AP Chemistry - Gases

The Kinetic-Molecular Theory of Gases

Any law is an empirical generalization which describes the results of several experiments. A law, however, only describes results; it does not explain why they have been obtained. A theory is a description which explains the results of experiments. The kinetic-molecular theory of gases is a theory of great explanatory power. We shall see how it explains the ideal gas law, which includes the laws of Boyle and of Charles; Dalton's law of partial pressures; and the law of combining volumes.
The kinetic-molecular theory of gases can be stated as four postulates:

    1.  A gas consists of molecules in constant random motion.
    2.  Gas molecules influence each other only by collision; they exert no other forces on each other.
    3.  All collisions between gas molecules are perfectly elastic; all kinetic energy is conserved.
    4.  The volume actually occupied by the molecules of a gas is negligibly small; the vast majority of the volume of the gas is empty space through which the gas molecules are moving.
 

These postulates, which correspond to a physical model of a gas much like a group of billiard balls moving around on a billiard table, describe the behavior of an ideal gas. At room temperatures and pressures at or below normal atmospheric pressure, real gases seem to be accurately described by these postulates, and the consequences of this model correspond to the empirical gas laws in a quantitative way.
Gas Pressure Explained
In the KMT, pressure is the force exerted against the wall of a container by the continual collision of molecules against it.  Newton's second law of motion tells us that the force exerted on a wall by a single gas molecule is equal to the mass of the molecule multiplied by the velocity of the molecule.  The molecule rebounds elastically and no kinetic energy is lost in a collision.  All the molecules in a gas do not have the same velocity. The average velocity is used to describe the overall energy in a container of gas.
The pressure-volume product of the ideal gas equation is directly proportional to the average velocity of the gas molecules.  If the velocity of the molecules is a function only of the temperature, then the KMT gives us Boyle's Law.
The statement that the pressure-volume product of an ideal gas is directly proportional to the total kinetic energy of the gas is also a statement of Boyle's Law, since the total kinetic energy of an ideal gas depends only upon the temperature.
Gas Temperature Explained
If you look at the ideal gas law equation carefully you will see that it shows that the total kinetic energy of a collection of gas molecules is directly proportional to the absolute temeprature of the gas.  The ideal gas law can be rearranged to give an explicit expression for temperature. Temperature is a function only of the mean kinetic energy, the mean velocity and the mean molar mass.
As the absolute temperature decreases, the kinetic energy must decrease and thus the mean velocity of the molecules must decrease also. At T=0, the absolute zero of temperautre, all motion of gas molecules would cease and the pressure would then also be zero  No molecules would be moving. Experimentally, the absolute zero of temperature has never been attained, although modern experiments have made it to temperatures as low as 0.01 K.
It has been necessary to use the average velocity of the molecules of a gas because the actual velocities are distributed over a very wide range.   This distribution, can be described by Maxwell's law of distribution of velocities, or you can think of a superhighway.  There is an average speed on the highway.  Some cars are travelling slower, some faster, some at exactly the right speed.  Even those on cruise control are never exactly on the right speed because of the discrepencies in the speed control on each individual engine and the ground geometry over which the car must be moved.
It is not necesary to use a Maxwell-Boltzmann distribution of velocities to explain either the nature of temperature or Charles' law, although it is the correct expression of the distribution. Charles' law can be obtained for any distribution in which the velocities of the gas molecules are a function of  the nature of the gas and the absolute temperature only.
Partial-Pressure Explained
Dalton's law of partial pressures follows from the KMT of gases.  If the gas molecules in a mixture are in constant and random motion and if there are no forces operating between the molecules except collisions, then on the average, the net effect of collisions with other molecules must be zero.  For this reason each gas acts as if it were present alone.
Velocity Explained
The root mean square velocity of a molecule can be obtained by using the formula

         vrms = (3RT/M)1/2
 

Example: Calculate the root-mean-square velocity of oxygen moleules at room temperature, 25oC.  M is the molarcular mass of oxygen which is 31.9998 g/mol; the molar gas constant is 8.314 J/mol K, and the temperature is 298.15 K.  The molecular mass must be divideved by 1000 to convert it into a usable form, therefore

vrms = (3(8.314)(298.15)/(0.0319998))1/2   = 481.2 m/s

So an oxygen molecule travels through the air at 481.2 m/s which is 1726 km/h, much faster than a jetliner can fly and faster than that of most rifle bullets.
 

