Chemistry 103A; Sections 5, 6, 7, 8; Lecture 36, 20 Nov 00

Recall:

We were talking about:

Kinetic Molecular Theory of Gases

According to the theoretical model of a gas provided by kinetic molecular theory the pressure of a gas comes from collisions of molecules with the walls of the container. Since each collision transfers momentum to the wall, and since there many collisions per second the net result is a force on the wall due to molecular collisions.

With a little bit of algebra we were able to write,

This result,
is the crucial relationship which connects the properties and motion of the molecules with the bulk thermodynamic properties of the gas.

The velocity, v, that occurs here is an average velocity.

(Actually, it is the "root-mean-square" average velocity, or the "rms" velocity. rms averages are used to talk about average values of quantities which can take on both positive and negative values. For example, the molecules in a gas are in random motion. On the average just as many molecules are moving to the right as there are molecules moving to the left. In our simplified model one sixth of the molecules were moving to the right and one sixth were moving to the left. Algebraically, molecules moving to the right have a positive velocity and molecules moving to the left have a negative velocity. If we were to average these velocities we would get zero, which means that the gas as a whole is not going anywhere. To get a measure of what velocities the molecules really have we square the velocities, take the average of these squares, and then take the square root of that average.

You have seen rms averages before. Ordinary household electricity is called "alternating current" because the voltage of a terminal alternates from positive to negative and back (at 60 times per second). The average voltage is zero, but if you stick your finger in a light socket you will find out that the voltage is not zero. Although the average voltage is zero, the rms average voltage is approximately 110 volts.)

The definition of the rms average is (using a bar over a quantity to indicate its average) is

 
Combining our expression for p with the ideal gas law
we can get several useful expressions:
from which we can find the average kinetic energy for a molecule,
The kinetic energy for N molecules is N times the kinetic energy for one molecule,
We can also calculate the average velocity (rms velocity) of a molecule by taking the square root of v2,
In calculating the average speed it is easier to use the formula weight of the molecules than mass of one molecule. Remembering that
,
we can rewrite the expression for average velocity as
.
The only thing we have to be careful about here is the make sure the formula weight is expressed in kg/mol instead of g/mol. R, of course, must be in SI units, namely 8.31451 J/K mol.
 

Diffusion

Diffusion is the movement of one gas through another gas when there is no convection, turbulence, or other currents or mixing. The rate of diffusion is proportional to the average velocity of the molecules.

Looking at our equation for average molecular velocities we see that the average velocity is inversely proportional to the square root of the formula weight. In equation form,

.
We will not need to know how to calculate absolute diffusion rates in this course, but we will need to be able to calculate relative diffusion rates. The ratio of the diffusion rate of gas A (Diffusion RateA) to the diffusion rate of gas B (Diffusion RateB) is, from the above discussion,
Example:

Calculate how much faster He will diffuse than oxygen gas at 298 K.
 

Effusion

Effusion is another word for leakage. Gases tend to leak out of containers, either through simple "pin holes," or through pores in the material of the container. For example, balloons filled with He gas tend to lose helium because the rubber of the balloon has tiny pores thorough which the helium can leak out. (Most party stores now sell balloons make of a mylar film because mylar has much smaller pores and the helium "effuses" out much slower. Effusion follows the same equation for relative effusion rates as diffusion,

Nonideal Gases

The ideal gas equation of state can be derived from the two assumptions that there are no intermolecular forces between the molecules and that the molecules are "point masses," that is, the molecules have no size. Neither of these assumptions is correct, although they are both approximately correct if the pressure is low and the temperature is high.

Johannes van der Waals found a way to correct for both of these assumptions. He put in two new terms. One of the terms corrects for the actual size of the molecules. He wrote V -nb in place of the volume, V. The nb term takes account of the fact that one molecule can't freely roam around the entire volume, V. It is excluded from portions of the volume which are occupied by other molecules. The parameter, b, is the "excluded volume" due to one mole of particles.

Also, the physical pressure felt by a molecule is larger than the pressure produced by the walls of the container. This is because the particles attract each other and so tend to draw molecules on the edge of the gas into the bulk gas. He added a term, an2/V2, to correct for this extra pressure. The van der Waals equation of state is then given by,

The parameters, a and b, are called the "van der Waals parameters." They are different for different gases. There are tables of van der Waals parameters in handbooks and a there is a small table on page 572 of your text.
 
 

Liquids and Solids

It is fair to ask why all substances are not gases. The answer is: "intermolecular forces." If it were not for the forces of attraction between molecules all substances would, indeed, be gases.

In our discussion of liquids and solids the principal consideration will be a competition between intermolecular forces and the kinetic energy of the molecules.

We know from kinetic molecular theory that the average kinetic energy of the molecules in a gas is proportional to the Kelvin temperature. It is also true that the average kinetic energy of molecules in a liquid or solid is proportional to the Kelvin temperature.

