (Fig. 2.2) .

In this case, the difference in pressure forces dF on the bases dS 1 and dS 2 must be compensated by the action of external field forces.

Total pressure difference dF = nGdV,

where G is the force acting on one particle from the external field.

The difference in pressure forces (by definition of pressure) dF = dPdxdy. Therefore, dP = nGdz.

It is known from mechanics that the potential energy of a particle in an external force field is related to the strength of this field by the relation .

Then the difference in pressure on the upper and lower bases of the allocated volume dP = - n dW p .

In a state of thermodynamic equilibrium of a physical system, its temperature T within the volume dV is the same everywhere. Therefore, we use the equation of state of an ideal gas for pressure dP = kTdn.

Solving the last two equalities together we get that

— ndW p = kTdn or .

After transformations we find that

where ℓ n n o is the integration constant (n o is the concentration of particles in that place in space where W p =0).

After potentiation, we get

Conclusion: in a state of thermodynamic equilibrium, the concentration (density) of particles of an ideal gas located in an external force field changes according to the law determined by formula (2.11), which is called Boltzmann distribution.

Taking into account (2.11), the probability function for the distribution of molecules in the gravity field takes the form

The probability of detecting a particle of an ideal gas in a volume dV located at a point determined by the radius vector can be represented as

For an ideal gas, pressure differs from concentration only by a constant factor kT (P=nkT).

Therefore, for such gases the pressure

Let us apply the Boltzmann distribution to atmospheric air located in the Earth's gravitational field.

The composition of the Earth's atmosphere includes gases: nitrogen - 78.1%; oxygen - 21%; argon-0.9%. The mass of the atmosphere is -5.15 × 10 18 kg. At an altitude of 20-25 km there is an ozone layer.

Near the earth's surface, the potential energy of air particles at a height is h W p = m o gh , where m o is the mass of the particle.

Potential energy at the Earth level (h=0) is zero (W p =0).

If, in a state of thermodynamic equilibrium, the particles of the earth’s atmosphere have a temperature T, then the change in atmospheric air pressure with height occurs according to the law

Formula (2.15) is called barometric formula; applicable for rarefied gas mixtures.

Conclusion: for the earth's atmosphere, the heavier the gas, the faster its pressure drops depending on the height, i.e. As altitude increases, the atmosphere should become increasingly enriched with light gases. Due to temperature changes, the atmosphere is not in an equilibrium state. Therefore, the barometric formula can be applied to small areas within which there is no change in temperature. In addition, the disequilibrium of the Earth's atmosphere is affected by the Earth's gravitational field, which cannot keep it close to the surface of the planet. The atmosphere dissipates faster, the weaker the gravitational field. For example, the earth's atmosphere dissipates quite slowly. During the existence of the Earth (

4-5 billion years) it lost a small part of its atmosphere (mainly light gases: hydrogen, helium, etc.).

The Moon's gravitational field is weaker than the Earth's, so it has almost completely lost its atmosphere.

The non-equilibrium of the earth's atmosphere can be proven as follows. Let us assume that the Earth’s atmosphere has reached a state of thermodynamic equilibrium and at any point in its space it has a constant temperature. Let us apply the Boltzmann formula (2.11), in which the role of potential energy is played by the potential energy of the Earth’s gravitational field, i.e.

where g is the gravitational constant; M s is the mass of the Earth; m o is the mass of an air particle; r is the distance of the particle from the center of the Earth.

When r ® ¥ W p =0. Therefore, the Boltzmann distribution (2.11) takes the form

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11.2 Law of distribution of ideal gas molecules in an external force field

When considering the kinetic theory of gases and Maxwell's distribution law, it was assumed that no forces act on gas molecules, with the exception of molecular impacts. Therefore, the molecules are evenly distributed throughout the vessel. In fact, the molecules of any gas are always in the gravitational field of the Earth. As a result, each molecule of mass m experiences the force of gravity f =mg.

