Ideal gas in an external potential field. Boltzmann's law for the distribution of particles in an external potential field
The barometric formula obtained in § 92
(see (92.4)) gives pressure as a function of height above the Earth's surface for an imaginary isothermal atmosphere. Let us replace the ratio in the exponent by the ratio equal to it ( is the mass of the molecule, k is the Boltzmann constant). In addition, we substitute in accordance with (86.7) instead of the expression and instead of - the expression Then reducing both parts of the equality by we come to the formula
(100.2)
Here - the concentration of molecules (i.e., their number per unit volume) at a height - the concentration of molecules at a height
It follows from formula (100.2) that as the temperature decreases, the number of particles at heights other than zero decreases, turning to zero at (Fig. 100.1). At absolute zero, all molecules would be located on the earth's surface.
At high temperatures, on the contrary, decreases slightly with height, so that the molecules are almost uniformly distributed along the height.
This fact has a simple physical explanation. Each specific distribution of molecules in height is established as a result of the action of two tendencies: 1) the attraction of molecules to the Earth (characterized by the force ) tends to place them on the surface of the Earth; 2) thermal motion (characterized by the value ) tends to scatter the molecules evenly over all heights. The larger and smaller T, the stronger the first tendency prevails, and the molecules condense near the surface of the Earth. In the limit at , thermal motion completely stops, and under the influence of attraction, the molecules are located on the earth's surface. At high temperatures, thermal motion predominates, and the density of molecules slowly decreases with height.
At different heights, a molecule has a different potential energy reserve:
Consequently, the distribution of molecules along the height is, at the same time, their distribution according to the values of potential energy. Taking into account (100.3), formula (100.2) can be written as follows:
where is the density of molecules in that place in space where the potential energy of the molecule matters - the density of molecules in the place where the potential energy of the molecule is zero.
From (100.4) it follows that the molecules are located with a higher density where their potential energy is less, and, conversely, with a lower density - in places where their potential energy is greater.
In accordance with (100.4), the ratio at the points where the potential energy of the molecule has the values is equal to
Boltzmann proved that distribution (100.4) is valid not only in the case of the potential field of terrestrial gravitational forces, but also in any potential field of forces for a collection of any identical particles in a state of chaotic thermal motion. Accordingly, the distribution (100.4) is called the Boltzmann distribution.
While Maxwell's law gives the distribution of particles over values kinetic energy, Boltzmann's law gives the distribution of particles over potential energy values. Both distributions are characterized by the presence of an exponential factor, the indicator of which is the ratio of the kinetic or, respectively, potential energy of one molecule to the value that determines the average energy of the thermal motion of the molecule.
According to formula (100.4), the number of molecules that fall within the volume located at a point with coordinates x, y, z is
We have received one more expression of the Boltzmann distribution law.
The Maxwell and Boltzmann distributions can be combined into one Maxwell-Boltzmann law, according to which the number of molecules whose velocity components lie in the range from to and coordinates in the range from x, y, z to is equal to
When deriving the basic equation of the molecular kinetic theory and the Maxwell distribution law, it was assumed that no external forces act on the molecules. Therefore, it could be assumed that the molecules are evenly distributed over the volume of the vessel.
In fact, the molecules of any gas are always in the Earth's gravitational field. If there was no thermal motion of atmospheric air molecules, then they would all fall to the Earth. If there were no gravitation, then atmospheric air would be dispersed throughout the universe. Thus, gravity and thermal motion bring the gas into a state in which its pressure and concentration of molecules depend on height.
The formula for the dependence of atmospheric pressure on altitude above the Earth's level is called the barometric formula. To derive the barometric formula, we introduce some assumptions:
We consider the gravitational acceleration to be practically constant and independent of height, since the atmospheric pressure becomes negligibly small already at a height of 100-200 km, which is much lower compared to the radius of the Earth;
The air temperature is assumed to be independent of altitude.
Atmospheric pressure is determined by the weight of the overlying layers of gas. Let's mentally select a vertical column of air (Fig. 18.1) with a base area S.
Let it be on top h gas pressure is p, and at height ( h+dh) the pressure is ( p+dp). Since the pressure decreases with increasing altitude, its increment will be negative ( dp< 0).
Pressure difference p And ( p+dp) is equal to the weight of the gas enclosed in a column of height dh, divided by the area S, that is
,
(18.1)
Where - air density per altitude h.
