Thermal Physics: Ideal Gas Equation and Laws

pV = nRT

That equation is gorram everything to this section.

Learn it. Love it.

Wear it into town.

Pressure, p
This is caused by the gas particles colliding with the wall of whatever container they're in. Measured in pascals, Pa

Volume, V
The space occupied by the gas. Measured in m³

Number of moles of gas, n
That's fairly self explanatory, really. Measured in numbers.

Molar gas constant, R
= 8.31 J mol^-1K^-1

Temperature, T
The measure of the average kinetic energy of the molecules of a gas. It is the absolute temperature and is measured in kelvin, K


Now, a little something about temperature scales. The temperature of gas is measured by the change in another substance at the same temperature. The obvious example of this is a glass thermometer, and the expansion of a length of mercury or alcohol within. This is known as a thermometric property.

The thermometric property is measured at two fixed points. Often this is at the change of state of a property. It is well known for example, that the Celsius temperature scale was made to be fixed at 0 °C and 100 °C, the then defined freezing and boiling point of water. So let's come back to our glass thermometers. They are designed so at 0 and 100 they are both accurate. It is assumed therefore that the two will progress linearly up and down the scale. Alas as is so often the way, things are a little more complicated.

Mercury and alcohol expand at different rates as they heat. The consequence of this is that while they both agree at 0 and 100 °C, these are the only points they can be guaranteed to agree on. But fear not. It was to overcome this that a standard scale was devised. May I thus reintroduce: Kelvin.

William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE

The Kelvin scale is actually based on our post subject, the behaviour of an ideal gas. It uses the pressure of an ideal gas as it's thermometric property, it's two fixed points being the point at which there is 0 pressure, known as absolute zero and the triple point of water, where water can exist as solid, liquid and gas, defined to be 273.16 degrees on the Kelvin scale.

Gas laws

Remember that equation I ordered you to remember? These 3 are what combine captain planet style to form it.

  1. Boyle's Law

For a fixed mass of an ideal gas at constant temperature p ∝ ¹⁄_V
∴ pV = constant  or p₁V₁ = p₂V₂

 2. Charles' Law

The volume of a fixed mass of an ideal gas at constant temperature is proportional to it's absolute temperature, V  ∝ T

 3. The pressure-temperature law

The pressure of a fixed mass of an ideal gas at constant volume is proportional to it's absolute temperature,  P  ∝ T.

Now, you do need to know these. At the very least, you need to associate the name with the key components, because then you can just look at the equation up top to work out what they're meant to be doing.

More constants!

The Avogadro constant

The corresponding law states that at the same temperature and pressure, equal volumes of gases contain equal numbers of molecules.  Lot of words, sure, but it's not that bad at all. Let's refer back to our beloved equation. Rearranging for number of moles give us:

n = ^pV⁄_RT

It is clearly apparent that if p, V and T are the same for two gases then yes, they will have the same number of moles.

We take this a step further by considering 12g of Carbon-12. It has as many atoms as there are particles in one mole of a substance. This gives us the Avogadro constant, N_A = 6.02 x 10²³

The Boltzmann constant

This is literally just a constant gained by taking our two current constants and dividing them:

k = ^R⁄_NA

k = 1.38 x 10^-23 JK^-1

• Back to Heat Capacity and it's related paraphernalia.
• Onwards, to kinetic theory!