Let us select a horizontal element of gas volume with height dh and base area S (Fig. 11.2). We assume that the gas is homogeneous and its temperature is constant. The number of molecules in this volume is equal to the product of its volume dV=Sdh by the number of molecules per unit volume. The total weight of molecules in the selected element is equal to

The action of the weight dF causes a pressure equal to

minus - because as dh increases, the pressure decreases. According to the basic equation of molecular kinetic theory

Equating the right-hand sides of (11.2) and (11.3), we obtain

Integrating this expression over the range from to h (respectively, the concentration changes from to n):

we get

Potentiating the resulting expression, we find

The exponent at exp has a multiplier that determines the increment in the potential energy of gas molecules. If you move a molecule from level to level h, then the change in its potential energy will be

Then the equation for the concentration of molecules is transformed to the form

This equation reflects Boltzmann's general law and gives the distribution of the number of particles as a function of their potential energy. It is applicable to any system of particles located in a force field, such as an electric one.

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Boltzmann distribution

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Let us assume that the gas is in an external potential field. In this case, a gas molecule of mass $m_0\ ,$ moving at a speed $\overrightarrow \ $has energy $_p$, which is expressed by the formula:

The probability ($dw$) of finding this particle in the phase volume $dxdydzdp_xdp_ydp_z$ is equal to:

The probability densities of the particle coordinates and its momenta are independent, therefore:

Formula (5) gives the Maxwell distribution for molecular speeds. Let us take a closer look at expression (4), which leads to the Boltzmann distribution. $dw_1\left(x,y,z\right)$ is the probability density of finding a particle in the volume $dxdydz$ near a point with coordinates $\left(x,y,z\right)$. We will assume that the gas molecules are independent and there are n particles in the selected volume of gas. Then, using the formula for adding probabilities, we get:

The coefficient $A_1$ is found from the normalization condition, which in our case means that there are n particles in the selected volume:

What is the Boltzmann distribution

The Boltzmann distribution is the expression:

Expression (8) specifies the spatial distribution of particle concentration depending on their potential energy. The coefficient $A_1$ is not calculated if it is necessary to know only the particle concentration distribution, and not their number. Let us assume that at the point ($x_0,y_ z_0$) the concentration $n_0$=$n_0$ $(x_0,y_ z_0)=\frac $ is given, the potential energy at the same point $U_0=U_0\left(x_0,y_ z_0\right).$ Let us denote the concentration of particles at the point (x,y,z) $n_0\ \left(x,y,z\right).\ $Substitute the data into formula (8), we obtain for one point:

for the second point:

Let's express $A_1$ from (9) and substitute it into (10):

Most often, the Boltzmann distribution is used in the form (11). It is especially convenient to choose a normalization at which $U_0\left(x,y,z\right)=0$.

Boltzmann distribution in a gravity field

The Boltzmann distribution in a gravity field can be written in the following form:

where $U\left(x,y,z\right)=m_0gz$ is the potential energy of a molecule of mass $m_0$ in the Earth's gravitational field, $g$ is the acceleration of gravity, $z$ is the height. Or for the gas density, distribution (12) will be written as:

Expression (13) is called the barometric formula.

When deriving the Boltzmann distribution, no restrictions were applied to the particle mass. Therefore, it is also applicable for heavy particles. If the particle mass is large, then the exponent changes rapidly with height. Thus, the exponent itself quickly tends to zero. In order for heavy particles to “not settle to the bottom,” it is necessary that their potential energy be low. This is achieved if the particles are placed, for example, in a dense liquid. Potential energy of a particle U(h) at a height h suspended in a liquid:

where $V_0$ is the volume of particles, $\rho $ is the density of particles, $_0$ is the density of the liquid, h is the distance (height) from the bottom of the vessel. Therefore, the distribution of the concentration of particles suspended in the liquid:

For the effect to be noticeable, the particles must be small. This effect is visually observed using a microscope.

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Average free path molecule is equal to the ratio of the path traveled by the molecule in 1 s to the number of collisions that occurred during this time: = / =1/(42r 2 n 0).

24. Internal energy of an ideal gas.

Internal energy is the sum of the energies of molecular interactions and the energy of thermal motion of molecules.

The internal energy of a system depends only on its state and is a unique function of the state.

