CHM2041
Molecular Motion

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 is negligible

• The average kinetic energy of the molecules does not change with time (as long as the temperature of the gas remains constant). Energy can be transferred between molecules during collisions but not lost because the collisions are perfectly elastic
• The average kinetic energy of the molecules is proportional to absolute temperature (A result of Thermodynamics). At a given temperature the molecules of all gases have the same average kinetic energy

Pressure

• The pressure of a gas is caused by collisions (momentum transfer) of the molecules of the gas with the walls of the container.
• The magnitude of the pressure is related to how hard and how often the molecules strike the wall

Absolute Temperature

• The absolute temperature of a gas is a measure of the average kinetic energy of its' molecules
• If two different gases are at the same temperature, their molecules have the same average kinetic energy
• If the temperature of a gas is doubled, the average kinetic energy of its molecules is doubled

Molecular Speed

• Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds, i.e. they exhibit a DISTRIBUTION of speeds; Some move fast, others relatively slowly. Collisions can change individual molecular speeds bu the distribution of speeds remains the same.
• At the same temperature, lighter gases move on average faster than heavier gases.

• At higher temperatures at greater fraction of the molecules are moving at higher speeds. This is important for activated chemical processes, reactions.
• !The average kinetic energy, e, is related to the root mean square (rms) speed u

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 average speed is:

• The root mean square speed is:

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

Gas Laws and Kinetic Theory

• At constant temperature, the average kinetic energy of the gas molecules remains constant
• Therefore, the rms speed of the molecules, u, also remains unchanged
• If the rms speed remains unchanged, but the volume increases, there will be fewer collisions with the container walls over a a given time:

• An increase in temperature means an increase in the average kinetic energy of the gas molecules, thus an increase in u
• At constant volume, the greater speed will mean more collisions per unit time and an increase in pressure
• If, instead, we allow the volume to change to maintain constant pressure, the volume must increase with increasing temperature to maintain constant pressure, which is just (Charles's law)

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

• At a given temperature, all gases have the same average kinetic energy and for a three dimensional gas this value is (3/2)RT. (what is the molar kinetic energy of a two dimesional gas trapped in the surface of a metal?)
• The rms velocity, u, in m/s, is simply

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!

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 19^th 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, r_i, for two gases labelled by i, are proportional to the ratio of the RMS speeds of the gases, u_i

Diffusion

• The relative rates of diffusion of two gases is also determined by the ratio of their average (rms) speeds

• The speed of molecules is quite high, but the rates of diffusion are slower than molecular speeds due to molecular collisions
• At the density of the atmosphere at sea level, each gas molecule experiences collisions at a rate of about 10¹⁰ (i.e. 10 billion) times per second
• Due to these collisions, the direction of a molecule of gas in the atmosphere is constantly changing, and the diffusion rate is much reduced from the instantaneous speed of the molecule

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

• The higher the density of gas, the smaller the mean free path (more likelyhood of a collision)
• At sea level the mean free path is about 60 nm
• At 100 km altitude the atmosphere is less dense and colder than where we live at the surface of the earth, the mean free path is about 0.1 m (about 1 million times longer than at sea level)