Replacing the density in this equation according to the formula obtained using the Clapeyron-Mendeleev equation (14.1):
we write expression (18.1) as
.
(18.2)
Assuming T=const(in accordance with the accepted assumptions) and integrating equation (18.2) over the height from 0 before h, we get
,
where do we find
,
(18.3)
Where p 0 - altitude pressure h = 0.
Expression (18.3) is called the barometric formula. It follows from it that the gas pressure decreases with increasing altitude the faster, the heavier the gas (the more ) and the lower the temperature. Figure 18.2 shows two dependencies of the form (18.3), corresponding to two gases with different molar masses 1 and 2 at T=const(pressure p 0 For h=0 both gases are assumed to be conditionally the same).
(18.4)
underlies the principle of operation of aviation altimeters (altimeters): by measuring pressure with a barometer, these devices show the value of altitude above sea level.
From formula (18.3), one can obtain the relationship between gas concentrations at different heights by substituting into it the equation of state of the gas in the form (15.26):
.
(18.5)
Replacing the ratio / R for a homogeneous gas to the ratio m/k (m is the mass of the molecule) and reducing both sides of the equality by kT, we get
, (18.6)
Where n 0 - concentration of gas molecules at h =0.
From expression (18.6) it follows that the heavier the gas (more m) and the lower its temperature T, the greater the concentration of molecules at the surface of the Earth compared to the concentration at a certain height (the predominance of the Earth's gravity over the thermal motion of molecules). Conversely, the lighter the gas and the greater its temperature, the more the thermal motion of molecules prevails over gravitation, and the concentration slowly decreases with increasing height.
A comparison of these dependences shows that the lower the gas temperature, the greater the inhomogeneity observed in the distribution of the concentration of gas molecules over height.
Work mgh in equation (18.6) is the potential energy W n one molecule in the Earth's gravitational field. Consequently, the distribution of molecules in height is, at the same time, their distribution in terms of values
potential energy:
.
(18.7)
The Austrian physicist L. Boltzmann proved that formula (18.7) is valid for any set of identical particles in a state of chaotic thermal motion in a potential field of any nature. In this connection, the function (18.7) is called Boltzmann distribution. Thus, the distribution (18.6) is a special case of the more general distribution (18.7). There is a great similarity between the Maxwell distribution (17.6) and Boltzmann (18.7): in both distributions, the exponent is the ratio of the energy of the molecule (in one case, potential, and in the other, kinetic) to the value kT, which determines the average kinetic energy of thermal chaotic motion.
Distributions (17.6) and (18.7) can be combined into one Maxwell-Boltzmann distribution, according to which the number of molecules whose velocity components lie in the range from 
to , and coordinates in the range from 
up to equals
Where 
.
From formula (18.8) it follows that 
is determined by the total energy of the molecule 
.
Thus, in a state with constant temperature, the velocities of molecules at each point in space are distributed according to Maxwell's law. Influence force field affects only the change in the concentration of molecules from point to point.
Potential energy distribution of gas molecules (Boltzmann distribution)
Ideal gas in an external force field
In ideal gases, molecules are considered non-interacting with each other by means of intermolecular force fields, and their potential energy does not appear in gas laws. However, in external force fields, this situation changes - the molecules acquire potential energy due to the action of external forces on them. This potential energy is taken into account in the laws of thermodynamics.
In the absence of external influences due to chaotic thermal motion, the gas evenly fills the volume provided to it. However, under external influences, the picture changes, and the potential energy affects the distribution of gas molecules in the space of the volume containing the gas.
Let us find the distribution of ideal gas molecules in a homogeneous, conservative, one-dimensional external force field (for example, the gravity field near the Earth's surface). Orienting the selected axis Z along the direction of the force action (vertically upwards, in our example) and we will look for the distribution of the concentration (and pressure) of molecules along this direction.
Let us single out two parallel planes (plates) in the gas with area S each oriented perpendicular to the axis Z with differential-small distance-gap d .z between them (Fig. 4.4). Due to the force acting on the molecules F(weight in the gravitational field) the pressure on the bottom plate will be greater than on the top. pressure difference dp equal to the forces acting on the plates from all molecules in the volume dV= related to their area S:
Where F(z)- force acting from the force field on one molecule located at the level zn(z)- concentration of molecules at the level Z-
Under given conditions, the force is conservative; this means that the force field is potential. Therefore, we can use the relationship between the force F(z) and potential energy U(z)
in the shape of 
(relation (1.33) in subsection 1.3.5). Now we can write
Rice. 4.4.
