The mole and molar mass

At the local office supply store, felt-tip pens are packed in dozens: $12$ units in each pack. Chemists do something similar with atoms and molecules, but they use a much larger "pack": the mole, as compared in the diagram.

1. a) How many particles does a mole contain? What is this number called?
2. b) How many molecules are in one mole of water? And how many atoms in one mole of gold?

In a spoonful of water, there are so many molecules that even if all of humanity started counting them one by one, they would never finish. Explain in your own words why chemists work with the mole instead of counting particles one by one.

In a workshop, they need to separate $1000$ identical screws and do not want to count them one by one. They know that each screw weighs $2\ \text{g}$.

1. a) How much do the $1000$ screws weigh in total?
2. b) Explain how they would use a scale to "count" the screws without counting them, and what this trick has to do with the concept of the mole.

Imagine that you could count the molecules in one mole of water one by one, at a rate of one molecule per second, without stopping even to sleep.

1. a) One year has about $3.2\times10^{7}\ \text{s}$. How many years would it take you to finish? Express the result in scientific notation. (Given: $N_{\text{A}} = 6.022\times10^{23}\ \text{mol}^{-1}$)
2. b) The age of the universe is about $1.4\times10^{10}$ years. Compare your result with it and draw a conclusion about how small molecules are.

Hemoglobin, the protein in blood that carries oxygen, has a molar mass of about $68000\ \text{g/mol}$.

1. a) Calculate the mass of a single molecule of hemoglobin, in grams and in scientific notation. (Given: $N_{\text{A}} = 6.022\times10^{23}\ \text{mol}^{-1}$)
2. b) In light of the result, explain why in the lab a single molecule is never weighed, but always a huge number of them.

It is estimated that across all the beaches and deserts on Earth, there are about $7.5\times10^{18}$ grains of sand.

1. a) How many times more particles are there in one mole than grains of sand on the whole Earth? (Given: $N_{\text{A}} = 6.022\times10^{23}\ \text{mol}^{-1}$)
2. b) In light of the result, explain why the mole is a unit designed for the microscopic world and not for counting macroscopic objects such as grains of sand.

In the periodic table, carbon appears with an atomic mass of $12\ \text{u}$. This means that an atom of carbon weighs $12\ \text{u}$, but also that one mole of carbon atoms weighs $12\ \text{g}$.

1. a) What is the molar mass of carbon? State the unit in which molar mass is expressed.
2. b) What is the molar mass of oxygen, whose atomic mass is $16\ \text{u}$?

Imagine one mole of helium atoms and one mole of lead atoms. Without doing any calculations, answer with reasons:

1. a) Which of the two samples contains more atoms?
2. b) Which of the two has more mass?

A jeweler has one mole of gold atoms ($\text{Au} = 197\ \text{u}$) and a bicycle manufacturer has one mole of aluminum atoms ($\text{Al} = 27\ \text{u}$).

1. a) Who has more atoms?
2. b) Who has more mass? How many grams does each one have?
3. c) Explain why one mole of each element does not weigh the same.

Water has the formula $\text{H}_2\text{O}$: each molecule contains $2$ hydrogen atoms and $1$ oxygen atom, as shown in the diagram.

Calculate the molar mass of water (atomic masses: $\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$).

Ammonia, $\text{NH}_3$, is manufactured in large quantities to produce fertilizers and cleaning products. Calculate its molar mass (atomic masses: $\text{N} = 14\ \text{u}$; $\text{H} = 1\ \text{u}$).

Hydrated copper sulfate, $\text{CuSO}_4\times5\text{H}_2\text{O}$, is a blue salt containing $5$ water molecules attached to each unit of $\text{CuSO}_4$; it forms when white anhydrous copper sulfate absorbs water, which is why the anhydrous salt is used in the school lab to show that air contains water vapor. The diagram shows its composition.

1. a) First calculate the molar mass of $\text{CuSO}_4$ (atomic masses: $\text{Cu} = 63.5\ \text{u}$; $\text{S} = 32\ \text{u}$; $\text{O} = 16\ \text{u}$).
2. b) Calculate the molar mass of the $5$ water molecules ($\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$).
3. c) What is the molar mass of hydrated copper sulfate?

A mineral contains an oxide with formula $\text{X}_2\text{O}_3$, where $\text{X}$ is an unknown metal. In the lab, students measure its molar mass and obtain $102\ \text{g/mol}$. The diagram shows the model of a unit of $\text{X}_2\text{O}_3$.

1. a) Calculate the atomic mass of element $\text{X}$ (atomic mass: $\text{O} = 16\ \text{u}$).
2. b) Look up in the periodic table which element it is and write its name and symbol.

Oxygen gas in the air is not made of individual atoms, but of $\text{O}_2$ molecules. A student says its molar mass is $16\ \text{g/mol}$ because the atomic mass of oxygen is $16\ \text{u}$. Explain the student's mistake and calculate the correct molar mass of $\text{O}_2$.

The gas that forms bubbles in soft drinks is carbon dioxide, $\text{CO}_2$. Calculate its molar mass (atomic masses: $\text{C} = 12\ \text{u}$; $\text{O} = 16\ \text{u}$).

Look at the formula for carbon dioxide, $\text{CO}_2$, and answer:

1. a) How many moles of oxygen atoms are there in $1\ \text{mol}$ of $\text{CO}_2$? And in $2\ \text{mol}$?
2. b) How many moles of atoms in total, adding carbon and oxygen, are there in $1\ \text{mol}$ of $\text{CO}_2$?

Methane, $\text{CH}_4$, is the main component of natural gas that reaches many homes. The diagram shows the model of its molecule.

1. a) How many atoms are in one molecule of methane?
2. b) How many moles of atoms are in $1\ \text{mol}$ of methane? How many atoms does this correspond to? (Given: $N_{\text{A}} = 6.022\times10^{23}\ \text{mol}^{-1}$)

Limestone, seashells and eggshells are made of calcium carbonate, $\text{CaCO}_3$. Calculate its molar mass (atomic masses: $\text{Ca} = 40\ \text{u}$; $\text{C} = 12\ \text{u}$; $\text{O} = 16\ \text{u}$).

Sulfuric acid, $\text{H}_2\text{SO}_4$, is one of the most manufactured compounds in the world: it is used to make fertilizers and is found in car batteries.

1. a) Calculate its molar mass (atomic masses: $\text{H} = 1\ \text{u}$; $\text{S} = 32\ \text{u}$; $\text{O} = 16\ \text{u}$).
2. b) What is the mass of half a mole of sulfuric acid?