This chapter is concerned with the most disorganized state of matter - gases. All gases share some common properties:

1. Gas molecules are in constant, rapid motion
2. Gas molecules are separated by relatively large distances (on a molecular scale). There are several consequences of this fact:
   1. gases have much smaller densities than liquids or solids (the density of a gas is normally measured in units of g/L, while that of liquids and solids is measured in g/mL)
   2. gases are compressible
   3. gases mix with one another easily
3. A sample of a gas retains neither volume nor shape - in other words, gases expand to occupy the available volume and they take the shape of the container which holds them
4. All gases mix with one another to form a homogeneous mixture (solution)
5. Gas molecules collide with the walls of the container which holds them, thus producing pressure

Pressure

Remember that pressure is force per unit area. It is easy to think of pressure as only force, but an example will show that this is incorrect. Picture a person holding a nail, point down, on a piece of wood. The person hits the nail with a hammer, using enough force to drive the nail into the wood. Now the person takes another nail, places it head down on the wood, and hits it with the same amount of force. Now the nail is not driven into the wood; the pressure is much less because the area of the nail head is so much larger than the area of the point of the nail.

Since anything with mass exerts a force as a result of gravitational attraction, the gases in the earth's atmosphere produce a force as well. This force is spread over the surface area of the earth, giving rise to atmospheric pressure. Since this pressure is usually measured using a barometer, it is sometimes called barometric pressure. The pressure of a closed sample of gas is measured with a manometer (shown in Figure 5.3.)

The SI unit of pressure is Newtons per meter squared, since the Newton (N) is the SI unit of force and the square meter is the SI unit for area. The unit N/m² is called a pascal (Pa). Although this is the accepted unit for pressure in the SI system of measurement, it is not a very convenient one, since it is a small unit: standard atmospheric pressure is 101.3 kilopascals. A more convenient unit is the atmosphere, atm. Standard atmospheric pressure is equal to one atm. Here are some equivalencies between pressure units:

Note that a bar is defined as 10⁵ Pa and that the plural of torr is torr, not torrs.

The Gas Laws

1. Boyle's Law: The volume of an ideal gas is inversely proportional to the pressure at constant temperature and number of moles.
   V₁ P₁ = V₂ P₂
2. Charles' Law: The volume of an ideal gas is directly proportional to the absolute temperature of the gas at constant pressure and number of moles.
   V₁ / T₁ = V₂ / T₂
3. Avogadro's Law: The volume of an ideal gas is directly proportional to the number of moles of the gas at constant temperature and pressure.
   V₁ / n₁ = V₂ / n₂
4. The combined gas law (constant number of moles):
   
   
5. The ideal gas law:
   PV=nRT, where R=0.08206 L atm/mol K.
   

Gas Stoichiometry

Stoichiometric calculations for gases in chemical reactions are performed just as you did in chapter 3 - the only difference is that the number of moles of the gas is not generally determined by dividing the mass of the gas by the molar mass (although this will, indeed, give moles). Usually, the mass of the gas sample is not given and moles must be determined in another way. The two most common methods of calculating the number of moles of a gas are:

1. Use the ideal gas equation and solve for n. Then use the given values for the pressure, volume, and temperature of the gas, with the known value of the universal gas constant R, to calculate the number of moles of the gas.
2. If the gas is at standard temperature and pressure (STP: 1 atm and 273 K), then use the relationship that one mole of an ideal gas at STP occupies a volume of 22.42 liters.

Gas Density

If you start out with the ideal gas equation and substitute grams/molar mass for moles, then rearrange to get

,

and then substitute d (density) for g/V, we have the relationship

You can use this equation to calculate the density of a gas from the pressure, temperature, and molar mass, or obviously the molar mass can be calculated if the density, pressure, and temperature are known. The equation above also tells us two important facts about the density of an ideal gas:

1. At constant temperature and pressure, the density of a gas is directly proportional to the molar mass. This means that the density of methane (CH₄, molar mass = 16 g/mol) is four times the density of helium (atomic weight = 4 g/mol) if the two gases are at the same T and P.
2. At constant molar mass (i.e., for a given gas) and pressure, density is inversely proportional to the absolute temperature. This explains the common observation that warm air rises and cold air falls.