Since the gas is ideal, its pressure is related to the concentration by equation (4.25), and the temperature is assumed to be the same at every point, so
Replacing the pressure change in (4.32) by (4.33), we obtain k^Tdn == -ndU. Separating the variables, we get 
. Integration
gives 
This equation can be rewritten as
And further 
Assuming that at the level taken as reference zero ( z= 0) the concentration is equal to and 0 , we get С = n 0 . Therefore, finally
The resulting relation relates the concentration of ideal gas molecules n(z) and his pressure p(z) with the potential energy of molecules U(z) in a force field with temperature T. This ratio is called Boltzmann distribution(or Boltzmann's law). The plot of Boltzmann's law is given for relative concentrations n(z)/n0 in fig. 4.5. It shows that a high concentration of molecules corresponds to the values of the coordinates z, where is the potential energy U(z) small. As the potential energy increases, the concentration of molecules decreases. At U(z) = kT the concentration of molecules is e times less than at the level where U(z) = 0.
Rice. 4.5. Dependence of the relative concentration of particles in the force field on the magnitude of the potential energy U(z)
Figure 4.5 shows a set of curves corresponding to different gas temperatures. As the temperature increases, the energy of the chaotic motion of molecules increases and the effect of temperature on the concentration decreases. Therefore, at high temperature, the concentration of molecules levels off, the gas evenly fills the entire volume. On the contrary, a decrease in temperature leads to a sharp dependence of the concentration on the potential energy. The influence of the force field is more pronounced.
Since concentration and pressure are proportional to each other, what was said earlier about concentration is true for pressure. In particular, taking into account (4.25), formula (4.34) can be rewritten as:
wherein p 0 And p(z) is the pressure at the points where the potential energy is zero and U(z), respectively.
Boltzmann distribution
Based on the basic equation of molecular kinetic theory: P = nkT, replace P And P0 in the barometric formula (2.4.1) on n And n 0 and get Boltzmann distribution For molar mass gas:
As the temperature decreases, the number of molecules at heights other than zero decreases. At T= 0 thermal motion stops, all molecules would be located on the earth's surface. At high temperatures, on the contrary, the molecules are almost evenly distributed along the height, and the density of the molecules slowly decreases with height. Because mgh- ϶ᴛᴏ potential energy U, then at different heights U=mgh- different. Therefore, (2.5.2) characterizes the distribution of particles according to the values of potential energy:
| , |
15 Surface tension- thermodynamic characteristic of the interface of two phases in equilibrium, determined by the work of reversible isothermokinetic formation of a unit area of this interface, provided that the temperature, volume of the system and chemical potentials of all components in both phases remain constant.
Surface tension has a double physical meaning - energy (thermodynamic) and force (mechanical). Energy (thermodynamic) definition: surface tension is the specific work of increasing the surface when it is stretched, provided that the temperature is constant. Force (mechanical) definition: surface tension is the force acting per unit length of a line that limits the surface of a liquid
Laplace's formula[edit | edit wiki text]
Consider a thin liquid film whose thickness can be neglected. In an effort to minimize its free energy, the film creates a pressure difference from different sides. This explains the existence of soap bubbles: the film is compressed until the pressure inside the bubble exceeds the atmospheric pressure by additional pressure films. Additional pressure at a point on the surface depends on the average curvature at this point and is given by Laplace formula:
Here are the radii of the principal curvatures at the point. Οʜᴎ have the same sign if the corresponding centers of curvature lie on the same side of the tangent plane at the point, and have a different sign if on the opposite side. For example, for a sphere, the centers of curvature at any point on the surface coincide with the center of the sphere, and therefore
It is important to note that for the case of the surface of a circular cylinder of radius, we have
capillary phenomena called the rise or fall of fluid in small diameter tubes - capillaries. Wetting liquids rise through the capillaries, non-wetting liquids descend.