The very high speed of gas molecules under normal room conditions would indicate that a gas molecule would travel across a room almost instantly.  In fact, gas molecules do not do so.   If a small sample of a very odorous (and poisonous) gas, H2S is released in one corner of a room, our noses will not detect it in another corner of the room for several minutes unless the air is vigorously stirred by a mechanical fan.  The slow diffusion of gas molecules which are moving very quickly occurs because the gas molecules travel only short distances in straight lines before they are deflected in a new direction by collision with other gas molecules.
The distance any single molecule travels between collisions will vary from very short to very long distances, but the average distance that a molecule travels between collisions in a gas can be calculated. This distance is called the mean free path of the gas molecules. If the root-mean-square velocity is divided by the mean free path of the gas molecules, the result will be the number of collisions one molecule undergoes per second.  This number is called the collision frequency of the gas molecules.
The postulates of the KMT of gases permit the calculation of the mean free path of gas molecules. The gas molecules are visualized as small hard spheres.  Without going into the mathmatical detail; as the temperature rises the mean free path increases; it also rises as the pressure decreases, and as the size of the molecules decrease.   Taking all this into account, the oxygen molecules from above have a mean free path of 67 nm.   Diffusion takes place slowly because even though molecules are moving very fast, they travel only short distances in any one straight line. 
Graham's Law of Effusion and Diffusion
Root-mean-square velocities of gas molecules are sometimes directly useful, but the comparison of velocities explains the results of, and is useful in, studies of effusion of molecules through a small hole in a container or diffusion of molecules through porous barriers. The comparison between two gases is most conveniently expressed as:
vrms(1)/vrms(2) = (M2/M1)1/2 = (d2/d1)1/2
This equation gives the velocity ratio in terms of either the molar mass ratio or the ratio of densities. The ratio of root-mean-square velocities is also the ratio of the rates of effusion, the process by which gases escape from containers through small holes, and the ratio of the rates of diffusion of gases.
This equation is called Graham's law of diffusion and effusion because it was observed by Thomas Graham (1805-1869) well before the kinetic-molecular theory of gases was developed. As an empirical law, it simply stated that the rates of diffusion and of effusion of gases varied as the square root of the densities of the gases. Graham's law is the basis of many separations of gases. The most significant is the separation of the isotopes of uranium as the gases 238UF6 and 235UF6. Fluorine has only one isotope, so the separation on the basis of molar mass is really a separation on the basis of isotopic mass. 
Example. The ratio of root-mean-square velocities of 238UF6 and 235UF6 can be calculated as follows. The molar mass of 238UF6 is 348.0343 and the molar mass of 238UF6 is 352.0412. The mass ratio is 1.011513 and the ratio of root-mean-square velocities is 1.00574. Although the difference is small, many kilograms of 235U have been separated using this difference in the gas-diffusion separation plant at Oak Ridge, Tennessee, U. S. A. This plant prepared the uranium for the Manhattan Project of the Second World War and produced the uranum used in the uranium atomic bomb dropped on Japan in 1945. 

Equations of State for Gases

The state of any amount of substance is something which chemists often find it necessary to specify clearly. Such a specification for even the simplest substance must include at least n, the amount of substance present; p, the pressure, and T, the temperature. In most cases it is found convenient to specify a standard state of a system as well. As a standard, the choice of n = 1 is universal; however, the pressure is commonly chosen as either p = 100 kPa (1 bar) or as p = 101.325 kPa (1 atm). The temperature is commonly chosen as either T = 298.15 K (25oC) or as T = 273.15 K (0oC). Throughout this text the values of p = 100 kPa and T = 298.15 K will be used as standard. For gases, other works may employ STP (standard temperature and pressure) as p = 101.325 kPa and T = 273.15 K, or SATP (Standard Ambient Temperature and Pressure) where p =101.325 kPa and T = 293.15 K.
Ideal Gas Law
The simplest known equation of state, which is an equation linking at least the three properties of temperature, pressure, and volume of a chemical system, is the ideal gas law, pV = nRT. This equation accurately describes an ideal gas, but it describes a real gas such as oxygen or carbon dioxide accurately only at pressures below atmospheric. As the pressure increases or the temperature decreases, real gases are found to deviate significantly from the behavior expected of ideal gases.
Many more complex equations of state have been proposed to describe the behavior of gases, liquids, and solids but their physical interpretation is often not obvious. We will consider only one of them, the van der Waals equation.
van der Waals Equation
In 1873, Johannes van der Waals, a physics professor at the University of Amsterdam, developed an equation to account more accurately for the behavior of real gases. It was considered a sufficiently important development to justify the award of the Nobel Prize in 1910. The van der Waals equation is the second most simple equation of state; only the ideal gas law is simpler. It is used to describe the behavior of gases when pressures are higher, or temperatures are lower, than those at which the ideal gas law is sufficiently accurate. The van der Waals equation describes the relationship between the physical quantities of pressure, temperature, and volume more accurately than does the ideal gas law. It does so, however, at the cost of a more complex equation and the use of a unique set of two van der Waals coefficients for each different gas. The usual form of the van der Waals equation is:
(p + (n2a/V2))(V - nb) = nRT
In the van der Waals equation, the term n2a/V2 reflects the fact that the attractive forces between molecules are not zero. The measured pressure is thus less than it should be because the attractive forces act to reduce it. The term nb reflects the fact that the volume of the molecules of a real gas is not zero, and so the volume in which the molecules may move is less than the total measured volume. A table of the van der Waals coefficients for several common gases is included in this section. 



Table: Van der Waals Coefficients of Selected Gases



Gas  a b

atm dm6/mol dm3/mol
ideal 0.0 0.0
He 0.034 0.0237
Ar 1.345 0.0322
O 1.360 0.0318
N 1.390 0.0391
CO2 3.592 0.0427
CH4 2.253 0.0428
H2 0.244 0.0266


Example. The volume of one mole of oxygen molecules is 31.8 cm3 according to the van der Waals coefficient values tabulated. If the molecular diameter is taken as 0.370 nm, an approximate molar volume would be NAd3, or 30.5 cm3.The volume actually occupied by one mole of oxygen gas at 25oC, according to the ideal gas law, is 24465 cm3. The molecules of the gas actually occupy only 0.13% of the total volume occupied by the gas at 25oC. 

Go to the Next Section on The Gas Empirical Laws