Moles

Contents

Masses, moles and molarity

The atomic mass unit (AMU) is defined as 1 ⁄ 12 the mass of an atom of 12C (i.e. roughly the mass of a proton or neutron). In biochemical circles, one AMU is often termed one Dalton (Da) = 1.66053873 × 10−27 kg. Atoms can be said to 'weigh' (or have a mass of) a certain number of Daltons.

A typical protein weighs:

Relative mass, Mr, is a second way of describing the mass of tiny things such as atoms. It is used because kilograms are not a very convenient unit of mass for such light entities, as can be seen for the protein mass above. The relative atomic mass (RAM) of an atom 'X' is the mass of an atom 'X' divided by mass of 12C, i.e. roughly the mass of the atom in 'proton masses'. Relative molecular mass (RMM), and relative formula mass (RFM, for things like KCl, where the chemical in question is not really a molecule) are similarly defined as the sum of the RAM of the constituent parts (i.e. RMM of methane, CH4, is RAM(C) + 4 × RAM(H) = 12 + 4 × 1 = 16).

Technically RAM, RFM and RMM have no units (since they are a ratio of two masses). However, it's often convenient to think of them as having the units of grams per mole (g mol−1), for reasons that will become clear when we do calculations. The RAM of an element will often be fractional: chlorine is a mixture of two isotopes, chlorine-35 (75%) and chlorine-37 (25%), hence the RAM for chlorine is usually quoted as ( 0.75 × 35 + 0.25 × 37 = ) 35.5. Mass spectrometry can be used to discover the relative abundances of the isotopes of an atom for such calculations.

A further complication in determining the atomic mass of an isotope is the existence of mass deficit. Mass deficit is the difference between the mass of what goes into making an atom, and the mass of the atom that results from their combination:

In forming bonds between the nucleons in a nucleus, and between the electrons and the nucleus, some mass is destroyed, and hence, by Einstein's formula E = m c2, energy is released. When we manipulate how electrons are arranged in atoms (chemistry), or how nucleons are arranged in nuclei (nuclear fission and fusion), we will alter the mass deficit, and hence take up or give out mass-energy. Electrons are light, and are held to the nucleus by a relatively weak force (electromagnetism), hence the amount of energy available to chemistry is relatively low, and the difference in mass between reactants and products in a chemical reaction is negligible. However, the force binding the much more massive nucleons within the nucleus is far stronger (it has to be to keep all those positively charged, repelling protons together), hence the energy available to nuclear fission and fusion is far higher. This is why a hydrogen bomb is so much more destructive than an similarly sized lump of TNT.

One atom is difficult to weigh, so another unit is used, the mole. 1 mole of something contains the same number of things (atoms/molecules/ions/tin cans/anything else) as there are atoms in 12 g of 12C. This number of things is called Avogadro's number, 6.022 × 1023 mol−1.

An atom of 12C weighs 12 Da. You need 12 grams to have 1 mole of it. Hence we can say (for practical purposes) that carbon-12 'weighs' 12 g mol−1. This is useful for calculations.

The concentration of a substance in (for example) water can be measured in grams per litre or similar. However, a 1 mg L−1 solution of a huge protein will contain considerably fewer molecules than a 1 mg L−1 solution of something lighter, like ammonia. We tend to use molarity to measure concentrations in biochemistry, since this measures the number of things in a solution, rather the the mass of things. One mole in one litre of solution is a 1 molar solution, 1 M = 1 mol L−1. Note that this is not the same as 1 mol dissolved in a litre of solvent (this is a 1 molal solution, and not much used), since dissolving solutes in solvents will usually slightly increase the volume of the solution.

Here is an example calculation using moles and molarity:

I want 0.5 litres of a 0.1 molar solution of sodium chloride. How many grams of salt do I need?

The RAM of sodium is 23.0 g mol−1 and the RAM of chlorine is 35.5 g mol−1.

A 0.1 M solution contains 0.1 mol L−1. Multiply the molarity (which is just a sort of concentration) by the RFM to get the concentration in g L−1.

Note how the mol and mol−1 cancel to leave you with just g L−1 left. Taking a similar approach, we want 0.5 L of this 5.85 g L−1 solution, so multiply these together to cancel the L with the L−1.

So we need 2.90 g of salt dissolved to 0.500 L with water to give us a 0.100 M solution.

With any calculation like this, you only need to remember two things.

Molarity triangle

Moles (mol)

Molarity (M)

Volume (L)

Remember M is the same as mol L−1 and that 1 L is 1000 mL, or (yuck) 1 dm3. To convert between molarity and volume and moles, cover up the bit of the triangle you want, and the arrangement of what is left will tell you what you need to do.

So to calculate the number of moles, cover up moles, and you are left with molarity and volume next to each other, which you should multiply together. 

To calculate molarity, cover it up, and you are left with moles on top of volume, so:

The other thing to remember is the

Relative mass triangle

Mass (g)

Relative mass (g mol−1)

Number of moles (mol)

Use this as before: number of moles = mass ⁄ relative mass, etc. Using these two triangles, you should be able to work out most calculations you'll come across in biochemistry.

Uncertainty & significance

When dealing with numbers in science, it is important to know how much uncertainty you have in your measurement. The number of significant figures (s.f.) you quote in a number reflects this uncertainty:

Note that to determine the number of s.f., you need to ignore any leading zeroes, then count the figures (including any trailing zeroes). An easy way to see how many s.f. we have is to use scientific notation:

A measurement is only as good as its weakest link:

Moles are rather large quantities, so you need to become familiar with smaller subdivisions of units generated using SI prefixes.

Conversion between these units is easy: you may find it helpful to think of:

So to convert 1 mL to litres.

Note how the mL on the top and bottom of the division cancel, and the L−1 at the bottom of the division becomes a L on the top of the result. You may find it easiest to convert everything into the base units (mol, L, g) even if you have to convert the result back to microlitres or similar later. It'll stop you making any cock-ups.

So, you can see that a number can be expressed in three equivalent ways.

These numbers are all equivalent, and they all have four significant figures. They all have an uncertainty of ± 0.000005 g. The general rule is there is an uncertainty of ± 0.5 of the least significant figure, so 12.4 has an uncertainty of 0.05, and 12 has an uncertainty of 0.5 . When doing calculations, never quote a number to more significant figures than the worst measurement you have made. So if you measure out 12.000 grams of salt and dissolve it to 1.0 L with water, you do not have a 12.000 g L−1 solution. You have a 12 g L−1 solution, because you have only measured two significant figures' worth of volume. Don't ever let me catch you quoting an absorbance to ten decimal places: there's no way you can possibly justify this. However, it's generally a good idea to do calculations to as many s.f. as you can, and then round off at the end: this saves compounding rounding errors.

Test yourself

  1. I wish to claim my inheritance early. How many grams of potassium cyanide (KCN) do I need to dissolve in water to make 250 mL of a 50 mM solution for the murders? The RMMs are K:39, C:12 and N:14.

Answers

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