Maxwell–Boltzmann distribution

The Maxwell - Boltzmann distribution or Maxwellian velocity distribution is a probability distribution of statistical physics and plays in thermodynamics, especially the kinetic theory of gases, an important role. Describing the statistical distribution of the amount of particle velocities in an ideal gas. It was named after James Clerk Maxwell and Ludwig Boltzmann, they have first derived in 1860. It results from the Boltzmann statistics.

Because of the simplifying assumptions of an ideal gas, the velocity distribution of the particles of a real gas shows deviations. However, in low density and high temperature, the Maxwell - Boltzmann distribution is sufficient for most analyzes.

• 2.1 Conclusions from the equations
• 2.2 Meaning of the thermodynamics

• 3.1 Most probable speed
• 3.2 Mean speed
• 3.3 Square averaged velocity
• 3.4 Harmonic Mean
• 3.5 Relations between the speeds

Derivation of the velocity distribution in the kinetic theory of gases

Derivation using the Boltzmann factor

The energy of a Teilchenzustands is by

Added, and the probability that it is open in the thermodynamic equilibrium state of the system of particles of a particle by the Boltzmann factor

What is needed is according to the proportion of molecules with magnitude of velocity in an interval. The corresponding density of states is to be determined from the assumption that the density of the three-dimensional space of the velocity components is constant. After all states have the same kinetic energy, the distance from the origin, so here fill a sphere the size. Then part of the volume element of the interval. Consequently, the proportion of molecules searched is equal to the product of the volume element, the constant for the whole voxel Boltzmann factor, and a constant scaling factor:

The normalization factor can be determined from this that the integral over the distribution has the value 1.

Derivation using the normal distribution of the components of the velocity

According to the kinetic theory of gases move in an ideal gas at temperature T ( in Kelvin ) are not all gas particles with the same speed, but randomly distributed at different speeds. There is no direction in space is preferred in this case. Mathematically, this is formulated so that the components of the velocity vector of the gas particles of mass m are independent and normally distributed, with the parameters

Here kB is Boltzmann's constant. The density of the distribution of the three-dimensional velocity space, here designated, thus obtained as the product of the distributions of the three components:

For the derivation of the Maxwell - Boltzmann distribution, one must above all particles with the same velocity amount integrate (or clear this " sum "). These lie on a spherical shell of radius v and infinitesimal thickness at a speed of 0:

It refers to the above-mentioned Integral over all vectors with sums in the interval. Since entering into the definition of only the squared magnitude of the velocities ( see definition above), which does not change in the infinitesimal interval, the integral is simply reshape:

Herein remains only the simple volume integral to be solved. It just gives the volume of the infinitesimal spherical shell and we obtain the desired Maxwell - Boltzmann distribution:

The importance and scope

Conclusions from the equations

• From the above equations it follows that the fraction f of the particles is directly proportional to velocity interval? V? V itself, as long as F (v) remains constant. If DELTA.v therefore increased slightly or concerns you more speeds in the interval, under the additional assumption temperature and molar mass are constant, so the number of particles contained in it rises to small deviations in proportion to DELTA.v. In other words, the distribution function is differentiable.
• The distribution function has a decreasing exponential function of the form ex, where x = Mv2/2RT. Since the expression x is directly proportional to the square of the particle velocity v2 at constant temperature and constant molar mass, it can be concluded from this that the exponential function and thus to a limited extent, the proportion of molecules for high speeds very small, and hence for small velocities very is large (for the exact context, see the pictures on the right ).
• For gases with a large molar mass M is the expression x, assuming a constant temperature, also very large and the exponential function takes consequently faster. This means that to meet the probability of heavy molecules at high speeds is very small and therefore very large for lighter molecules with a low molecular mass ( see picture above right).
• In the opposite case of a large temperature and a constant molar mass of the expression x is very small and the exponential function is therefore at an increasing speed faster towards zero. At a very high temperature, the proportion of particles is therefore lower than at a lower temperature (see figure below right).
• The lower the speed, the stronger will the quadratic expression V2 from outside the exponential function. This means that the proportion of faster molecules at low speeds decreases more rapidly than the speed itself, but also in return, that it increases quadratically with an increase in speed.

All other sizes require that the proportion of particles at a certain speed is always moving in the interval between zero and one ( [0,1] ). The two pictures on the right illustrate the dependence of the Maxwell - Boltzmann distribution of particle mass and temperature of the gas. As the temperature T increases, the average speed and the distribution is wider at the same time. With increasing particulate mM, however, the average velocity decreases, and the velocity distribution is narrower at the same time. This relationship between particle velocity and temperature or particle velocity and particle mass / molar mass here is also quantitatively describable. See the section rms speed.