Capillarity(from lat. capillaris - hair; hence comes the previously encountered in the Russian-speaking scientific literature term capillarity), capillary effect- a physical phenomenon consisting in the ability of liquids to change the level in tubes, narrow channels freeform, porous bodies. In the field of gravity (or forces of inertia, for example, when centrifuging porous samples), the liquid rises when the channels are wetted with liquids, for example, water in glass tubes, sand, soil, etc. The decrease in liquid occurs in tubes and channels, not wetted by a liquid, such as mercury in a glass tube.
cohesion(from lat. cohaesus - connected, linked), adhesion of molecules (ions) of a physical body under the influence of attractive forces.
adhesion of parts of the same homogeneous body (liquid or solid). Caused by chem. the connection between the particles (atoms, ions) that make up the body and the intermol. interaction. The work of cohesion is called. free energy of dividing the body into parts and removing them to such a distance when the integrity of the body is violated.
Adhesion(from lat. adhaesio- sticking) in physics - adhesion of surfaces of dissimilar solid and / or liquid bodies. Adhesion is due to intermolecular interactions (van der Waals, polar, sometimes - the formation chemical bonds or mutual diffusion) in the surface layer and is characterized by specific work, which is extremely important for the separation of surfaces. In some cases, adhesion may be stronger than cohesion, that is, adhesion within a homogeneous material, in such cases, when a tearing force is applied, a cohesive gap occurs, that is, a gap in the volume of the less strong of the contacting materials.
Boltzmann distribution - concept and types. Classification and features of the category "Boltzmann distribution" 2017, 2018.
Molecules of an ideal gas, free from external influences, are uniformly distributed over the entire occupied volume due to thermal motion. In an external field, a force acts on the molecules, and the distribution of particles over the volume becomes inhomogeneous. The law of change....
The basic equation of the molecular kinetic theory connects the parameters of the state of a gas with the characteristics of the movement of its molecules, i.e., it establishes the relationship between the pressure and volume of the gas and the kinetic energy forward movement its molecules. For output... .
Educational and material base of UKP. Educational and material base of UMTs GOChS. Structure of UMB GO and RSChS. The composition of the educational and material base for training various groups population in the area of security... .
barometric formula. is the Boltzmann distribution of particles in an external force field. z is the height above the ground. is the concentration of molecules at those points where the potential energy is zero. n0 is the concentration of molecules at the earth's surface. - addiction ... [read more] .
Based on the basic equation of the molecular kinetic theory: P = nkT, we replace P and P0 in the barometric formula (2.4.1) by n and n0 and obtain the Boltzmann distribution for the molar mass of gas: (2.5.1) where n0 and n are the number of molecules in a unit volume at a height h = 0 and h. Because... .
Until now, we have not taken into account the existence of an external force field (for example, gravitational). In the absence of a field, gas molecules are uniformly distributed over the entire volume, i.e. the density of the gas in the volume is constant. If a force field acts, then the particle density and gas pressure ... .
1. 4. Barometric formula.
When deriving the basic equation of the molecular kinetic theory, it was assumed that if external forces do not act on the gas molecules, then the molecules are uniformly distributed over the volume. However, the molecules of any gas are in the potential gravitational field of the Earth. Gravity, on the one hand, and thermal motion of molecules, on the other hand, lead to a certain stationary state of the gas, in which the concentration of gas molecules and its pressure decrease with height. We derive the law of change in gas pressure with height, assuming that the gravitational field is uniform, the temperature is constant, and the mass of all molecules is the same. If the atmospheric pressure at height h is equal, then at height h + dho it is equal to p + dp (Fig. 1.2). When dh> 0,dр< 0, т.к. давление с высотой убывает. Разность давлений р и (р +dр) равна hydrostatic pressure a column of gas abcd enclosed in a volume of a cylinder with a height dh and an area with a base equal to one. This is written in the following form: p- (p+dp) =gρdh, -dp=gρdh or dp= -gρdh, where ρ is the gas density at altitude h. Let's use the ideal gas equation of state pV=mRT/M and express the density ρ=m/V=pM/RT. Substitute this expression into the formula for dр:
dp= -pMgdh/RT or dp/p= -Mgdh/RT
Integration given equation gives the following result: Here C is a constant and in this case it is convenient to denote the constant of integration by lnC. Potentiating the resulting expression, we find that
Under the condition h=0, we obtain that C=p 0 , where p 0 is the pressure at height h=0.
