Kinetic theory

The kinetic theory of gases is a theory that explains the macroscopic properties of gases by consideration of their composition at a molecular level.

 Contents

Postulates

The fundamental principles of the kinetic theory are given in the form of 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.
• The collisions between gas molecules are elastic.
• The total volume of the gas molecules is negligible compared to the volume of the container.
• The forces of attraction between the 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 the kinetic theory as arising from the force exerted by collisions of gas molecules with the walls of the container. The derivation of the mathematical expression for pressure is given below:

Consider a gas with N molecules, each of mass m, enclosed in a cuboidal container of volume V. Suppose that a gas molecule collides with a wall of the container which is perpendicular to the x co-ordinate 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 given by

[itex]2mv_x[itex]

where vx is the x-component of the initial velocity of the particle.
Now, force is the rate of change of momentum. The particle under consideration impacts with the wall once every 2l/vx time units, where l is the length of the container. Therefore the force due to this particle is

[itex]mv_x *v_x \over l[itex]

and the total force on the wall is

[itex]m\sum_j v_{jx}^2 \over l[itex]

where the summation is over all the gas molecules in the container. Since the particles are moving randomly in all directions, and since

[itex] v^2 = v_x^2 + v_y^2 + v_z^2 [itex]

for each particle, the expression for the total force becomes

[itex]m\sum_j v_j^2 \over 3l[itex]

This can be written as

[itex]Nmv_{rms}^2 \over 3l[itex]

where vrms is the root mean square velocity of the gas. Therefore, pressure, the force per unit area, equals

[itex]Nmv_{rms}^2 \over 3Al[itex]

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

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

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

[itex] P = {1 \over 3} \rho\ v_{rms}^2[itex]

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 (1/2 mvrms2), which is a microscopic property.

Note that the product of pressure and volume is simply two-third 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:

Rms speeds of molecules etc.

From the kinetic energy formula we find:

[itex]v_{rms}^2[itex] = 24,940 T / molecular mass

with v in m/s and T in kelvins.

For standard temperature the 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.

• Introduction (http://www.ucdsb.on.ca/tiss/stretton/chem1/gases9.html) to the kinetic molecular theory of gases, from The Upper Canada District School Board
• Java animation (http://comp.uark.edu/~jgeabana/mol_dyn/) illustrating the kinetic theory from University of Arkansas
• Flowchart (http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/ktcon.html) linking together kinetic theory concepts, from HyperPhysics
• Interactive Java Applets (http://www.ewellcastle.co.uk/science/pages/kinetics.html) allowing high school students to experiment and discover how various factors affect rates of chemical reactions.

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