Internal energy of an ideal gas is proportional to the mass of the gas and its thermodynamic temperature.

Gas work during expansion.

Let there be a gas in the cylinder under the piston, occupying a volume V under pressure p. Area of the piston S. The force with which the gas presses on the piston, F=pS. When the gas expands, the piston is raised to a height dh, and the gas does work A=Fdh=pSdh. But Sdh=dV is an increase in gas volume. Therefore, the elementary work is A=pdV. Full work A, performed by the gas when its volume changes from V1 to V2, we will find by integration

The result of integration depends on the process occurring in the gases.

In an isochoric process, V=const, therefore, dV=0 and A=0.

For an isobaric process p=const, then

Work during isobaric expansion of a gas is equal to the product of the gas pressure and the increase in volume.

In an isothermal process T=const. p=(mRT)/(MV).

Amount of heat.

The energy transferred to gas by heat exchange is called amount of heat Q.

When an infinitesimal amount of heat Q is imparted to the system, its temperature will change by dT.

26. Heat capacity C of a system is a quantity equal to the ratio of the amount of heat Q imparted to the system to the change in temperature dT of the system: C=Q/dT.

Distinguish specific heat capacity (heat capacity of 1 kg of substance) c=Q/(mdT) and molar heat capacity(heat capacity of 1 mole of substance) c=Mc.

For different processes occurring in thermodynamic systems, the heat capacities will be different.

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Let an ideal gas be in some force field, for example, in a gravity field. Since external forces act on the gas molecules in this case, the gas pressure will not be the same everywhere, but will vary from point to point.

In the simplest case, the field forces have a constant direction, characterized by the z axis. Let two areas of unit area be oriented perpendicular to the z axis and located at a distance dz from each other. If the gas pressures on both sites are equal to p and p + dp, then the pressure difference should obviously be equal to the total force acting on gas particles contained in the volume of a parallelepiped with a unit base and height d z. This force is equal Fn d z, Where n– density of molecules (i.e. their number per unit volume), a F– force acting on one molecule at a point with coordinate z. That's why

d p = nF d z.

Strength F is related to the potential energy U(z) of the molecule by the relation F = - dU/dz, so

d p = – n d z d U/d z= – n d U.

Since the gas is assumed to be ideal, then p = nkT. If the temperature of the gas at different points is the same, then

d p = kT d n.

Pressure difference d p in both cases it is determined by the difference in height. That's why

and finally

Here n 0 is a constant representing the density of molecules at the point where U = 0.

The resulting formula connecting the change in the density of a gas with the potential energy of its molecules is called the Boltzmann formula. Pressure differs from density by a constant factor kT, therefore the same equation is valid for pressure

In the case of a gravitational field near the earth's surface, the potential energy of a molecule at a height z is equal to U = mgz, where m is the mass of the molecule. Therefore, if we consider the temperature of the gas to be independent of altitude, then the pressure r on top z will be related to pressure r 0 on the Earth's surface by the relation

This formula is called the barometric formula. It is more convenient to represent it in the form

where m is the molecular weight of the gas, R is the gas constant.

This formula can also be applied in the case of a mixture of gases. Since the molecules of ideal gases practically do not interact with each other, each gas can be considered separately, that is, a similar formula applies to the partial pressure of each of them. The greater the molecular weight of a gas, the faster its pressure decreases with altitude. Therefore, as the altitude increases, the atmosphere becomes increasingly enriched in light gases: oxygen, for example, decreases in the atmosphere faster than nitrogen.

It should be borne in mind, however, that the applicability of the barometric formula to the real atmosphere is very limited, since the atmosphere is not actually in thermal equilibrium and its temperature varies with altitude.