This expression is called the barometric formula. It allows you to find atmospheric pressure as a function of altitude, or altitude if the pressure is known.
Figure 1.3 shows the dependence of pressure on altitude. An instrument for determining altitude above sea level is called an altimeter or altimeter. It is a barometer calibrated in terms of altitude.
1. 5. Boltzmann's law on the distribution of particles in an external potential field. @
If we use the expression p = nkT, then we can bring the barometric formula to the form:
h 
where n is the concentration of molecules at a height h, n 0 is the same at the Earth's surface. Since M \u003d m 0 N A, where m 0 is the mass of one molecule, and R \u003d kN A, then we get П \u003d m 0 gh is the potential energy of one molecule in the gravitational field. Since kT~‹ε post ›, then the concentration of molecules at a certain height depends on the ratio P and ‹ε post ›
The resulting expression is called the Boltzmann distribution for the external potential field. It follows from it that at a constant temperature, the density of the gas (which is related to the concentration) is greater where the potential energy of its molecules is less.
1. 6. Maxwell's distribution of ideal gas molecules over velocities. @
When deriving the basic equation of the molecular kinetic theory, it was noted that molecules have different velocities. As a result of multiple collisions, the velocity of each molecule changes with time in absolute value and in direction. Due to the randomness of the thermal motion of molecules, all directions are equally probable, and the mean square velocity remains constant. We can write down
The constancy of ‹υ kv › is explained by the fact that a stationary velocity distribution of molecules that does not change with time is established in the gas, which obeys a certain statistical law. This law was theoretically derived by D.K. Maxwell. He calculated the function f(u), called the velocity distribution function of molecules. If we divide the range of all possible velocities of molecules into small intervals equal to du, then for each interval of speed there will be a certain number of molecules dN(u) having a speed enclosed in this interval (Fig.1.4.).
The function f(v) determines the relative number of molecules whose velocities lie in the interval from u to u + du. This number is dN(u)/N= f(u)du. Using the methods of probability theory, Maxwell found a form for the function f(u)
This expression is the law on the distribution of molecules of an ideal gas by velocities. The specific form of the function depends on the type of gas, the mass of its molecules and temperature (Fig. 1.5). The function f(u)=0 at u=0 and reaches a maximum at some value of u in, and then asymptotically tends to zero. The curve is asymmetric about the maximum. The relative number of molecules dN(u)/N whose velocities lie in the interval du and equal to f(u)du is found as the area of the shaded strip with base dv and height f(u) shown in Fig. 1.4. The entire area bounded by the f (u) curve and the abscissa axis is equal to one, because if you sum up all the fractions of molecules with all possible speeds, you get one. As shown in Fig. 1.5, with increasing temperature, the distribution curve shifts to the right, i.e. the number of fast molecules increases, but the area under the curve remains constant, because N = const.
The speed u at which the function f(u) reaches its maximum is called the most probable speed. From the condition that the first derivative of the function f(v) ′ = 0 is equal to zero, it follows that
In figure 1.4. one more characteristic is noted - the arithmetic mean velocity of the molecule. It is determined by the formula:
An experiment conducted by the German physicist O. Stern experimentally confirmed the validity of the Maxwell distribution (Figure 1.5.). The Stern device consists of two coaxial cylinders. A platinum wire coated with a layer of silver passes along the axis of the inner cylinder with a slot. If current is passed through the wire, it heats up and the silver evaporates. Silver atoms, flying out through the slot, fall on the inner surface of the second cylinder. If the device rotates, then the silver atoms will not settle against the gap, but will be displaced from the point O for a certain distance. The study of the amount of sediment makes it possible to estimate the distribution of molecules by velocities. It turned out that the distribution corresponds to the Maxwellian one.
barometric formula. Boltzmann distribution
The basic equation of the molecular kinetic theory connects the parameters of the state of a gas with the characteristics of the motion of its molecules, i.e., it establishes the relationship between the pressure and volume of the gas and the kinetic energy of the translational motion of its molecules.
t 0 m o v - (- m o v) = 2m o v.
