# On a functional equation arising in the kinetic theory of gases

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Part I: Existence, Arch. Part II: The full initial value problem, Arch. Arkeryd and C. Cercignani, On a functional equation arising in the kinetic theory of gases, Rend. Lincei s. MathSciNet Google Scholar. Cercignani, On the Boltzmann equation for rigid spheres, Transp. Theory Stat. Google Scholar. Cercignani, Are there more than five linearly independent collision invariants for the Boltzmann equation?

Statistical Phys. Cercignani, Ludwig Boltzmann. The man who trusted atoms, Oxford University Press, Oxford Di Perna and P. Lions, On the Cauchy problem for Boltzmann equations, Ann.

Grad, On the kinetic theory of rarified gases, Comm. Pure Appl. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in a vacuum, Commun. Pulvirenti, Global validity of the Boltzmann equation for two-and three-dimensional rare gas in vacuum: Erratum and improved result, Comm.But here, we will derive the equation from the kinetic theory of gases.

The kinetic theory of gases is a very important theory which relates macroscopic quantities like pressure to microscopic quantities like the velocity of gas molecules. This equation is applicable only for ideal gases, but be approximated for real gas under some conditions. In German chemist August Kronig had developed the simple gas model. He had only considered the translational motion of gas particles in his model. Later, in Rudolf Clausius independent of Kronig developed a sophisticated version of the kinetic theory of gases.

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Clausius had not only considered translational motion but also the rotational and vibrational motion of gas molecules. The behaviour of gases is simplified significantly by making many assumptions. This is necessary to avoid complexity.

### Elementary Properties of the Solutions

The assumptions in the theory are as follows:. Visualise a cube in space as shown in the figure below. L be the length of the cube and Area, A. V be the volume of the cube. The cube is filled with an ideal gas of pressure Pat temperature T. Let n and N be the moles and the number of molecules of the gas in the cube. For simplicity, we will start with the x -direction as depicted in the below figure. This is the first collision. The total distance travel in the first collision is L 0 and the time taken is t 1.

This is also shown in the figure below. Also, the distance travel by the molecule between the 2 nd and 3 rd collision is the same 2 L. This two quantities, time interval and distance travel, will remain the same for all successive collisions. So, we can calculate the speed v x which is distance travel divided time interval for all collisions as follows:.

This is the time period for a collision, and the reciprocal of it is frequency f of collisions. So, the force exerted by the molecule on the wall F mw is opposite of the force exerted by the wall on the molecule F wm. In the beginning, we have considered only one molecule; So, the above force equation is for only one molecule. The force by N molecules moving in x -direction is.

By knowing the force on the wall, we can determine pressure Pwhich is force per unit area. In equation 4we have only incorporated the x direction. We will replace it with a more convenient quantity: temperature which can easily measure by a thermometer. The kinetic energy equation from classical mechanics is half of the mass times square of velocity. Here, k is the Boltzmann constant and its approximate value is 1.

Substituting in equation the above equation. In the above equation, the ratio is the moles of gas n. Assumptions in Ideal Gas Equation The behaviour of gases is simplified significantly by making many assumptions.

The assumptions in the theory are as follows: The gas comprises of very small particle known as molecules. The molecules of the gas are solid rigid identical spheres.

All the molecules of the gas have the same mass. The volume of each molecule is negligible in comparison to the total volume of the gas. Or in other words, the molecules do not occupy space relative to the size of a container in which they are stored. The intermolecular forces among the molecules are zero.The kinetic theory of gases is a scientific model that explains the physical behavior of a gas as the motion of the molecular particles that compose the gas. In this model, the submicroscopic particles atoms or molecules that make up the gas are continually moving around in random motion, constantly colliding not only with each other but also with the sides of any container that the gas is within.

It is this motion that results in physical properties of the gas such as heat and pressure. It can also in many ways be applied to fluids as well as gas. The example of Brownian motiondiscussed below, applies the kinetic theory to fluids. The Greek philosopher Lucretius was a proponent of an early form of atomism, though this was largely discarded for several centuries in favor of a physical model of gases built upon the non-atomic work of Aristotle. Without a theory of matter as tiny particles, the kinetic theory did not get developed within this Aristotlean framework. The work of Daniel Bernoulli presented the kinetic theory to a European audience, with his publication of Hydrodynamica.

At the time, even principles like the conservation of energy had not been established, and so a lot of his approaches were not widely adopted. Over the next century, the kinetic theory became more widely adopted among scientists, as part of a growing trend toward scientists adopting the modern view of matter as composed of atoms. One of the lynchpins in experimentally confirming the kinetic theory, and atomism is general, was related to Brownian motion.

This is the motion of a tiny particle suspended in a liquid, which under a microscope appears to randomly jerk about. In an acclaimed paper, Albert Einstein explained Brownian motion in terms of random collisions with the particles that composed the liquid. This paper was the result of Einstein's doctoral thesis work, where he created a diffusion formula by applying statistical methods to the problem. A similar result was independently performed by the Polish physicist Marian Smoluchowski, who published his work in Together, these applications of kinetic theory went a long way to support the idea that liquids and gases and, likely, also solids are composed of tiny particles.

The kinetic theory involves a number of assumptions that focus on being able to talk about an ideal gas. The result of these assumptions is that you have a gas within a container that moves around randomly within the container. When particles of the gas collide with the side of the container, they bounce off the side of the container in a perfectly elastic collision, which means that if they strike at a degree angle, they'll bounce off at a degree angle.

