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Background:

Ideal Gas Law:

Pressure x Volume = Moles x Ideal Gas Constant x Temperature

Substituting in variables, the formula is:

PV=nRT

Explanation and Discussion:

The Ideal Gas Law may be the largest and most complex of the gas laws. This is in part because of the number of variables in the equation, and in part to the abstraction of an "ideal" gas that the law is built on. The Ideal Gas Law is also designed as a sort of umbrella for Boyle's, Charles', and Avogadro's laws.

First, we'll go over the parts of the equation, PV=nRT. P is pressure. Pressure can be in either atmospheres (atm) or kilopascals (kPa). V is volume in liters (L). n is the number of moles of the gas. Because moles of a substance are determined by mass divided by molecular mass, it can create an interesting variant we will discuss later. R is the Ideal Gas Constant. Depending on whether atmosphers or kilospascals were used, the value is either 0.0821 L-atm/mol-K or 8.31 L-kPa/mol-K, respectively. Temperature is in absolute degrees Kelvin.

An interesting aspect of the Ideal Gas Law is its flexibility. It contains elements that allow you to solve for other quantities, such as density or molecular mass. To solve for molecular mass:

PV=mass/mol. mass x RT - change moles to mass(m) in grams divided by molecular mass in grams
mol. mass x PV = mRT - multiply by molecular mass
molecular mass = mRT/PV - divide by pressure and volume.

We can also see density in that last equation, m/V (grams/liter). The same equation, but with density(d) in place of mass per volume (m/V), is:

molecular mass = dRT/P

To solve just for density, the equation would become:

density = (molecular mass x pressure)/(constant x temperature)

So far, we have been skirting the concept of an ideal gas. What exactly is an ideal gas? An ideal gas is one that exactly conforms to the kinetic theory. The kinetic theory, as stated by Rudolf Clausius in 1857, has five key points. These are:

1. Gases are made of molecules in constant, random movement. Gases like Argon have 1-atom molecules.
2. The large portion of the volume of a gas is empty space. The volume of all gas molecules, in comparison, is negligible.
3. The molecules show no forces of attraction or repulsion.
4. No energy is lost in collision of molecules; the impacts are completely elastic.
5. The temperature of a gas is the average kinetic energy of all of the molecules.

Non-Ideal Behavior

The Kinetic Theory makes several assumptions about an ideal gas. These cause problems because real gases are not ideal. The main causes of error are related to pressure and temperature.

Pressure
At high pressures, the behavior of real gases changes dramatically from that predicted by the Ideal Gas Law. Under 10 atmospheres of pressure or less, Ideal Gas Law predictions are very close to real amounts and do not generate serious error.

Temperature
When the temperature of a gas is close to its liquefaction point, the behavior is very different from Ideal Gas Law predictions. With increasing temperatures, the Ideal Gas Law predictions become close to real values.

Why?
The answer is simple: ideal gases have molecular volume and show no attraction between molecules at any distance; real gas molecules have volume and show attraction at short distances. Let us first consider what pressure does. Pressure at high degrees will bring the molecules very close together. This causes more collisions and also allows the weak attractive forces to come into play. With low temperatures, the molecules do not have enough energy to continue on their path to avoid that attraction.

The van der Waals Equation

In order to overcome the errors in the Ideal Gas Law, Johannes van der Waals developed an equation to predict the behavior of real gases. Johannes van der Waals' equation included corrections for the finite volume of the molecules of the gas and the attractive forces between the molecules. Two new constants, a and b, were added. The corrected equation is:

P = (nRT)/(V-nb) - (n2a)/(V2)

The correction nb subtracts the volume of the molecules. b is measured in liters/mole. The correction with a reflects the strength of attraction and is measured in liters2-atmosperes per moles2.

The equation is generally put in the form:

Values of a and b are different for each gas. The values of a and b generally increase with increased mass of the molecule and complexity of the molecule.

Calculations:

For an example calculation, check the Dalton's Law of Partial Pressure page.

Continued Study
You can test yourself.

Sources:
Brown, Theodore L., H. Eugene LeMay, Jr. and Bruce E. Burston, Chemistry: The Central Science, Englewood Cliffs, NJ: Prentice Hall, Inc., 1994

Dorin, Henry, Peter E. Demmin, and Dorothy L. Gabel. Prentice Hall Chemistry: The Study of Matter, Needham, Massachusetts and Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1989.

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