During the time Dt of the platform DS, only those molecules are reached that are enclosed in the volume of a cylinder with a base DS and a height v Dt. The number of these molecules is P D.S. v Dt (n - number of molecules per unit volume). To simplify calculations, the chaotic motion of molecules is replaced by motion along three mutually perpendicular directions, so that at any moment "time, 1/3 of the molecules move along each of them, and half of the molecules move along this direction in one direction, half in the opposite direction. Then the number of impacts molecules moving in a given direction, about the pad ds will be 1/6 P DSvDt. . m o v 1/6 P DSvDt = 1/3 m o v 2 DSDt
R= F/DS=P/(DSDt)=1/3 m o v 2 (1),
(because F=dP/dt).
If the gas is in volume V contains N
(2)
p = 1/3 m o v sq 2 (3)
Given that P= N/V, we get pV = 1/3 Nm o v kv 2
or pV = 2/3 N (m o v square 2/2)= 2/3 E(4),
Where E - the total kinetic energy of the translational motion of all gas molecules.
Expression (4) (i.e. pV = 2/3E) or its equivalent (3) is called the basic equation of the molecular kinetic theory of ideal gases. Exact calculation, taking into account the movement of molecules in all possible directions, gives the same formula.
p= n kT, and on the other p = 1/3 m o v
(5),
since the molar mass m = m 0 N A, Where t 0 - the mass of one molecule, N A - Avogadro constant, To= R/N A
The average kinetic energy of the translational motion of one molecule of an ideal gas, using that p= n kT, and p = 1/3 m o v square 2 is equal to
e= m o v sq. 2/2=3/2kT
Those. it is proportional to the thermodynamic temperature and depends only on it. Thus, thermodynamic temperature is a measure of the average kinetic energy of the translational motion of ideal gas molecules.
When deriving the basic equation of the molecular-kinetic theory of gases and the Maxwellian distribution of molecules by velocities, it was assumed that external forces do not act on gas molecules, therefore the molecules are uniformly distributed over the volume. However, the molecules of any gas are in the potential gravitational field of the Earth. Gravity, on the one hand, and thermal motion of molecules, on the other hand, lead to a certain stationary state of the gas, in which the gas pressure decreases with height.
Let us derive the law of change in pressure with height, assuming that the gravitational field is uniform, the temperature is constant, and the mass of all molecules is the same. If atmospheric pressure is high h equals R, then at the height h+ dh it is equal to p + dp(at dh> ABOUT dp< 0, так как давление с высотой убывает). Разность давлений R And p + dp equal to the weight of the gas enclosed in the volume of a cylinder with a height dh
p - (p + dp)= ρ gdh,
h. Hence,
dp=- ρ gdh.(1)
pV = m/mRT, where m is the mass of gas, m-
r= m/V= pm/(RT).
Substituting into (1), we obtain
or 
h, and the pressure on hpo.
(2),
since m = m 0 N A, And R= kN A, Where t o - the mass of one molecule, N A - Avogadro constant.
Expression (2) is called barometric formula. It allows you to find the atmospheric pressure depending on the altitude (or, by measuring the pressure, find the altitude). From this formula it follows that the pressure decreases with height the faster, the heavier the gas.
The barometric formula (2) can be transformed using the expression R = PCT:
(3)
Here n h, A no- concentration of particles at height h=0.
It follows from formula (3) that as the temperature decreases, the number of molecules at a certain height h decreases. At T=0 all molecules would be on the surface of the earth. Gravity tends to lower the molecule to the ground, and thermal motion scatters them along the heights, so the distribution of molecules in the atmosphere with height is determined by the balance of these tendencies.
Considering that m o gh= P
(4)
Expression (4) is called Boltzmann distribution in an external potential field
If the particles have the same mass and are in a state of chaotic thermal motion, then the Boltzmann distribution (4) is valid in any external potential field, and not only in the field of gravity.
BAROMETRIC FORMULA. BOLTZMANN DISTRIBUTION
The barometric formula is the dependence of the pressure or density of a gas on altitude in a gravitational field.
For an ideal gas that has a constant temperature and is in a uniform gravitational field (at all points in its volume, the acceleration due to gravity is the same), the barometric formula has the following form:
where is the gas pressure in a layer located at a height, is the pressure at zero level (), is the molar mass of the gas, is the universal gas constant, is the absolute temperature. It follows from the barometric formula that the concentration of molecules (or gas density) decreases with height according to the same law:
where is the mass of a gas molecule, is the Boltzmann constant.