An interesting conclusion can be drawn from Boltzmann's formula if we try to apply it to the atmosphere at any distance from the Earth. At very large distances from the earth's surface under U need to understand not mgz, a the exact value of the particle’s potential energy

where g is the gravitational constant, M is the mass of the Earth and r is the distance from the center of the Earth. The validity of this expression can be easily verified by differentiation by distance (F = - dU/dr) and subsequent comparison with the law of universal gravitation. Substituting this energy into the Boltzmann formula gives the following expression for the gas density:

where n ¥ now denotes the gas density at the location where U=0 (i.e. at an infinite distance from the Earth). If r equal to the radius of the Earth R, we get the relationship between the density of the atmosphere on the Earth’s surface n 0 and at infinity n ¥:

According to this formula, the density of the atmosphere at an infinitely large distance from the Earth should be different from zero. Such a conclusion, however, is absurd, since the atmosphere is of terrestrial origin, and a finite amount of gas cannot be distributed over an infinite volume with a density that never disappears. The conclusion obtained is explained by the fact that the atmosphere was assumed to be in a state of thermal equilibrium, which is not true.

This result shows that the gravitational field cannot keep the gas in a state of equilibrium at all, and therefore the atmosphere must continuously dissipate in space. In the case of the Earth, this dissipation is extremely slow, and throughout its existence the Earth has not lost any noticeable fraction of its atmosphere. But, for example, in the case of the Moon, with its much weaker gravitational field, the loss of the atmosphere occurred much faster, and as a result, the Moon currently no longer has an atmosphere.

Average free path molecule is equal to the ratio of the path traveled by the molecule in 1 s to the number of collisions that occurred during this time: = / =1/(42r 2 n 0).

24. Internal energy of an ideal gas.

Internal energy is the sum of the energies of molecular interactions and the energy of thermal motion of molecules.

The internal energy of a system depends only on its state and is a unique function of the state.

Internal energy of an ideal gas is proportional to the mass of the gas and its thermodynamic temperature.

Gas work during expansion.

Let there be a gas in the cylinder under the piston, occupying a volume V under pressure p. Area of the piston S. The force with which the gas presses on the piston, F=pS. When the gas expands, the piston is raised to a height dh, and the gas does work A=Fdh=pSdh. But Sdh=dV is an increase in gas volume. Therefore, the elementary work is A=pdV. We find the total work A performed by the gas when its volume changes from V1 to V2 by integration

The result of integration depends on the process occurring in the gases.

In an isochoric process, V=const, therefore, dV=0 and A=0.

For an isobaric process p=const, then

Work during isobaric expansion of a gas is equal to the product of the gas pressure and the increase in volume.

In an isothermal process T=const. p=(mRT)/(MV).

Amount of heat.

The energy transferred to gas by heat exchange is called amount of heat Q.

When an infinitesimal amount of heat Q is imparted to the system, its temperature will change by dT.

26. Heat capacity C of a system is a quantity equal to the ratio of the amount of heat Q imparted to the system to the change in temperature dT of the system: C=Q/dT.

Distinguish specific heat capacity(heat capacity of 1 kg of substance) c=Q/(mdT) and molar heat capacity(heat capacity of 1 mole of substance) c=Mc.

For different processes occurring in thermodynamic systems, the heat capacities will be different.

Boltzmann distribution

Boltzmann distribution, statistically equilibrium distribution function over the momenta p and coordinates r of particles of an ideal gas, the molecules of which move according to the laws of classical mechanics, in an external potential field:

Here p 2 /2m is the kinetic energy of a molecule of mass m, U(ν) is its potential energy in an external field, T is the absolute temperature of the gas. The constant A is determined from the condition that the total number of particles in different possible states is equal to the total number of particles in the system (normalization condition).
The Boltzmann distribution is a special case of the canonical Gibbs distribution for an ideal gas in an external potential field, since in the absence of interaction between particles, the Gibbs distribution decomposes into the product of the Boltzmann distribution for individual particles. The Boltzmann distribution at U=0 gives the Maxwell distribution. The distribution function (1) is sometimes called the Maxwell-Boltzmann distribution, and the Boltzmann distribution is the distribution function (1), integrated over all particle momenta and representing the number density of particles at point ν:

where n 0 is the number density of particles of the system in the absence of an external field. The ratio of particle number densities at different points depends on the difference in potential energy values at these points