The component of their velocity perpendicular to the side of the container changes direction but retains the same magnitude. The kinetic theory of gases is significant, in that the set of assumptions above lead us to derive the ideal gas law, or ideal gas equation, that relates the pressure pvolume Vand temperature Tin terms of the Boltzmann constant k and the number of molecules N. The resulting ideal gas equation is:. Share Flipboard Email. Andrew Zimmerman Jones.

Math and Physics Expert.The aim of kinetic theory is to account for the properties of gases in terms of the forces between the molecules, assuming that their motions are described by the laws of mechanics usually classical Newtonian mechanics, although quantum mechanics is needed in some cases. The present discussion focuses on dilute ideal gases, in which molecular collisions of at most two bodies are of primary importance.

Only the simplest theories are treated here in order to avoid obscuring the fundamental physics with complex mathematics. The ideal gas equation of state can be deduced by calculating the pressure as caused by molecular impacts on a container wall.

The calculation is significant because it is basically the same one used to explain all dilute-gas phenomena. A molecule experiences a change in momentum when it collides with a container wall; during the collision an impulse is imparted by the wall to the molecule that is equal and opposite to the impulse imparted by the molecule to the wall.

The sum of the impulses imparted by all the molecules to the wall is, in effect, the pressure. Consider a system of molecules of mass m traveling with a velocity v in an enclosed container. In order to arrive at an expression for the pressure, a calculation will be made of the impulse imparted to one of the walls by a single impact, followed by a calculation of how many impacts occur on that wall during a time t.

## Kinetic Theory Of Gases

Although the molecules are moving in all directions, only those with a component of velocity toward the wall can collide with it; call this component v zwhere z represents the direction directly toward the wall. Not all molecules have the same v zof course; perhaps only N z out of a total of N molecules do.

To find the total pressure, the contributions from molecules with all different values of v z must be summed. A molecule approaches the wall with an initial momentum m v zand after impact it moves away from the wall with an equal momentum in the opposite direction, - m v z. The number of impacts on a small area A of the wall in time t is equal to the number of molecules that reach the wall in time t. Since the molecules are traveling at speed v zonly those within a distance v z t and moving toward the wall will reach it in that time. Thus, the molecules that are traveling toward the wall and are within a volume A v z t will strike the area A of the wall in time t.

On the average, half of the molecules in this volume will be moving toward the wall. Equating these two expressions, the time factor t cancels out. Because there are different values of v z 2 for different molecules, the average value, denoted v z 2is used to take into account the contributions from all the molecules.

Since the molecules are in random motion, this result is independent of the choice of axis. The gas is in equilibriumso it must appear the same in any direction, and the average velocities are therefore the same in all directions—i. To rewrite this in molar units, N is set equal to n N 0 —i.

Any energy residing in the internal motions of the individual molecules is simply carried separately without contributing to the pressure.

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Average molecular speeds can be calculated from the results of kinetic theory in terms of the so-called root-mean-square speed v rms. Molecule-molecule collisions were not considered in the calculation of the expression for pressure even though many such collisions occur. Such collisions could be ignored because they are elastic; i. Two molecules therefore continue to carry the same momentum to the wall even if they collide with one another before striking it.

The ideal gas equation of state remains valid as the density is decreased, even holding for a free-molecule gas. The equation eventually fails as the density is increased, however, because other molecules exert forces and change the rate of collisions with the walls. It was not until the mid- to late 19th century that kinetic theory was successfully applied to such calculations as gas pressure.Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container.

Express the relationship between the pressure and the average kinetic energy of gas molecules in the form of equation. In Newtonian mechanics, if pressure is the force divided by the area on which the force is exerted, then what is the origin of pressure in a gas? What forces create the pressure? We can gain a better understanding of pressure and temperature as well from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container, as illustrated in the figure below.

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Translational Motion of Helium : Real gases do not always behave according to the ideal model under certain conditions, such as high pressure. Here, the size of helium atoms relative to their spacing is shown to scale under atmospheres of pressure.

Since the assumption is that the particles move in random directions, if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value of the squared velocity along each direction must be same. This does not mean that each particle always travel in 45 degrees to the coordinate axes. Pressure : Pressure arises from the force exerted by molecules or atoms impacting on the walls of a container.

This is a first non-trivial result of the kinetic theory because it relates pressure a macroscopic property to the average translational kinetic energy per molecule which is a microscopic property. A gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution. The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution illustrated in.

The distribution has a long tail because some molecules may go several times the rms speed. The most probable speed v p at the peak of the curve is less than the rms speed v rms. As shown in, the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.

Maxwell-Boltzmann Distribution at Higher Temperatures : The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures. Maxwell-Boltzmann Distribution : The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas. Maxwell-Boltzmann distribution is a probability distribution. It applies to ideal gases close to thermodynamic equilibrium, and is given as the following equation:. Derivation of the formula goes beyond the scope of introductory physics.

It can also be shown that the Maxwell—Boltzmann velocity distribution for the vector velocity [ v xv yv z ] is the product of the distributions for each of the three directions:. This makes sense because particles are moving randomly, meaning that each component of the velocity should be independent.

Usually, we are more interested in the speeds of molecules rather than their component velocities. The Maxwell—Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above.But if you incorporate the task of making the request into your regular business routine, you will begin to accumulate an impressive amount of reviews.

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