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!

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Thermal Physics: Heat Capacity

In Britain's own peculiar way it is bloody freezing, in the middle of June. So, what better time than now to do a little something on thermal physics.

As with most physics, this all comes down to energy, and the transfer, exchange and general movement of same. The effect of this is generally kind of obvious. The more thermal energy something has, the larger it's temperature is. What's interesting is that various materials have a greater inclination to be heated that others. This is easier demonstrated. Find a metal surface, and consider it's temperature. Unless you already know what I'm about to get at and are purposefully being difficult by considering the inside of an oven, you should agree that said metal is at room temperature. Now if you touch said metal, you should notice that it is quite cold. "Gosh James, how can this be?" I hear you ask. Well, as we all know, heat tends to flow to colder areas. Things like to reach an equilibrium, that's just they way of things. So when you touch metal, you aren't feeling cold metal as much as you are feeling the heat energy in your fingers being SUCKED OUT FROM YOUR BODY.

Dramatic reconstruction

The way we measure how willing objects are to drain the life from your very body is called the specific heat capacity, c. This is defined as the energy required to cause a temperature rise of 1K per unit mass, usually 1 kg. From it we can work out a bunch of stuff with the equation:

ΔQ = mcΔT

In words, the change in thermal energy of an object is equal to the mass by the SHC by the change in temperature. T is measured in kelvins really but given that 1 K and 1 °C is exactly the same it doesn't really matter once you stick that Δ there.

All important equations are given to you but I feel it's helpful to get to know them beforehand. So let me throw this one at you:

 ΔQ = ml

This is where we bring in latent heat, l. Again this comes with a "specific" alternative. Two actually, and as such we have two definitions to remember. Both are pretty similar however, so I don't imagine you will have too much trouble.
Specific latent heat of fusion

the energy required to chance 1 kg of solid into 1kg of liquid, with no change in temperature.

Specific latent heat of vaporisation

the energy required to change 1 kg of liquid into 1 kg of gas with no change in temperature.

What this means for you is that when working out the energy required to  get a substance from one temperature to another, and there is a state change in between, you are going to need to make more than one calculation. One for the first temperature increase, one for the state change, then another for any further temperature increase. Keep yourself aware of this and you should be fine.

On the practicality side of things, this area of physics is widely used, mainly in the area of cooling, by dissipating thermal energy. It can been seen in power stations, and closer to home, us when we sweat.

Continue into the world of ideal gases

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