Suppose we start with a substance, say water, in the gas phase. In the gas phase the kinetic energy dominates the intermolecular forces. The intermolecular forces are not strong enough to overcome the kinetic energy so that the gas molecules are free to "roam" in space. If the temperature is high enough the effect of intermolecular forces is negligible so that the gas is essentially ideal. As we lower the temperature the influence of the intermolecular forces will become more pronounced so that gas may no longer be ideal.

As we continue to lower the temperature the kinetic energy of the molecules decreases. There will come a temperature where the kinetic energy is no longer able to overcome the intermolecular forces and the gas condenses into a liquid.

However, the kinetic energy is high enough that a reasonable fraction of the molecules may still be able to escape the surface of the liquid. We say that the liquid has a "high vapor pressure."

(The vapor pressure of a liquid or solid is a measure of the ability of molecules to escape from the surface of the liquid or solid into the gas phase. We will discuss vapor pressure further below.) In the liquid the molecules are close together so that the average distance between adjacent molecules is about the same as the diameter of the molecules. However, the molecules are still free to "tumble over each other" and the substance remains a liquid.

But as we lower the temperature the kinetic energy of the molecules decreases so that the number of molecules able to escape from the surface of the liquid decreases. We say that the vapor pressure is decreasing.

As we continue to lower the temperature we will reach a stage where the intermolecular forces dominate the kinetic energy to the point that the molecules are no longer free to "tumble over each other." The liquid freezes to a solid.

In a solid the molecules still have kinetic energy, but the motion is limited to molecules vibrating about an equilibrium position in the crystal.

If we continue to lower the temperature the kinetic energy decreases and reaches a minimum at absolute zero.

(Classical mechanics would say that all motion stops at 0 K, but quantum mechanics tells us that this never happens. According to quantum mechanics there is still a residual motion at absolute zero, called "zero point motion" or "zero point energy.") Intermolecular Forces

There are three major classes of intermolecular forces between molecules of a pure substance.

1. Dispersion Forces.

Dispersion forces occur between ALL molecules. They are generally the weakest of the intermolecular forces and they go by several different names. Dispersion forces, London dispersion forces, van der Waals forces, induced dipole-induced dipole forces, and so on.

Dispersion forces are produced by the nuclei and charge cloud of one molecule interacting with the nuclei and charge cloud of another molecule.

Dispersion forces depend on the number of electrons in the molecule (and to a lesser degree on the shape of the molecule). As a rule, the size (volume) of a molecule depends on the number of electrons. Large molecules have large dispersion forces and small molecules have small dispersion forces. For example H2 molecules have much smaller dispersion forces than, say an I2 molecule. That is because the I2 molecule has many more electrons than a H2 molecule and these electrons take up more space.

Roughly speaking, dispersion forces are proportional to the square of the volume of the molecule.

Repeat: all molecules have dispersion forces.

 2. Dipole-dipole forces. Only polar molecules have dipole-dipole forces. That is, the molecules must have a permanent dipole moment.

Dipole-dipole forces are generally stronger than dispersion forces. However, dipole-dipole forces are proportional to the square of the dipole moment of the molecule. That means that molecules with very small dipole moments (like CO) may have relatively small dipole-dipole forces.

Dipole-dipole forces originate through the interaction of the positive and negative ends of one polar molecule with the positive and negative ends of the other polar molecule.

Molecules with dipole-dipole forces also have dispersion forces, but usually the dipole-dipole forces will dominate.

 3. Hydrogen bonding forces. Hydrogen bonding intermolecular forces are the strongest of the intermolecular forces between neutral molecules.

In order for a molecule to exhibit hydrogen bonding forces the molecule must have an N- H bond, an O- H bond, or a F- H bond. (There is only one molecule which has an F- H bond, so we don't have to worry too much about F- H bonds.)

"Hydrogen bonding forces" is an unfortunate name because it can lead to confusion with covalent bonds of hydrogen with other elements. Hydrogen bonding forces are not covalent bonds.

N, O, and F are very electronegative elements. When hydrogen is bonded to one of these elements the covalent bond is polar and the hydrogen end is positive.

Hydrogen bonded forces originate from the interaction of the positive H end of an O- H or N- H bond with the lone pair of electrons on an O or an N atom in the other molecule.

Note: H2O and NH3 have hydrogen bonding forces. CH4 does not have hydrogen bonding forces.

 Approximate energies of intermolecular interactions

Interaction                                     Energy (kJ/mol)

Covalent bond (for comparison)          200 - 1000

Hydrogen bonding                               20 - 40

Dipole-dipole                                      0.4 - 4

Dispersion                                        0.004 - 0.8

Examples:

We will give lots of examples in class.