Sorting the maths out in VCE Chemistry - worksheet included!

[Phenomenol]

I haven't posted in a while, but I'll try to be more active with answering questions and posting my resources from here on out.
One of the students I tutor IRL has asked today, "how do I do calculations when there's no clear formula to use?"
In my opinion there are two important things to look out for:

1. You should manipulate your given values in a way that the units of the resultant answer make sense. One other fine user here has termed this technique as "unit hacking".

For example, if the question asks for a quantity of energy, and you are given a value in kJ g^-1, and another value in g, chances are you need to multiply these two numbers together regardless of the context of the question! This is of course since the units will combine in the following manner:

kJ g^-1 × g = kJ.

So, the resulting answer is a quantity of energy.

Here's a slightly more involved example: Suppose it is given that complete combustion of 2.3 L of fuel releases 14 MJ of energy. What is the volume of fuel required to produce 5.0 MJ of energy?

We should start off by calculating the proportion between the volume/energy quantities. We could either take the energy released per 1L of fuel, or the volume of fuel required to produce 1 MJ energy. (often one part of the question will ask to calculate such a value first). Below is the first option:

Energy released per 1 L of fuel = 14 MJ ÷ 2.3 L = 6.1 MJ L^-1

Notice the units here make sense! I just calculated an energy/volume proportion. Now... how do I apply this proportion to the other value, "5.0 MJ", to give a quantity of volume? I could divide one by the other, vice versa, or multiply them together. Let's try all of them!

Volume of fuel required to produce 5.0 MJ = 6.1 MJ L^-1 × 5.0 MJ = 3.0 × 10¹ MJ² L^-1

Volume of fuel required to produce 5.0 MJ = 6.1 MJ L^-1 ÷ 5.0 MJ = 1.2 L^-1

Volume of fuel required to produce 5.0 MJ = 5.0 MJ ÷ 6.1 MJ L^-1 = 0.82 L

The last line is the correct working as it gives the correct units for the answer. With enough practice, you can hopefully see which option to go with from looking at the units on the values you have!

2. Think logically whether calculated values should be qualitatively higher or lower than those given.

A common error is incorrect application of an amount ratio.

Here's a simple example: 2Al(OH)₃(s) +  3H₂SO₄(aq) → Al₂(SO₄)₃(aq) + 6H2O(l)

Suppose you had n(H₂SO₄) = 0.034 mol and you were to apply an amount ratio to calculate n(Al(OH)₃).

First off, the amount ratio we are concerned with is 3/2 or 2/3. Let's say we're not sure which way around it is yet.

Then, in that case, we can read off the equation: if I have 3 mol of H₂SO₄, then how much Al(OH)₃ would completely react with it? Well... looking at the equation, we should say that 2 mol will! Is this more or less in amount than the amount of H₂SO₄ we started with? Less!

Okay, so if I have 0.034 mol of H₂SO₄ then I am expecting less than 0.034 mol of Al(OH)₃.

So my working will be: n(Al(OH)₃ = 2/3 × 0.034 mol = 0.023 mol.

And, it would not be:  n(Al(OH)₃ = 3/2 × 0.034 mol = 0.051 mol.

This could be quite obvious to some people, but others may find being reminded of such simple approaches useful.

Let's return to a previous example.

Suppose it is given that complete combustion of 2.3 L of fuel releases 14 MJ of energy. What is the volume of fuel required to produce 5.0 MJ of energy?

Am I expecting a quantity of energy greater than or less than 2.3 L? If 2.3 L of fuel releases 14 MJ of energy, will more or less than 2.3 L release 5.0 MJ of energy? It makes sense that it will require less fuel when spelled out this way. Thinking this out could help with spotting simple errors that might be passed on unnoticed (i.e. in this case if your calculated value is greater than "2.3").
These are my personal thoughts on the matter. These techniques are of course not going to be suitable for everyone, but for people that may have a bit of trouble with maths-focused questions in this subject, it might hopefully help with developing your own shortcuts.

Lastly, I have attached a (rather tough) worksheet that should hopefully give you some good practice with calculations whether or not you follow the mental checks I have suggested. I will probably upload answers later this week. Thanks for reading!

Edit: I have been pretty occupied by my own work recently... if someone requests for solutions I will still try to get them up then. 

[Vaike]

How did I not see this post earlier!  Fantastic advice Phenomenol! These kinds of reasoning is really an awesome tool to have in VCE chemistry, they make figuring out what calculations to perform a lot easier. If anyone wants to learn more about 'unit hacking', just google dimensional analysis (the real name for it lol).