The barometric formula can be obtained from the law of distribution of ideal gas molecules over velocities and coordinates in a potential force field (see Maxwell-Boltzmann statistics). In this case, two conditions must be satisfied: the constancy of the gas temperature and the uniformity of the force field. Similar conditions can be met for the smallest solid particles suspended in a liquid or gas. Based on this, the French physicist J. Perrin in 1908 applied the barometric formula to the height distribution of emulsion particles, which allowed him to directly determine the value of the Boltzmann constant.
The barometric formula shows that the density of a gas decreases exponentially with altitude. Value 
, which determines the density decay rate, is the ratio of the potential energy of particles to their average kinetic energy, which is proportional to . The higher the temperature, the slower the decrease in density with height. On the other hand, an increase in gravity (at a constant temperature) leads to a much greater compaction of the lower layers and an increase in the density difference (gradient). The force of gravity acting on the particles can be changed due to two quantities: acceleration and particle mass.
Consequently, in a mixture of gases located in a gravitational field, molecules of different masses are distributed differently in height.
The actual distribution of pressure and air density in the earth's atmosphere does not follow the barometric formula, since within the atmosphere the temperature and gravitational acceleration change with altitude and latitude. In addition, atmospheric pressure increases with the concentration of water vapor in the atmosphere.
The barometric formula underlies barometric leveling - a method for determining the height difference between two points by the pressure measured at these points ( and ). Since the atmospheric pressure depends on the weather, the time interval between measurements should be as short as possible, and the measurement points should not be located too far from each other. The barometric formula is written in this case as: (in m), where is the average temperature of the air layer between the measurement points, is the temperature coefficient of the volumetric expansion of the air. The error in calculations using this formula does not exceed 0.1-0.5% of the measured height. The Laplace formula is more accurate, taking into account the influence of air humidity and the change in the acceleration of free fall.
In the presence of a gravitational field (or, in general, any potential field), the force of gravity acts on gas molecules. As a result, the concentration of gas molecules turns out to be height-dependent in accordance with the Boltzmann distribution law:
n = n0exp(-mgh / kT)
where n is the concentration of molecules at height h, n0 is the concentration of molecules at entry level h = 0, m - mass of particles, g - free fall acceleration, k - Boltzmann's constant, T - temperature.
barometric formula. Boltzmann distribution.
The basic equation of the molecular kinetic theory connects the parameters of the state of a gas with the characteristics of the motion of its molecules, i.e., it establishes the relationship between the pressure and volume of the gas and the kinetic energy of the translational motion of its molecules.
To derive the equation, consider the one-atomic ideal gas. Let us assume that gas molecules move randomly with the same speed v, the number of mutual collisions between gas molecules is negligible compared to the number of impacts on the walls of the vessel, the collisions of molecules with the walls of the vessel are absolutely elastic. Let us single out some elementary area DS on the vessel wall (Fig. 1) and calculate the pressure exerted on this area. In each collision, a molecule of mass t 0 transmits an impulse to the vessel wall m o v - (- m o v) = 2m o v.
During the time Dt of the platform DS, only those molecules are reached that are enclosed in the volume of a cylinder with a base DS and a height v Dt. The number of these molecules is P D.S. v Dt (n - number of molecules per unit volume). To simplify calculations, the chaotic motion of molecules is replaced by motion along three mutually perpendicular directions, so that at any moment "time, 1/3 of the molecules move along each of them, and half of the molecules move along this direction in one direction, half in the opposite direction. Then the number of impacts molecules moving in a given direction, about the pad ds will be 1/6 P DSvDt. . When colliding with the platform, these molecules will give it momentum P=2 m o v 1/6 P DSvDt = 1/3 m o v 2 DSDt
Then the pressure of the gas exerted by it on the wall of the vessel,
R= F/DS=P/(DSDt)=1/3 m o v 2 (1),
(because F=dP/dt).
If the gas in the volume V contains N molecules moving with different speeds, then we can consider the root-mean-square velocity characterizing the entire set of gas molecules.
(2)
Equation (1), taking into account (2), takes the form
p = 1/3 m o v sq 2 (3)
Given that P= N/V, we get pV = 1/3 Nm o v kv 2
or pV = 2/3 N (m o v square 2/2)= 2/3 E(4),
Where E - the total kinetic energy of the translational motion of all gas molecules.