where ΔU= U(ν 1)-U(ν 2). In particular, from (3) follows a barometric formula that determines the height distribution of gas in the gravitational field above the earth's surface. In this case, ΔU=mgh, where g is the acceleration of gravity, m is the mass of the particle, h is the height above the earth’s surface. For a mixture of gases with different particle masses, the Boltzmann distribution shows that the distribution of partial particle densities for each component is independent of the other components. For a gas in a rotating vessel, U (r) determines the potential of the field of centrifugal forces U (r)=-mω 2 r 2 /2, where ω is the angular velocity of rotation. The separation of isotopes and highly dispersed systems using an ultracentrifuge is based on this effect.
For quantum ideal gases, the state of individual particles is determined not by momenta and coordinates, but by quantum energy levels Ε i of the particle in the field U(r). In this case, the average number of particles in the i-th quantum state, or the average occupation number, is equal to:

where μ is the chemical potential determined from the condition that the total number of particles at all quantum levels Ε i is equal to the total number of particles N in the system: Σin i =N. Formula (4) is valid at such temperatures and densities when the average distance between particles is significantly greater than the de Broglie wavelength corresponding to the average thermal velocity, i.e. when it is possible to neglect not only the force interaction of particles, but also their mutual quantum mechanical influence (there is no quantum gas degeneration (see Degenerate gas). Thus, the Boltzmann distribution is a limiting case of both the Fermi-Dirac distribution and the Bose-Einstein distribution for low-density gases.

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MOLECULAR PHYSICS AND THERMODYNAMICS

Due to the chaotic movement, changes in the position of each particle (molecule, atom, etc.) of a physical system (macroscopic body) are in the nature of a random process. Therefore, we can talk about the probability of detecting a particle in a particular region of space.

From kinematics it is known that the position of a particle in space is characterized by its radius vector or coordinates.

We will measure the vector interval by volume dV=dxdydz.

Probability density (probability function of the distribution of radius vector values)

The particle at a given moment in time is actually located somewhere in the specified space, which means the normalization condition must be satisfied:

Let us find the particle distribution probability function f() for a classical ideal gas. The gas occupies the entire volume V and is in a state of thermodynamic equilibrium with temperature T.

In the absence of an external force field, all positions of each particle are equally probable, i.e. gas occupies the entire volume with the same density. Therefore f() = c onst.

Using the normalization condition we find that

If the number of gas particles is N, then the concentration n = N/V.

Therefore, f(r) =n/N.

Conclusion: in the absence of an external force field, the probability dW() of detecting an ideal gas particle in a volume dV does not depend on the position of this volume in space, i.e. .

Let us place an ideal gas in an external force field.

As a result of the spatial redistribution of gas particles, the probability density f() ¹ c onst.

In this case, the difference in pressure forces dF on the bases dS 1 and dS 2 must be compensated by the action of external field forces.

Total pressure difference dF = nGdV,

where G is the force acting on one particle from the external field.

The difference in pressure forces (by definition of pressure) dF = dPdxdy. Therefore, dP = nGdz.

It is known from mechanics that the potential energy of a particle in an external force field is related to the strength of this field by the relation .

Then the difference in pressure on the upper and lower bases of the allocated volume dP = - n dW p .

Solving the last two equalities together we get that

— ndW p = kTdn or .

After transformations we find that

where ℓ n n o is the integration constant (n o is the concentration of particles in that place in space where W p =0).

After potentiation, we get

Taking into account (2.11), the probability function for the distribution of molecules in the gravity field takes the form

The probability of detecting a particle of an ideal gas in a volume dV located at a point determined by the radius vector can be represented as

For an ideal gas, pressure differs from concentration only by a constant factor kT (P=nkT).

Therefore, for such gases the pressure

Let us apply the Boltzmann distribution to atmospheric air located in the Earth's gravitational field.

Near the earth's surface, the potential energy of air particles at a height is h W p = m o gh , where m o is the mass of the particle.

Potential energy at the Earth level (h=0) is zero (W p =0).

If, in a state of thermodynamic equilibrium, the particles of the earth’s atmosphere have a temperature T, then the change in atmospheric air pressure with height occurs according to the law

Formula (2.15) is called barometric formula; applicable for rarefied gas mixtures.

