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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Decision Maths: List of Definitions

The D1 exam is but a week off, friends, and hopefully you've been practising all the algorithms and stuff. There is one aspect that I'm having trouble with, so I thought I'd try and right up a little revision sheet for us all, unless you're reading this in the future, in which case I have undoubtedly become a shining blogging god, and thanks to my own last post no one ever took a Decision Maths exam again, instead opting for nicer, more productive modules like Mechanics, or Further Maths. Indeed, anyone taking a D𝒙 exam is actually brought into a psychologist's lab to help further understand why someone would subject themselves to such a thing. Good luck, strange person I just made up. May you find peace in your future.


Anyway, the thing I want to work on right now is definitions. I haven't had to properly equate lists of uncommon words and phrases with arbitrarily specific meanings since learning all the Italian for Music Theory, so I'm not looking forward to it. Best get it done though. I'll try to keep things minimal, and only go into the particularly arcane ones. You should be able to remember something like "A subgraph is part of a graph" yourself. For the more tricky ones I'll try to single out the words to remember that might help jog your memory.

Graphs and Networks

Graphs understandably hold a whole bunch of the words you need to remember, taking up most of the course as they do.

Path - A finite sequence of edges such that the end vertex of one edge in the sequence is the start vertex of the next, and in which no vertex appears more than once

Ok, you see what I mean? It's not enough to just say that it's a bunch of connected vertices. You have to be ultra specific, to the point of obfuscation. I'm pretty sure Decision is Mathematicians' way of trolling Biology and Law students.

Keywords: finite sequence, no more than once

Walk - a path where you are permitted to return to vertices more than once.

So, a path can only visit each point once, whereas if you are walking along said path, you can go wherever the devil you like.

And with scenes like this, why not?

Keywords: more than once

Cycle - A closed path

a path where you could conceivably go around in a circle if you followed it. These are the things you want to avoid in them there Prim and Kruskal problems.


Spanning tree - A subgraph of graph G containing all the vertices of graph G and is a tree.

Define tree with tree. Stay classy Decision Maths.


Keywords: subgraph, tree


Isomorphic Graph - Shows the same information but differently.

If you're coming at this from a more scientific background you should recognise the prefix iso, meaning "equal". Isotope is a variant on an element that occupies the same τόπος (tópos, “place”) on the periodic table. Isomer is something that has the same molecular formula, but a different structure (μέρος (meros, “part”))¹ .

Keywords: Iso-same, morphic-shape

Complete graph - a graph in which every vertex is directly connected by an edge to each of the other vertices.

Everything is connected to everything, lines are everywhere.

Keywords: Every vertex, connected

Bipartite graphs 

This is a fairly important part, as almost every bipartite question I've done has asked for some definition or another. As such I've seperated it from the other graph definitions.

Not to be confused with...

Bipartite graph - consists of two sets of vertices X and Y. The edges only join vertices in X to vertices in Y, not vertices within a set.

Pretty simple one, it's just describing the set up of your average bipartite problem.

Alternating path - starts at an unmatched node on one side of a bipartite graph and finishes at an unmatched node on the other side. It uses arcs that are alternatively 'not in' and 'in' the initial matching.

This is another one of those definitions that really takes the piss. It basically means that an alternating path is a path that alternates, specifically between the two sets of nodes. It is what you end up with after applying the maximum matching algorithm.
It is probably best split into it's two parts. Try to associate everything in red with "path" and everything in blue with "alternating". The second one should be easier as "alternatively" is right in there already.

Keywords: Starts unmatched, finishes unmatched, initial matching

Matching - The one-to-one pairing of some or all of the elements of on set, X, with elements of a second set, Y.

The main thing I can recommend for this one is DO NOT FORGET the "one-to-one" bit. That is worth , well, probably only a single mark, but any mark you can grab in this exam is a good one. It is simply not enough to just say that it's the pairing of the two sets.


Keywords: one-to-one pairing, some or all


Complete matching - This is the same as a compete graph but with matching replacing the word graph, and taking into account the two separate sets. It is possible that you cannot obtain a complete matching if some tosser in the question refuses to do some task or another. In this case you simply get as many pairs are you can and call the a maximal matching.

Route Inspection (aka Chinese postman problem)

I prefer to think of this as the envelope problem, because it reminds me of that puzzle we all did when younger of trying to draw an envelope without pen leaving paper or redrawing lines. It didn't take too long to work out that this was impossible without adding an extra line to make it look like an open envelope. If there is one thing I can say I enjoyed about D1 it's the joy of finally knowing why this was. Anyone, on to the definitions.

Valency - This is essentially the number of arcs that are connected, or "incident" to a particular vertex. This can be either odd or even, because, you know, integers work that way.

Eulerian - A graph has this property if all the valencies in it was even. If precisely 2 valencies are odd, and the others even then this is known as being semi-Eulerian

Leonhard Euler. This guy has had an insane amount of stuff named after him

Traversable - A graph is traversable if it is possible to traverse every arc once without removing pen from paper. This fits with the Eulerian definition as is a graph is Eulerian, it is traversable. Indeed, if a graph is semi-Eulerian is it semi-traversable, with the start and end points being the two odd valencies.

Critical path analysis

Source node - The first one.

Sink node - The last one.

Dummy activity - A virtual line with no time value, used to show that an activity depends on another before starting, if said activity has already been drawn leading to another node.

Early event time - The earliest time of arrival at the event allowing for the completion of all preceding activities. Calculated during the forward pass/scan.



Late event time - The latest time that the event can be left without extending the time needed for the project. Calculated during the backwards pass/scan.



Critical activity - If any increase in an activity's duration with cause an increase in the total duration of the project then it is a critical activity. The path leading from source node to sink node following these activities is the critical path.



And that, as they say, is that. I've left out linear programming as that is basically algebra, and we all know algebra terms. Inequalities and all that. I have on the other hand added a few potentially superfluous ones just for completions sake. I can't say I've heard much call for a definition of source/sink node. 



Good luck, all.






¹I could go on, and I really want to because etymologies are fascinating, but I kinda committed to this revision thing.

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Decision Maths Tips and Tricks

Welcome to my help with Decision Maths post. all of us who are doing this horrible excuse for a maths module know just how irritating it is. What seems like common sense is wrong, and the mark schemes are notoriously nit-picky. Well all hope is not lost, as I guide you through the points that will lead you to exam success!

• Practise everything over and over and over and over and over...(etc)
• Memorise the definitions of anything and everything
• Hope that the mark scheme isn't a bitch.

Aaaand that's it. Seriously. Just remember to write everything out in the way you have been taught and have practised. So break out the past papers, look out the book examples, and do them all. Then do them all again. And if you have the time, again.

This poorly scanned rubbish is your enemy. Also maybe procrastination. This stuff is all kinds of boring.

This is not a maths exam, whatever it says on the front. This is a test of your ability to learn by rote. Sucks, I know, but there it is. It's really easy to make marks, but it is just as easy to lose them. You forget one thing, make one tiny mistake, and that single lost mark suddenly becomes a whole load more lost down the line. It's like a terrible, horrifying mark scheme avalanche. You will either do really well, or fail horribly.

Best make sure we do really well, eh?

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