Expression (4) (ᴛ.ᴇ. pV = 2/3E) or its equivalent (3) is called the basic equation of the molecular-kinetic theory of ideal gases. Exact calculation, taking into account the movement of molecules in all possible directions, gives the same formula.
Considering that on the one hand p= n kT, and on the other p = 1/3 m o v sq 2, we get the expression for the root mean square speed
(5),
since the molar mass m = m 0 N A, Where t 0 - the mass of one molecule, N A - Avogadro constant, To= R/N A. From here it is easy to find that at room temperature the oxygen molecules have a root-mean-square velocity of 480 m/s.
The average kinetic energy of the translational motion of one molecule of an ideal gas, using that p= n kT, and p = 1/3 m o v square 2 is equal to
e= m o v sq. 2/2=3/2kT
Those. it is proportional to the thermodynamic temperature and depends only on it. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, thermodynamic temperature is a measure of the average kinetic energy of the translational motion of ideal gas molecules.
When deriving the basic equation of the molecular-kinetic theory of gases and the Maxwellian distribution of molecules over velocities, it was assumed that external forces do not act on gas molecules, and therefore the molecules are uniformly distributed over the volume. In this case, the molecules of any gas are in the potential field of gravity of the Earth. Gravity, on the one hand, and thermal motion of molecules, on the other hand, lead to a certain stationary state of the gas, in which the gas pressure decreases with height.
Let us derive the law of change of pressure with height, assuming that the gravitational field is uniform, the temperature is constant, and the mass of all molecules is the same. If the atmospheric pressure is at altitude h equals R, then at the height h+ dh it is equal to p + dp(at dh> ABOUT dp< 0, так как давление с высотой убывает). Разность давлений R And p + dp is equal to the weight of the gas enclosed in the volume of a cylinder with a height dh with a base area equal to unit area:
p - (p + dp)= ρ gdh,
where ρ is the gas density at altitude h. Hence,
dp=- ρ gdh.(1)
Using the ideal gas equation of state pV = m/mRT, where m is the mass of gas, m- molar mass of gas), we find that the density of the gas is equal to
r= m/V= pm/(RT).
Substituting into (1), we obtain
or
We integrate this equation, taking into account the fact that p is the pressure at altitude h, and the pressure on h\u003d 0 (on the surface of the earth) is equal to po.
(2),
since m = m 0 N A, And R= kN A, Where t o - the mass of one molecule, N A - Avogadro constant.
Expression (2) is called barometric formula. It allows you to find atmospheric pressure based on altitude (or, by measuring pressure, find altitude). From this formula it follows that the pressure decreases with height the faster, the heavier the gas.
The barometric formula (2) can be transformed if we use the expression R = PCT:
(3)
Here n- concentration of particles at height h, A no- concentration of particles at height h=0.
It follows from formula (3) that as the temperature decreases, the number of molecules at a certain height h decreases. At T=0 all molecules would be on the surface of the earth. Gravity tends to lower the molecule to the ground, and thermal motion scatters them along the heights, in connection with this, the distribution of molecules in the atmosphere with height is determined by the balance of these tendencies.
If we take into account that m o gh= P is the potential energy of a molecule in the gravitational field, then the formula can be rewritten.
(4)
Expression (4) is called Boltzmann distribution in an external potential field. It follows from it that at a constant temperature the density of a gas is greater where the potential energy of its molecules is less.
If the particles have the same mass and are in a state of chaotic thermal motion, then the Boltzmann distribution (4) is valid in any external potential field, and not only in the field of gravity.
Let an ideal gas be in the field of conservative forces under conditions of thermal equilibrium. In this case, the gas concentration will be different at points with different potential energies, 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 with decreasing height, because the bottom layer has to support the weight of all the atoms located above.
Based on the basic equation of molecular kinetic theory: P = nkT, replace P And P0 in the barometric formula (2.4.1) on n And n 0 and get Boltzmann distribution for the molar mass of gas:
As the temperature decreases, the number of molecules at heights other than zero decreases. At T= 0 thermal motion stops, all molecules would settle down on the earth's surface. At high temperatures, on the contrary, the molecules are almost uniformly distributed along the height, and the density of the molecules slowly decreases with height. Because mgh is the potential energy U, then at different heights U=mgh- different. Therefore, (2.5.2) characterizes the distribution of particles according to the values of potential energy:
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.
