Physics uses simplification to arrive at quick, moderately accurate answers for problems. Small friction sources and tiny masses may be discounted. In textbooks students are introduced to *massless strings* and *frictionless pulleys*.

For chemists, there is no single equation that conveys the behavior of every gas. Still, the basic properties gases do share in common are combined to produce the simplistic* equation of state*. We will use this equation to determine the ideal gas law molecular weight.

## Deriving the Ideal Gas Law

When we compress a gas, its volume shrinks. That is, the pressure is proportional to the inverse of the volume.

P ∝ 1/V

This has been known for hundreds of years as* Boyle’s Law*.

In addition, as temperature increases, gas volume increases (think hot-air balloon).

T ∝ V

This, too, has long been known and has been assigned a name. It is *Charles’ Law*.

Now obviously, the more gas there is, the greater its volume will be. Gas is often listed in terms of moles*, n.

n ∝ V

Obviously, there is no name for this relationship.

We now combine above. Since we have not been discussing equalities, we wind up with a “proportion” equation, not an “equals” equation. We need to introduce a constant to get around this. We call it the Ideal Gas constant, R¹. Its value is 0.082 liter-atmospheres per mole-degrees Kelvin. This allows us to state the Ideal Gas Law in exact mathematical form,

PV = nRT

## Simple Ideal Gas Law Application

The ideal gas law equation is useful to describe the physical characteristics of simple non-interactive gases.

The number of moles, n, equals the weight of gas present divided by the molar mass (atomic or molecular weight²). For example: at standard temperatures and pressures (STP), one mole of a gas is maintained at 273 degrees Kelvin, at a pressure of one atmosphere. What is the volume this gas occupies?

Rewriting our formula, we get,

V = nRT/P

This works out to approximately 22.414 liters under these conditions.

## Ideal Gas Law Molecular Weight

The number of moles, n, can be converted into the mass divided by the molar mass (essentially, the *molecular weight*), m/M. Then,

PV = (m/M)(RT)

Rearranging, the above equation becomes,

M = (m/PV)(RT)

Plugging in the correct values, we should be able to determine the ideal gas law molecular weight of a gas.

## The Units We Choose

At standard temperatures and pressures (STP), for instance, the pressure is one atmosphere. The volume is 22.414 liter. The temperature is 273° Kelvin. Under these conditions, the mass of the gas equals its molar mass. To demonstrate this, we evaluate 4.000 grams of helium under these conditions. We calculate its molar mass,

M = [4.000/(1)(22.414)][(0.082)(273)] = 3.995 grams.

The actual M (atomic weight) for helium is 4.003 grams.

Other and varied complications are listed in the references for those desiring to pursue the ideal gas law further.

## The Ideal Gas Law in Terms of Density

For the purpose of showing how other forms of the ideal gas law can be derived, consider the following:

PV = m/MRT

m/V = RT/MP

δ = RT/MP

Another (if strange) way of viewing things is the* concentration*, C, of a gas in space—the number of moles of gas in an otherwise empty volume. The ideal gas law is then written,

n/V = P/RT or

C = P/RT

## When the Ideal Gas Law is Insufficient

Results from application of the ideal gas law will never match reality, since no gas is an “ideal gas.” An ideal gas requires *no* interaction of its component atoms or molecules. The size of an “ideal atom” is *zero*. Clearly this is not reality either. Nevertheless, the law is a good one. It is not a bad approximation for non-reactive gases that consist of very tiny particles, as is the case for helium. The ideal gas law molecular weight determination is an excellent instructional aid for beginning students.

¹ The value of R is closely associated with Avogadro’s Number. See references.

² A simple gas may consist of atoms, such as He, or small molecules, such as H₂.

**Note:** You might also enjoy Hydrogen Metal?

**References:**

- University of Waterloo: The Ideal Gas Law
- AUS-e-TUTE: Chemistry Tutorial – Ideal Gas Law
*YouTube*: Find the Density of a Gas (Ideal Gas Law)

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Reminds me of school, many , many years ago!