Molecular Motion

Has been presented as a compliation of empirical observation, i.e. the historically significant Gas Laws, but does The Ideal Gas equation have some deeper, more fundamental 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.  Usually there is a great distance between each other, so the molecules travel in straight lines between abrupt collisions at the walls and between each other.  These collisions randomize the motion of the molecules. Most of the collisions between molecules are binary, in that only two molecules are involved.
2. The volume of the molecules of the gas is negligible compared to the total volume in which the gas is contained.  A common bond length between atoms is about 10^-10 m or 1 Angstrom. Small molecules are therefore on the order of 10 Angstroms in diameter, or less than 10^-24 Liters in Molecular Volume, quite tiny indeed!  Remember, however that there can be a great many molecules in the sample of gas, perhaps on the order of a mole, or 6 x 10²³.  So that when concentrations of molecules exceed about 1 mol/liter, then the approximation that the volume of ALL the molecules in the container is much less than the volume of the container itself, fails.  In the case of an ideal gas, we will assume that molecules are point masses, i.e., the volume of a mole of gas molecules (as if they were at rest) is zero, so molecular and container volumes never become comparable.
3. Attractive forces between gas molecules are negligible.  We know that if these forces were significant, the molecules would stick together.  This happens when it rains and gaseous water molecules stick together to form a liquid.  Water vapor is a condensible gas, and this shows us that gas molecules are sticky, but at a high enough temperature they form only a permanent gas, because their stickiness can be considered negligible.  We will assume that in an ideal gas, molecular attractive forces are not just small, but identically zero. 

• The average kinetic energy of the molecules does not change with time.  The molecules bounce and bounce but, on average, do not slow down 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 (not sticky)
• The average kinetic energy of the molecules is proportional to absolute temperature (A result of Thermodynamics). At a given temperature the molecules of all species of gas, no matter what size shape or weight, have the same average kinetic energy.

• The pressure of a gas is manifested at the boundary of the vessel it is confined in, and 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. Check out this simulation of molecules in motion in a gas.

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 absolute temperature of a gas is doubled, the average kinetic energy of its molecules is doubled.
• If the temperature approaches absolute zero, the kinetic energy of the molecules approach zero and they 'stop'

Molecular Speed, Averaging, and The Maxwell-Boltzmann Speed Distribution

• 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 change individual molecular speeds but the distribution of speeds remains the same.
• At the same temperature, lighter gases move, on average, faster than heavier gases.

The following graph shows the Distribution of Speeds for Gases, i.e.the fraction of the sample of gas molecules that have a given speed is shown by the height of the curve above the speed axis.  There are no molecules exactly at rest.

• 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

Where does this number lie on our Graph of the speed distribution?  To find out, the root mean square 'average' of the distribution must be taken (defined) in a specific way.  It is defined exactly how it sounds.  First your square all the speeds, then average those numbers, and take the square root of that average.  The mean of a distribution is just the average of all the numbers in the distribtion.  The most probable value of a distribution is also exactly what it sounds like, the speed of that the largest fraction of molecules are travelling.  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 mean (simple average) speed is:

• The root mean square speed is:

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

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

• 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 (i.e. the number and strength of 'hits' per wall), 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 calculating.

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 be inversely proportional to the average speed of the molecules or proportional to the square root of the ratio of the molecular masses..

The ratio of effusion rates, r_i, for two gases labelled by i, is 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 higher the density of gas, the smaller the mean free path (more likelyhood of a collision). The larger the molecules, the smaller the mean free path. The mean free path depends on the number density of the gas molecules and their size --- and nothing else
• At sea level the mean free path of atmospheric gases is about 60 nm
• At 100 km altitude, the atmosphere is less dense than where we live at the surface of the earth, and the mean free path is about 0.1 m (about 1 million times longer than at sea level)