Kinetic theory

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 molecule, 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:

The gas consists of very small particles, each of which has a mass.
These molecules are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container.
The collisions of gas particles with the walls of the container holding them are perfectly elastic.
The interactions between molecules are negligible. They exert no forces on one another except during collisions.
The total volume of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is relatively large compared to their size.
The average kinetic energy of the gas particles depends only on the temperature of the system.
Quantum-mechanical effects are negligible. This means that the interparticle distance is much larger than the thermal de Broglie wavelength and the molecules can be treated as classical objects.

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:

\Delta p_x = p_i - p_f = m v_x - -m v_x = 2 m v_x\,

where v_x is the x-component of the initial velocity of the particle.

The particle impacts the wall once every 2l/v_x time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/v_x time units, only the momentum change on one wall is considered in the derivation so that the particle produces a momentum change once every 2l/v_x time units on a particular wall.

\Delta t = \frac{2l}{v_x}

Since force is the rate of change of momentum, the force due to this particle is:

F = \frac{\Delta p}{\Delta t} = \frac{2 m v_x}{\frac{2l}{v_x}} = \frac{m v_x^2}{l}

The total force acting on the wall is: propotional to the variety of the molecular structure of an atom of a certain element or compound.

F = \frac{m\sum_j v_{jx}^2}{l}

where the summation is over all the gas molecules in the container.

The magnitude of the velocity for each particle will follow:

v^2 = v_x^2 + v_y^2 + v_z^2

Now considering the total force acting on all six walls, adding the contributions from each direction we have:

Total Force = 2 \cdot \frac{m}{l}(\sum_j v_{jx}^2 + \sum_j v_{jy}^2 + \sum_j v_{jz}^2) = 2 \cdot \frac{m}{l} \sum_j (v_{jx}^2 + v_{jy}^2 + v_{jz}^2) = 2 \cdot \frac{m \sum_j v_{j}^2}{l}

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:

F = \frac{1}{6} (2 \cdot \frac{m \sum_j v_{j}^2}{l}) = \frac{m \sum_j v_{j}^2}{3l})

The quantity $\frac{1}{N}\sum_j v_{j}^2$ can be written as $\overline{v^2}$ , where the bar denotes an average, in this case an average over all particles. This quantity is also denoted by $v_{rms}^2$ where $v_{rms}$ is the root-mean-square velocity of the collection of particles.

Thus the force can be written as:

F = \frac{Nmv_{rms}^2}{3l}

Pressure, which is force per unit area, of the gas can then be written as:

P = \frac{F}{A} = \frac{Nmv_{rms}^2}{3Al}

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

P = {Nmv_{rms}^2 \over 3V}

where V is the volume. Also, as Nm is the total mass of the gas, and mass divided by volume is density

P = {1 \over 3} \rho\ v_{rms}^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 (1/2mv_rms²), which is a microscopic property. Note that the product of pressure and volume is simply two thirds of the total kinetic energy.

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:

per mole: 12.47 J
per molecule: 20.7 yJ = 129 μeV

At standard temperature (273.15 K), we get:

per mole: 3406 J
per molecule: 5.65 zJ = 35.2 meV

RMS speeds of molecules

From the kinetic energy formula it can be shown that

v_{rms}^2

= 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:

thermal neutrons 2610 m/s at 275 K or 2 °C

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.

References

The Mathematical Theory of Non-uniform Gases : An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases Sydney Chapman, T. G. Cowling

External links

Introduction to the kinetic molecular theory of gases, from The Upper Canada District School Board
Java animation illustrating the kinetic theory from University of Arkansas
Flowchart linking together kinetic theory concepts, from HyperPhysics
Interactive Java Applets allowing high school students to experiment and discover how various factors affect rates of chemical reactions.
Molecular kinetic theory fundamentals

Gases | Thermodynamics

Gourt Home::Gourt :: Kinetic theory