From WolfWikis

Conversions

Scientific numbers consist of four parts

1. a numerical part
   1. its sign (or phase, if the number is complex)
   2. its magnitude
   3. it uncertainty (indicating how well we know its magnitude)
2. a dimensional part
   1. this part is indicated in units

In CH 101 we will not deal with complex numbers and the uncertainty part is also for later, so we just have to deal with (positive or negative) magnitudes and dimensions.

The dimensional part can be absent, some numbers in science are dimensionless and there are procedures to make numbers dimensionless as well. Usually however we need to keep track of both the numerical and the dimensional part, i.e. we need to do double bookkeeping. Just multiplying or dividing numbers will easily lead to error! In fact it is often the dimensions that tell you whether to multiply or divide! Make them your friends!

Basic units and prefixes

Some dimensions have basic units like meters [m], Kelvins [K] or kilograms [kg]. Even these can require some conversion, particularly if they have a prefix:

1 kg = 1,000 g

1 kg = 1,000,000 mg

1 kg = 1,000,000,000 μg

1 kg = 1,000,000,000,000 ng

1 kg = 1,000,000,000,000,000 pg

1 kg = 1,000,000,000,000,000,000 fg

In the SI -system of units the prefixes are ordered in factors of 1000, i.e. you can shift the decimal point by three decimal places by using the next prefix. This is very handy as it reduces the chance someone misreads a number by a decimal place.

Apart from the prefixed units, the same dimension can have more than one unit, often these are older units than the SI ones, that are still in use.

For pressure e.g. we have:

Si unit: 1 Pa (Pascal)

cgs unit: 1 bar = 100 kPa

older unit: 1 atm = 101.25 kPa

still older unit: 760 Torr = 1 atm

(Luckily bars and atmospheres are only different by a percent and a quarter...)

Here in the US you may need a little more conversion than elsewhere in the world as units like gallons, miles and ounces are still in use. In US science however the English units are usually avoided.

Composite units

Many units are actually ratios of two other units, e.g.

Molar mass M: [g]/[mol]

Speed of car: [mi]/[h]

Speed of bullet: [m]/[s]

If you use M in a conversion always make sure you know whether to multiply or divide. This is where double bookkeeping comes in.

Notice that we can put the same prefix in the numerator and denominator without changing anything:

1[g]/[mol] = [1000 mg] / [1000 mmol] = [1,000,000 μg] / [1000,000 μmol] = [1,000,000,000 ng] / [1,000,000,000 nmol]

1[g]/[mol] = [1000 mg] / [1000 mmol] = [1,000,000 μg] / [1000,000 μmol] = [1,000,000,000 ng] / [1,000,000,000 nmol]

1[g]/[mol] = [mg/mmol] = [μg/μmol] = [ng/nmol]

A special case are two and three dimensional units like square or cubic meters. The latter dimension (volume) can also have its own unit (liters)

1 liter = 1 dm³

A decimeter (1 [dm] = 0.1 [m] or 1[m]=10[dm]) is no longer a valid SI unit, but the liter 1l is...

1 m³ = 1000 l

Notice that the conversion factor also involves a cube power: 1000 = 10³.

A handy thing to remember is:

1 ml = 1 cm³

Conversions and the importance of double bookkeeping

As you can see speed was given in both [mi.]/[h] (mph) and in meters per second. We can convert between these units as follows:

1 [mi.] = 1.6 [km], so the conversion factor is 1.6/1 [km/mi]

1 hour = 60*60=3600 seconds, so the conversion factor is 3600 [s/h]

Say the car's speed is 50 mph what is that in [m/s]?

Let's get rid of the miles first. As the conversion factor has mi. in the denominator we can just multiply:

50* 1.6 [mi/h][km/mi.] = 80 [km/h]

Notice how the miles can now be canceled. This actually confirms that we must multiply and provide a much needed check on our calculation

Next step is to get rid of the hours. If we multiply

[km/h]*[s/h]

we get [km.s/h²] that is not what we want! So in this case we must divide:

[km/h]/[s/h]= [km/h]*[h/s]= [km/h]*[h/s] = [km/s]

This tell us the right operation:

80 [km/h] / 3600 [s/h] = .02222 [km/s] = 22.2 [m/s]

The last step makes use of the fact that 1 [km] = 1000 [m] and allows the reader to have a better idea of what the number stands for.

Notice that if we had not kept track of the unit that we may very well have ended up with nonsense!

Conversions in chemistry

A lot of conversions in chemistry center on:

1. from mass to moles and back using the molar mass in [g/mole] or Dalton
2. from moles to numbers of particle and back using Avogadro's number 6.202 10²³ [mole^-1]
3. from masses/moles/number of particles to concentrations in [mol/l] etc.
4. from volumes of gases at given P and T to moles, masses, numbers of molecules etc. (1 mole at P=1 atm and 25^oC = 22.4 liter)

Exercises

Exercise 1

Convert into moles (or [mmol] milimoles etc.) the following weights

1. 1.2 gram of B₂H₆
2. 1.2 gram of B₂O₃
3. 1.2 gram of B₂Te₃
4. 2 mg of Ti₂Te₃
5. 2 μg of SiCl₄
6. 63.546 g of Cu

Exercise 2 Convert into grams (or kg or mg etc.)

1. 2 mmol of Sc₂Te₃
2. 2.34 μmol of Cs₂Te₃
3. 2 mol of polyethylene with a molar mass of 120,000 Dalton
4. 0.234 mol of Sc₃LaGe₄O₁₄

Exercise 3 How many molecules or formula units are there in

1. 1.6 mol of acetone
2. 1.7 nmol of Ag
3. 0.55 fmol of Si

Exercise 4 How many molecules or formula units are there in

1. 12 fg of carbon-12
2. 6 mg of Sc₂Te₃
3. 1 g of polyethylene with a molar mass of 120,000 Dalton

Exercise 5 How many atoms of oxygen are there in

1. 0.234 mol of Sc₃LaGe₄O₁₄
2. 2 mg of Ti₂Te₃
3. 1.2 gram of B₂O₃