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Kinetic theory attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions between molecules moving about with a certain velocity. Kinetic theory is also known as kinetic-molecular theory or collision theory.
History In 1738, Swiss physician and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.• Postulates The kinetic theory for ideal gases makes the following assumptions: In addition, if the gas is in a container, the collisions with the walls are assumed to be instantaneous and elastic. More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions. In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number. The kinetic theory has also been extended to include inelastic collisions in granular matter by Jenkins and others. Pressure Pressure is explained by kinetic theory as arising from the force exerted by colliding gas molecules onto the walls of the container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is: where vx is the x-component of the initial velocity of the particle. The particle impacts the wall once every 2l/vx time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/vx time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2l/vx time units. The force due to this particle is: The total force acting on the wall is: where the summation is over all the gas molecules in the container. The magnitude of the velocity for each particle will follow: Now considering the total force acting on all six walls, adding the contributions from each direction we have: where the factor of two arises from now considering both walls in a given direction. Assuming there are a large number of particles moving sufficiently randomly, the force on each of the walls will be approximately the same and now considering the force on only one wall we have: The quantity can be written as , where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by where is the root-mean-square velocity of the collection of particles. Thus the force can be written as: Pressure, which is force per unit area, of the gas can then be written as: where A is the area of the wall of which the force exerted on is considered. Thus, as cross-sectional area multiplied by length is equal to volume, we have the following expression for the pressure where V is the volume. Also, as Nm is the total mass of the gas, and mass divided by volume is density ho v_^2 where ρ is the density of the gas. This result is interesting and significant, because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule (1/2mvrms2), which is a microscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy. Number of collisions with wall One can calculate the number of atomic or molecular collisions with a wall of a containter per unit area per unit time. Assuming an ideal gas, a derivations of this* results in an equation for total number of collisions per unit time per area: A = Temperature The above equation tells us that the product of pressure and volume per mole is proportional to the average molecular kinetic energy. Further, the ideal gas equation tells us that this product is proportional to the absolute temperature. Putting the two together, we arrive at one important result of the kinetic theory: average molecular kinetic energy is proportional to the absolute temperature. The constant of proportionality per degree of freedom is 1/2 times Boltzmann's constant. This result is related to the equipartition theorem. Monatomic gasses have 3 degrees of freedom. As noted in the article on heat capacity, diatomic gasses should have 7 degrees of freedom, but the lighter gasses act as if they have only 5. Thus the kinetic energy per kelvin (monatomic ideal gas) is: At standard temperature (273.15 K), we get: RMS speeds of molecules From the kinetic energy formula it can be shown that = 3kT / molecular mass with v in m/s, T in kelvins, and k is Boltzmann's Constant. For standard temperature, root mean square speeds are: The most probable speeds are 81.6% of these (e.g. for thermal neutrons 2131 m/s), and the mean speeds 92.1%, see also distribution of speeds. See also | ||||||||
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