Dalton's Law of Partial Pressures

Dalton's law states that the total pressure in a mixture of gases is the sum of the partial pressures of the component gases. This law actually results in two equations which are useful in this chapter:

1. P_T = p_A + p_B + p_C + … + p_N

   Note that total pressure is designated with an upper-case P and the partial pressures are designated with lower-case p's. This is a very simple concept: since pressure is caused by the collision of molecules with the container, each component contributes its own pressure, and the total pressure is the result of all of the partial pressures.

2. p_A = P_TX _A , where X stands for the Greek letter chi, which represents the mole fraction of component A of the mixture.

One of the most common uses of the first equation is to determine the partial pressure of a gas in a mixture given the total pressure and the pressure of the other gas(es). For example, if a gas is collected over water, there is actually a mixture of two gases in the sample: the gas which was collected and water vapor. If the total pressure inside the collection vessel is 750 torr, and the temperature of the water is 25 °C, then the partial pressure of the gas is the total pressure minus the vapor pressure of water at that temperature:

Kinetic Molecular Theory (KMT) of Gases

According to this model, gases are assumed to obey the following postulates:

1. The volume of gas molecules is zero. This is obviously false, since molecules do occupy volume. What this really means is that the volume of a gas sample (i.e., the volume available to a gas) is so large compared to the volume of the molecules themselves that the molecular volume can be ignored. Don't forget that if the volume of a gas is one liter, the vast majority of that volume is empty space; the volume of the molecules themselves is negligibly small. For example, the total volume taken up by one mole of nitrogen molecules is 0.0391 liters (see Table 5.3). But the total volume of a mole of nitrogen at STP is 22.4 liters. Thus, the percentage of the total volume due to the molecular volume is
   

   This means that 99.8% of the volume is empty space. This postulate isn't valid under extremely high pressure conditions, since molecules are compressed under these conditions, making the molecular volume a larger percentage of the total volume.

2. Gas particles are in constant, rapid motion, and the collisions of the molecules with the walls of the container give rise to the pressure exerted by the gas.
3. Collisions between particles are perfectly elastic; i.e., molecules do not interact with each other at all; they behave like miniature billiard balls. In essence, this postulate means that there are no attractive forces between gas molecules. In reality, this postulate is false. All gas molecules are attracted to one another, some more so than others. Under normal temperature and pressure conditions, however, the attractive forces between molecules don't have much time to act, since the molecules are far apart from one another and moving past one another very quickly. At extremely low temperatures, however, the molecules are moving slowly enough that the intermolecular forces have time to act on the molecules. Furthermore, at extremely high pressures, the molecules are much closer to one another, making intermolecular attractions more likely. Thus, this postulate is accurate for normal temperatures and pressures, but isn't realistic for high pressure/low temperature conditions.
4. The average kinetic energy of a gas is directly proportional to the absolute temperature. KE_avg=(3/2) kT.

Section 5.8 discusses real gases -- in other words, conditions under which some of the postulates above are not valid - and, therefore, conditions under which the ideal gas equation isn't valid. Make sure you understand this section, including the meaning of the van der Waal's constants a and b.

Another useful expression which can be derived from the KMT is that for calculating the root-mean-square speed of an ideal gas molecule:

When using this equation, be sure to use the proper units to get a rms speed with units of meter per second. This means that R needs to be expressed as 8.314 Joules/mol K, temperature must naturally be in Kelvin, and the molar mass needs to have units of kg/mole, not grams/mole. This is because a Joule is defined as a kg m²/s² . Plugging in a molar mass of 0.028 kg/mol for nitrogen and a typical room temperature of 25°C, we get a root-mean-square speed of

This is equivalent to almost 1200 miles per hour! Gas molecules are indeed in constant, rapid motion.

Graham's Law of Effusion

The mixing of two gases is called diffusion. The escape of a gas into a vacuum through a tiny orifice is called effusion. There is an important difference between the two terms. Diffusion involves the collision of gas molecules. Even though gases move very quickly, diffusion is a relatively slow process because the molecule moves only a very small distance (the mean free path) before colliding with another molecule and perhaps changing direction. Thus, although a gas may move at a thousand miles per hour, it may take many minutes for it to move across a room.

Effusion is different, since there are no molecular collisions after the molecule escapes from the container. The rate of effusion for a molecule is related to the average speed of the molecules; the faster ones simply have a greater chance of escaping from the hole. The relative rates of effusion for two gases is given by Graham's Law of Effusion:

where MM represents the molar mass of the gas.

Last modified June 16, 1997