Kinetic theory

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Kinetic theory, or kinetic-molecular theory attempts to explain the macroscopic properties of gases by considering their composition at a molecular level.

1 Postulates
2 Pressure
3 Temperature
4 RMS speeds of molecules
5 See also
6 External links

Postulates

The fundamental aspects of kinetic theory are given by several postulates:

Gases are composed of molecules in constant, random motion; the moving particles constantly collide with each other and with the walls of the container containing the gas
The collisions between gas molecules are elastic
The total volume of the gas molecules is negligible compared to the volume of the entire container
The interactions between molecules are negligible

The above postulates accurately describe the behavior of ideal gases. Real gases approach ideality under conditions of low density and high temperature.

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

2 m v x

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

Since force is the rate of change of momentum and the particle under consideration impacts with the wall once every 2l/v_x time units (where l is the length of the container), the force due to this particle is

$mv_x \cdot v_x \over l$

and the total force acting on the wall is

$m\sum_j v_{jx}^2 \over l$

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

Since the particles are moving randomly in all directions, and since for each particle

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

the expression for the total force becomes

$m\sum_j v_j^2 \over 3l$

This can be written as

$Nmv_{rms}^2 \over 3l$

where v_rms is the root mean square velocity of the gas.

Therefore, pressure, the force per unit area, equals

$Nmv_{rms}^2 \over 3Al$

where A is the area of the wall.

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 is 3/2 times Boltzmann's constant, which is the ratio of the gas constant R to Avogadro's number (independent of the gas). This result is related to the equipartition theorem.

Thus the kinetic energy per kelvin 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

Examples:

hydrogen (molecular mass = 2): 1703 kJ/kg
nitrogen (molecular mass = 28): 122 kJ/kg
oxygen (molecular mass = 32): 106 kJ/kg

RMS speeds of molecules

From the kinetic energy formula it cna be shown that

$v_{rms}^2$ = 24,940 T / molecular mass

with v in m/s and T in kelvins.

For standard temperature, root mean square speeds are:

thermal neutrons 2610 m/s
hydrogen 1846 m/s
nitrogen 493 m/s
oxygen 461 m/s

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.

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.de:Kinetische Gastheorie

es:Teoría cinética it:Teoria cinetica dei gas sl:Kinetična teorija plinov sv:Kinetiska gasteorin

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Categories: Gases | Thermodynamics