Importance for the thermodynamics

The Maxwell - Boltzmann distribution, for example, explains the process of evaporation. For example, wet laundry to dry at temperatures of 20 ° C, as there is a small proportion of molecules with the required high speed available in this distribution curve, which can be solved from the liquid bandage. It will therefore always be some molecules even at low temperatures that are fast enough to overcome the attractive forces by their neighbors and move from the liquid or solid state to a gaseous state, which is referred to as evaporation or sublimation. Conversely, there are also among the relatively fast particles of the gas always a few who do not have sufficient speeds and therefore again change from the gaseous to the liquid or solid state, which is known as condensation or Resublimation. These processes are summarized under the term of the phase transition, with a dynamic equilibrium is established between particles that enter the gas phase and particles that emerge from the gas phase, in that there is no interference from the outside. This is object of study of equilibrium thermodynamics, therefore it is called also thermodynamic equilibrium. The particles of the gaseous phase in the equilibrium state in this case exert a pressure, which is referred to as the saturation vapor pressure. Is plotted the phase behavior of substances in the phase diagram.

See also: equation of state, fundamental equation, Thermodynamic Potential, Ideal gas, real gas, triple point, critical point

Particle velocities

All distributions is assumed that a reference point is selected, which is not moving, otherwise would have no symmetry of the velocity distribution and the mass of gas moves as a whole.

Most probable speed

The most probable velocity

Is the speed at which the distribution function has its maximum value. It can be calculated from the requirement. here is the particle mass and the molar mass of the substance.

Medium speed

The average speed is defined by:

Here, () the individual velocities of the particles and their total number.

As a solution of the integral is obtained:

Rms speed

The rms speed is defined by:

From the kinetic theory of gases, the following state equation:

The empirically determined equation of state of ideal gases is here:

Substituting the expression pV equal one obtains:

Changed by the square root of one finally obtains:

It is found that the root mean square velocity of the gas is directly proportional to the square root of the temperature, in that the molar mass remains constant, but this is generally the case.

From this it can be derived under the assumption of constant molar mass, an important principle:

A doubling of the temperature on the Kelvin scale leads to an increase of the root mean square by a factor of speed.

Through this basic relationship, the dependence of temperature on the velocity of the particles can not only qualitatively but also quantitatively derived. The temperature is thus defined in this way by the kinetic theory of gases. Assuming a constant temperature and a variable molar mass is shown in this connection in the same form, the relationship between it and the root mean square velocity, both of which, however, in contrast to the temperature are inversely proportional to each other, as can be seen from the above equation.

The same result can also be when you F (v ) is substituted into the following equation and then integrated:

However, the root mean square velocity is also a measure of the average kinetic energy ( E kin ) of the molecules:

Changed resulting therefrom:

Harmonic mean

For purposes of the peak hours, etc. you need called another mean, harmonic mean. The harmonic mean is defined by:

Here, () the individual velocities of the particles and their total number.

By substituting and and integrating we obtain:

Or

Relations between the velocities

In the picture to the right the Maxwell - Boltzmann velocity distribution for nitrogen ( N2) is shown at three different temperatures. It is also the most probable speed and the average speed located. Always applies that the most likely velocity is less than the average speed. General:

The relationship between the velocities resulting from this:

Derivation of the canonical ensemble

The Maxwell - Boltzmann distribution can be derived using the methods of statistical physics. One considers an N- particle system with the Hamiltonian

To derive only the assumption is made that the potential U conservative, ie the independently. Therefore, the following derivation is also valid for real gases.

Let the system in the canonical state with the phase space density

And the canonical partition function

The parameter is proportional to the inverse temperature

The expected value of a classical observable is given by

For the transformation of probability densities: Given is a random variable and a probability density and a picture. Then the probability of the random variable Diche.

Now we can calculate the probability density function for the impulse of any particle of the system. After the above transformation theorem:

All local integrations can be shortened, and any momentum integrations for. Thus, the integration only remains.

To evaluate this expression one uses in the numerator the convolution property of the delta function

The denominator is integrated over a Gaussian function; the integration in three dimensions can be reduced to a one-dimensional integral with

The probability density is obtained for the pulse of any particle:

The pre-factor substantially equal to the thermal de Broglie wavelength.

In order to estimate the probability density for the rate of speed with the transform set determine

The integration is carried through in spherical coordinates, and uses the relationship

Now again the convolution property of the delta function to use

Here is the Heaviside step function, which dissolves the probability for negative amount speeds.

If, for one comes to the Maxwell - Boltzmann distribution