The Physics
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Opus in profectus

# The Atomic Nature of Matter

## Practice

### practice problem 1

Verify the following claims about the mole.
1. A mole of dollar bills would make a stack almost 7000 light years high.
2. A mole of mini marshmallows would cover the Earth to a depth of about 1 kilometer.
3. A mole of standard men would weigh about 7 Earth masses.
4. A mole of food calories could feed every person on Earth for about 94 million years.
5. Counting a mole of anything at a rate of one object per second would take about 19 quadrillion years.

#### solution

Avogadro's number is huge! The Avogadro constant in the International System of Units is an astonishing 6.02 × 1023 objects per mole.

1. A mole of dollar bills would make a stack almost 7000 light years high.
2. A mole of mini marshmallows would cover the Earth to a depth of about 1 kilometer.
3. A mole of standard men would weigh about 7 Earth masses.
4. A mole of food calories could feed every person on Earth for about 94 million years.
5. Counting a mole of anything at a rate of one object per second would take about 19 quadrillion years.

### practice problem 2

How many moles of water are in a one liter bottle of water?

#### solution

The metric system of units (the precursor to the International System of Units) was devised so that one liter of liquid water would have a mass of one kilogram. While this is no longer explicitly true, it's close enough for most purposes.

An atom of water consists of two hydrogen atoms bound to one oxygen atom — thus the chemical symbol H2O. A hydrogen atom basically has a mass of 1 atomic unit and an oxygen atom has a mass of 16 atomic mass units. Thus, a single water molecule has a mass of 18 atomic mass units. Consult a periodic table whenever you need information like this.

 2 × (mH = 01 u) mO = 16 u mH2O = 18 u

The mass of a molecule (or atom) in atomic mass units is the same as the mass of a mole of that molecule in grams. Divide the mass of a liter of water in grams by the mass of a mole of water in grams and you get the number of moles in a liter of water. To three significant digits (my favorite number of significant digits), the answer is…

 1000 g/L = 55.6 mol/L 18 g/mol

But the mass of a typical hydrogen atom is not exactly 1 u and the mass of a typical oxygen atom is not exactly 16 u. To make things even more complex, every real sample of an element is always a mixture of isotopes (elements with the same number of protons, but a different numbers of neutrons). Whole number masses are nice for calculation, but not always realistic. Again consult a periodic table, but this time use the average atomic mass reported with all its glorious precision.

 2 × (mH = 1.0080 u) mO = 15.999 u mH2O = 18.015 u

But wait, there's more. The density of water varies with temperature. How cold or hot is your water? Let's say the bottle in question is not quite fridge temperature. Something reasonably cool, say 10 °C. Then 1 liter of water weighs less than 1 kilogram. Consult a table of densities whenever you need information like this. Calculating an answer from numbers with 5 significant digits means we should state our answer with 5 significant digits.

 999.70 g/L = 55.493 mol/L 18.015 g/mol

To me, both these answers are 55½ mol/L. For an introductory physics textbook like this one, an answer like that makes the most sense.

### practice problem 3

Write something different.

Answer it.

### practice problem 4

Write something completely different.

Answer it.