§4 Maxwell's law of distribution of velocities and energies
The distribution law of ideal gas molecules in velocity,
theoretically derived by Maxwell in 1860 determines how many molecules
dN homogeneous (p = const) monatomic ideal gas of the total number N of molecules per unit volume at a given temperature T speeds in the range of inmates v to v + dv.
To derive the function of the velocity distribution f(v) equal to the ratio of the number of molecules dN, the speed of which lie in the interval v ÷ v + dv to the total number of molecules N and the size of the interval dv
Maxwell used two sentences:
directions in space are equivalent and therefore any direction of the
particle, ie, any direction of the velocity is equally likely. This
property is sometimes called the property of the isotropic distribution
b) moving along three mutually perpendicular axes that are independent, ie x-component of velocity vx does not depend on what the meaning of its components or vy, vz. And then the output f(v) is initially for one component vx, and then extended to all the coordinates of speed.
It is also believed that the gas consists of a large number N of identical molecules are in a state of random thermal motion at the same temperature. Force fields do not apply to gas.
Function f (v) determines the relative number of molecules dN (v) / N speeds are in the interval from v to v + dv (eg gas has N = 106 molecules, with dN = 100 molecules have a velocity of v = 100 and v + dv = 101 m /s (dv = 1 m / s) then
Using the methods of probability theory, Maxwell found the function f (v) - the distribution of ideal gas molecules in velocity:
f (v) depends on the type of gas (the mass of the molecule) and the state variable (temperature T)
f (v) depends on the ratio of the kinetic energy of a molecule corresponding to selected speed vto the value of kT characterizes the average thermal energy of the gas molecules.
For small v and the function f(v) changes almost on a parabola v2. With an increase in the factor decreases faster than the multiplier , ie a max function f (v).
The speed at which the distribution function of molecules of an ideal
gas at the maximum speed is called the most probable velocity vmp from the condition:
therefore, with increasing temperature the most probable speed
increases, but the area of ??S, bounded by the curve of the
distribution function remains unchanged, since the normalization
condition since the probability of a certain event is 1), so that the temperature distribution curve f(v) will stretch and fall.
In statistical physics,
the average value of any quantity is defined as the integral from 0
to infinity, the product of the probability density for this value
Then the arithmetic average velocity of the molecules
and integrating by parts received
Speed, characterizing the state the gas
§ 5 Experimental verification of the Maxwell distribution law – Stern experience
Along the axis
of the inner cylinder tight platinum wire, covered with a layer of
silver, which is heated by the current. When heated, the silver
evaporates; the silver atoms are emitted through the gap and onto the
inner surface of the second cylinder. If both cylinders are fixed, all
atoms regardless of their speed fall in the same place B. When
rotating cylinder with an angular velocity ω silver atoms get into the points B ', B'' and so on. The magnitude of ω, the distance ? and displacement x = BB 'can calculate the velocity of the atoms belonging to a point B'.
getting blurred. Exploring the thickness of the deposited layer can
estimate the distribution of the velocity, which corresponds to a
§6 Barometric formula. The Boltzmann distribution
Up to now, we have considered the behavior of an ideal gas not liable
to attack to external force fields. From experience it is well known
that the action of external forces, a uniform distribution of particles
in space can be broken. Since gravity molecules tend to fall to the
bottom of the vessel. Intense thermal motion prevents precipitation of,
and the molecules are distributed so that their concentration
gradually decreases with increasing height.
the variation of pressure with height assuming that the gravitational
field is uniform, the temperature is constant and the mass of all the
molecules of the same. If the atmospheric pressure at a height h equal to p, then at a height h + dh is equal to p + dp (with dh> 0, dp <0, since p decreases with increasing h).
The pressure difference at the heights h and h + dh, we can determine as the weight of the air molecules enclosed in a volume with a base area equal to 1 and a height dh.
ρ - density at the height h, and since , then ρ = const.
Of Mendeleev-Clapeyron equation.
With the change in height from h1 to h2 pressure changes from p1 to p2
Potentiated this expression
Barometric formula shows how the pressure changes with altitude
n - the concentration of molecules at a height
n0 - the concentration of molecules at a height h = 0.
the potential energy of the molecules in a gravitational field
Boltzmann distribution law in an exterior potential field. From it follows that at T = const the density of gas is more there where potential energy of molecules is less.
§ 7 The experimental determination of the Avogadro constant
Perrin (French scientist) in 1909, studied the behavior of Brownian
particles in the emulsion gamboge (tree sap) with dimensions were
examined with a microscope, which had a depth of field - 1 mm. Moving
the microscope in the vertical direction could be to investigate the
distribution of Brownian particles in height.
Having applied to them a Boltzmann distribution law it is possible to write down
where m- particle mass
m1 - weight of the displaced fluid.
If n1 and n2 concentration of particles at levels h1 and h2, and
Value is in good agreement with the reference value,
which confirms the Boltzmann distribution of particles.