Gas Laws

Ideal Gas Law

The Ideal Gas Law (IGL) is a fairly fundamental physical concept. If you confine a gas and compress it, all the molecules get closer together, making it denser -- like squishing a balloon. If you heat a gas in a confined space, its pressure will go up. Conversely, if you heat a gas in an unconfined space, its density will go down -- like with a hot air balloon.

All of these changes are described (approximately) by the IGL. The IGL is an "equation of state" for idealized gasses in relatively "normal" conditions. It is usually first presented to students in the form -

$$PV = n\mathbb{R}T$$

where \(P\) is pressure (pascals), \(V\) is volume (cubic meters), \(n\) is the number of moles (unitless), \(\mathbb{R}\) is the universal gas constant (about 8.314 for SI units), and \(T\) is temperature (kelvin).

However, the above form is often fairly inconvenient for many engineering applications. It's usually not really practical to directly measure in any way the number of moles of a substance passing through an engine, or a compressor, or whatever else.

To solve this, one can simply multiply \(n\), the number of moles, by \(M\), the molar mass of the gas. This gives a mass \(m\), which is often more convenient. To maintain equality in the gas law, the molar mass has to be divided out somewhere else; typically, since it's approximately constant for any given type of gas, it's combined with the other constant in the equation - the gas constant \(\mathbb{R}\). This creates a new constant, typically known as the "specific gas constant," which is constant for a given gas but different between gasses.

$$\frac{\mathbb{R}}{M} = R$$

(here, the universal gas constant is denoted \(\mathbb{R}\), while the specific gas constant is \(R\))

Any given gas has a different value for \(R\), depending on its properties. The value for atmospheric air is about 287.0 \(\frac{J}{kg \cdot K}\) in SI units.

This simplification yields the equation

$$PV = mRT$$

Since mass over volume gives density, this is another simplification that can be made, yielding the equation

$$P = \rho RT$$

Finally, this form is what is often used, especially in aerospace contexts.