Molecular Motion


Kinetic-Molecular Theory

The ideal gas equation

pV = nRT

Has been presented as an empirical observation. Does it have meaning?

The Kinetic-Molecular Theory ("the theory of moving molecules"; Rudolf Clausius, 1857)

  1. Gases consist of large numbers of molecules (or atoms, in the case of the noble gases) that are in continuous, random motion
  2. The volume of all the molecules of the gas is negligible compared to the total volume in which the gas is contained
  3. Attractive and repulsive forces between gas molecules are negligible

Pressure


Absolute Temperature


Molecular Speed


The average of a distribution must be taken (defined) in a specific way. In general the mean, the root mean square and the most probable value in a distribution are all different.

A Note on Distributions a simple numerical example:

Suppose we have four molecules in our gas sample. Their speeds are 3.0, 4.5, 5.2 and 8.3 m/s.




The rms speed as well as the entire distribution of speeds of gas molecules are a function of temperature. Below, the blue line is a cold gas and the red line is a hot gas. Note that the rms speed, u as well as the entire speed distribution changes with temperature for a given gas.
The rms speed for a given speed distribution
(which is determined by the temperature and molecular weight of the gas) is greater in magnitude than the most probable speed or the mean speed.


Trick question: What is the mean velocity of the molecules in a gas at any temperature.

Gas Laws and Kinetic Theory

Therefore, the pressure will decrease (Boyle's law)



The Ideal Gas Equation of State follows directly from the Kinetic Theory of Gases. Here is a Pseudo-Derivation


Molecular Effusion and Diffusion

Kinetic-molecular theory states that the average kinetic energy of a mole of molecules molecules is proportional to absolute temperature, and the proportionality constant is R, the universal gas constant

(1/2)Mu2 = (3/2)R T

where M is the molar mass in kg/mole, R is the gas constant in J/K.mole, and T is the absolute temperature in K.

Numerical Example:

Calculate the rms speed, u, of an N2 molecule at room temperature (25°C) Be careful of your UNITS!

T = (25+273)K = 298K
M = 28 g/mol = 0.028 kg/mol
R = 8.314 J/mol K = 8.314 kg m2/s2 mol K
u = 515 m/s


Note: this is equal to 1,150 miles/hour!

Effusion

The escape of a gas through a tiny pore or pinhole in its container is called EFFUSION.

The effusion rate, r, has been found to be inversely proportional to the square root of its molar mass: (Why?)


Thus, comparison of the effusion rates of two gases with different masses will follow the relation:

This effect was observed in the 19th century by Graham and is sometimes called GRAHAM's LAW A note on Rates and Times
The effusion time (the time it takes for a certain amount of gas to escape a vessel) is inversely proportional to the effusion rate (the amount of gas effusing from the hole per unit time). Be careful that you understand whether it is a rate or a time that you are tyring to find out.

Gas may effuse, but for this to happen a molecule must pass through a pore or pinhole and escape to the outside. In effect, a molecule must 'collide' with an escape hole. The number of such collisions will be linearly proportional to the average speed of the molecules in the gas and thus the effusion rate. The effusion time should the be inversely proportional to the average speed

The ratio of effusion rates, ri, for two gases labelled by i, are proportional to the ratio of the RMS speeds of the gases, ui



Diffusion

Similarly to effusion, the process of diffusion is the spontaneous intermingling (mixing) of dissimilar gases (fluids) that are initially spatially separated. If you put a drop of ink in a glass of water and you see the ink gradually spread out to fill the glass, this is diffusion


The average distance traveled by a molecule between collisions is called the mean free path


See some multiple choice problems relating to gases.
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PJ Brucat // University of Florida