4-5 billion years) it lost a small part of its atmosphere (mainly light gases: hydrogen, helium, etc.).

The Moon's gravitational field is weaker than the Earth's, so it has almost completely lost its atmosphere.

where g is the gravitational constant; M s is the mass of the Earth; m o is the mass of an air particle; r is the distance of the particle from the center of the Earth.

When r ® ¥ W p =0. Therefore, the Boltzmann distribution (2.11) takes the form

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11.2 Law of distribution of ideal gas molecules in an external force field

The action of the weight dF causes a pressure equal to

minus - because as dh increases, the pressure decreases. According to the basic equation of molecular kinetic theory

Equating the right-hand sides of (11.2) and (11.3), we obtain

Integrating this expression over the range from to h (respectively, the concentration changes from to n):

we get

Potentiating the resulting expression, we find

Then the equation for the concentration of molecules is transformed to the form

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Boltzmann's law on the distribution of particles in an external potential field

Based on the basic equation of molecular kinetic theory: P = nkT, replace P And P0 in the barometric formula (2.4.1) at n And n 0 and we get Boltzmann distribution for molar mass of gas:

Since a , then (2.5.1) can be represented in the form

Figure 2.11 shows the dependence of the concentration of various gases on altitude. It can be seen that the number of heavier molecules decreases faster with height than light ones.

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Barometric formula is the dependence of gas pressure or density on height in a gravitational field. For an ideal gas having a constant temperature T and located in a uniform gravitational field at all points of its volume, the acceleration of gravity g is the same barometric formula has the following form: where p the gas pressure in a layer located at a height h p0 pressure at zero level h = h0 M molar mass of gas R gas constant T absolute temperature. From the barometric formula it follows that the concentration of molecules n or...

45.Barometric formula. Boltzmann's law for the distribution of particles in an external potential field.

Barometric formuladependence of gas pressure or density on height in the gravitational field. For ideal gas , having a constant temperature T and located in a uniform gravitational field (at all points of its volumeacceleration of gravity g the same), the barometric formula is as follows:

where p gas pressure in a layer located at a height h , p 0 pressure at zero level ( h = h 0 ), M molar mass of gas, R gas constant, T absolute temperature. From the barometric formula it follows that the concentration of molecules n (or gas density) decreases with height according to the same law:

where M molar mass of gas, R gas constant.

The barometric formula shows that the density of a gas decreases exponentially with altitude. Magnitude, which determines the rate of density decline, is the ratio of the potential energy of particles to their average kinetic energy, proportional to kT . The higher the temperature T , the slower the density decreases with height. On the other hand, an increase in gravity mg (at a constant temperature) leads to a significantly greater compaction of the lower layers and an increase in the density difference (gradient). Gravity acting on particles mg can change due to two quantities: acceleration g and particle masses m.

Consequently, in a mixture of gases located in a gravitational field, molecules of different masses are distributed differently in height.

Based on the basic equation of molecular kinetic theory: P = nkT , replace P and P 0 in the barometric formula (2.4.1) at n and n 0 and we get Boltzmann distributionfor molar mass of gas:

(2.5.1)

where n 0 and n - number of molecules in a unit volume at height h = 0 and h.

Since a , then (2.5.1) can be represented in the form

(2.5.2)

As the temperature decreases, the number of molecules at heights other than zero decreases. At T = 0 thermal movement stops, all molecules would be located on the earth's surface. At high temperatures, on the contrary, the molecules are distributed almost evenly over the height, and the density of the molecules slowly decreases with height. Because mgh this is potential energy U , then at different heights U = mgh different. Consequently, (2.5.2) characterizes the distribution of particles according to potential energy values:

(2.5.3)

this is the law of particle distribution according to potential energies Boltzmann distribution. Here n 0 number of molecules per unit volume where U = 0.

Figure 2.11 shows the dependence of the concentration of various gases on altitude. It can be seen that the number of heavier molecules decreases faster with height than light ones.

Rice. 2.11

From (2.5.3) it can be obtained that the ratio of the concentrations of molecules at points with U 1 and i>U 2 are equal to:

(2.5.4)

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