9.4-9.6

kinetic molecular theory (KMT) five postulates

1. Gases are composed of molecules that are in continuous motion, travelling in straight lines and changing direction only when they collide with other molecules or with the walls of a container. 2. The molecules composing the gas are negligibly small compared to the distances between them. 3. The pressure exerted by a gas in a container results from collisions between the gas molecules and the container walls. 4. Gas molecules exert no attractive or repulsive forces on each other or the container walls; therefore, their collisions are elastic (do not involve a loss of energy). 5. The average kinetic energy of the gas molecules is proportional to the kelvin temperature of the gas.

effusion

A process involving movement of gaseous species similar to diffusion is effusion, the escape of gas molecules through a tiny hole such as a pinhole in a balloon into a vacuum (

KMT conceptually explains the behavior of a gas Avogadro's law

At constant pressure and temperature, the frequency and force of molecule-wall collisions are constant. Under such conditions, increasing the number of gaseous molecules will require a proportional increase in the container volume in order to yield a decrease in the number of collisions per unit area to compensate for the increased frequency of collisions

KMT conceptually explains the behavior of a gas • Dalton's Law

Because of the large distances between them, the molecules of one gas in a mixture bombard the container walls with the same frequency whether other gases are present or not, and the total pressure of a gas mixture equals the sum of the (partial) pressures of the individual gases.

KMT conceptually explains the behavior of a gas Boyle's law

If the gas volume is decreased, the container wall area decreases and the molecule-wall collision frequency increases, both of which increase the pressure exerted by the gas

KMT conceptually explains the behavior of a gas Amontons's law

If the temperature is increased, the average speed and kinetic energy of the gas molecules increase. If the volume is held constant, the increased speed of the gas molecules results in more frequent and more forceful collisions with the walls of the container, therefore increasing the pressure

KMT conceptually explains the behavior of a gas Charles's law

If the temperature of a gas is increased, a constant pressure may be maintained only if the volume occupied by the gas increases. This will result in greater average distances traveled by the molecules to reach the container walls, as well as increased wall surface area. These conditions will decrease the both the frequency of molecule-wall collisions and the number of collisions per unit area, the combined effects of which balance the effect of increased collision forces due to the greater kinetic energy at the higher temperature

The kinetic energy (KE) of a particle of mass (m) and speed (u) is given by:

KE = 1 2 mu^2

If temperature decreases

KEavg decreases, more molecules have lower speeds and fewer molecules have higher speeds, and the distribution shifts toward lower speeds overall, that is, to the left.

The diffusion rate depends on several factors:

The diffusion rate depends on several factors: the concentration gradient (the increase or decrease in concentration from one point to another); the amount of surface area available for diffusion; and the distance the gas particles must travel. Note also that the time required for diffusion to occur is inversely proportional to the rate of diffusion, as shown in the rate of diffusion equation.

Describe characteristics of a gas at high pressures

The gas therefore becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not proportional as predicted by Boyle's law

mean free path

The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be hundreds of times the diameter of the molecule

Graham's law of effusion:

The rate of effusion of a gas is inversely proportional to the square root of the mass of its particles: rate of effusion ∝ 1 /√ℳ

Where is this change in pressure or volume more pronounced?

This change is more pronounced at low temperatures because the molecules have lower KE relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another.

What does this force of attraction do to the molecules?

This force pulls the molecules a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (at constant pressure)

One way in which the accuracy of PV = nRT can be judged is by comparing the actual volume of 1 mole of gas (its molar volume, Vm) to the molar volume of an ideal gas at the same temperature and pressure. This ratio is called the compressibility factor (Z) with:

Z = molar volume of gas at same T and P/molar volume of ideal gas at same T and P = (PVm/RT) measured

at high pressures, the molecules of a gas

are crowded closer together, and the amount of empty space between the molecules is reduced. At these higher pressures, the volume of the gas molecules themselves becomes appreciable relative to the total volume occupied by the gas.

In a gas sample, individual molecules have widely varying speeds; however, because of the vast number of molecules and collisions involved, the molecular speed distribution and average speed are

constant.

At relatively low pressures

gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas.

At higher pressures,

however, the force of attraction is also no longer insignificant

van der Waals equation

improves upon the ideal gas law by adding two terms: one to account for the volume of the gas molecules and another for the attractive forces between them.

the ideal gas equation functions well when

intermolecular attractions between gas molecules are negligible and the gas molecules themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of low pressure and high temperature.

kinetic molecular theory (KMT)

is a simple microscopic model that effectively explains the gas laws described in previous modules of this chapter.

root mean square velocity of a particle, urms

is defined as the square root of the average of the squares of the velocities with n = the number of particles

If the temperature of a gas increases

its KEavg increases, more molecules have higher speeds and fewer molecules have lower speeds, and the distribution shifts toward higher speeds overall, that is, to the right.

Maxwell-Boltzmann distribution

molecular speed distribution is known as a Maxwell-Boltzmann distribution, and it depicts the relative numbers of molecules in a bulk sample of gas that possesses a given speed

In a closed environment, diffusion will ultimately result in

n equal concentrations of gas throughout, as depicted in Figure 9.27. The gaseous atoms and molecules continue to move, but since their concentrations are the same in both bulbs, the rates of transfer between the bulbs are equal (no net transfer of molecules occurs).

diffusion

process by which molecules disperse in space in response to differences in concentration

We are often interested in the rate of diffusion, the amount of gas passing through some area per unit time:

rate of diffusion = amount of gas passing through an area/unit of time

This means that if two gases A and B are at the same temperature and pressure, the ratio of their effusion rates is inversely proportional to the ratio of the square roots of the masses of their particles:

rate of effusion of A rate of effusion of B = √ℳB/√ℳA

The gaseous atoms or molecules are, of course, unaware of any concentration gradient, they simply move randomly—

regions of higher concentration have more particles than regions of lower concentrations, and so a net movement of species from high to low concentration areas takes place.

real gases approximate this behavior at

relatively low pressures and high temperatures.

Particles of a hypothetical ideal gas have no

significant volume and do not attract or repel each other

Although diffusion and effusion rates both depend on the molar mass of the gas involved,

their rates are not equal; however, the ratios of their rates are the same.