Exercises: Quantitative calculations in everyday life

Calculate the percentage composition of water, $\text{H}_2\text{O}$ (atomic masses: $\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$). Check that the two percentages add up to $100\ \%$.

Carbon dioxide emitted by cars is the main greenhouse gas. Calculate the mass percentage of carbon and oxygen in $\text{CO}_2$ (atomic masses: $\text{C} = 12\ \text{u}$; $\text{O} = 16\ \text{u}$).

Ammonia, $\text{NH}_3$, is the raw material for the nitrogenous fertilizer industry. Calculate its mass percentage of nitrogen (atomic masses: $\text{N} = 14\ \text{u}$; $\text{H} = 1\ \text{u}$).

A bag of fertilizer from a farm supply store has the label "NPK $15$-$15$-$15$" printed on it, indicating $15\ \%$ by mass of nitrogen, $15\ \%$ of phosphorus(V) oxide, and $15\ \%$ of potassium oxide.

1. a) Explain what the three numbers on the label mean.
2. b) If the bag contains $20\ \text{kg}$ of fertilizer, how many kilograms of each nutrient does it provide?

Glucose has the molecular formula $\text{C}_6\text{H}_{12}\text{O}_6$.

1. a) What is its empirical formula? Explain what information each of the two formulas gives.
2. b) Give an example of a compound whose empirical and molecular formulas coincide and explain why this occurs.

An iron ore arrives at the steel mill with a purity of $80\ \%$ in iron(III) oxide.

1. a) How many kilograms of iron(III) oxide are in $50\ \text{kg}$ of ore?
2. b) What does the remaining $20\ \%$ refer to?

The marble used in countertops is calcium carbonate, $\text{CaCO}_3$. Calculate its percentage composition (atomic masses: $\text{Ca} = 40\ \text{u}$; $\text{C} = 12\ \text{u}$; $\text{O} = 16\ \text{u}$).

Table salt is sodium chloride, $\text{NaCl}$. Calculate the mass percentage of sodium and chlorine (atomic masses: $\text{Na} = 23\ \text{u}$; $\text{Cl} = 35.5\ \text{u}$). Nutritionists warn about the sodium in salt: explain, using your results, why $5\ \text{g}$ of salt is not $5\ \text{g}$ of sodium.

A tank contains $90\ \text{g}$ of water. Knowing that the percentage composition of water is $11.1\ \%$ of hydrogen and $88.9\ \%$ of oxygen, as summarized by the bar:

1. a) How many grams of hydrogen and how many of oxygen does the tank contain?
2. b) Check that the two masses you calculated add up to the total mass of the water.

Urea, $\text{CO}(\text{NH}_2)_2$, is the most widely used nitrogenous fertilizer in the world (atomic masses: $\text{C} = 12\ \text{u}$; $\text{O} = 16\ \text{u}$; $\text{N} = 14\ \text{u}$; $\text{H} = 1\ \text{u}$).

1. a) Calculate its molar mass and its mass percentage of nitrogen.
2. b) How many kilograms of nitrogen does a bag of $25\ \text{kg}$ of urea provide?

The analysis of a laboratory moisturizer shows a percentage composition of $40.0\ \%$ of carbon, $6.7\ \%$ of hydrogen, and $53.3\ \%$ of oxygen (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Calculate the moles of each element present in $100\ \text{g}$ of the compound.
2. b) Deduce its empirical formula.

A gaseous hydrocarbon has a percentage composition of $85.7\ \%$ carbon and $14.3\ \%$ hydrogen, and its molar mass is $42\ \text{g/mol}$ (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$).

1. a) Determine its empirical formula.
2. b) Determine its molecular formula. What additional piece of data was needed compared to the previous part?

A mine supplies an iron ore with a purity of $80\ \%$ iron(III) oxide, $\text{Fe}_2\text{O}_3$ (atomic masses: $\text{Fe} = 56\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Calculate the percentage by mass of iron in the pure oxide.
2. b) How many kilograms of iron can be obtained, at most, from one metric ton ($1\,000\ \text{kg}$) of this ore?

Copper(II) sulfate pentahydrate, $\text{CuSO}_4 \times 5\ \text{H}_2\text{O}$, is the blue salt used in crystallization labs. Its molar mass is $249.5\ \text{g/mol}$ and that of water is $18\ \text{g/mol}$. The figure shows what "pentahydrate" means:

1. a) Calculate the percentage by mass of water of crystallization in the hydrated salt.
2. b) If $50\ \text{g}$ of the blue salt is heated until completely dehydrated, what mass of water is given off?

A farmer buys NPK fertilizer $15$-$15$-$15$ in $40\ \text{kg}$ bags.

1. a) How many kilograms of nitrogen does each bag contain?
2. b) If their plot needs an input of $12\ \text{kg}$ of nitrogen, how many bags should they apply? How many kilograms of phosphorus(V) oxide and potassium oxide will they be adding in the process?

Sodium nitrate, $\text{NaNO}_3$, known as "Chilean nitrate", was one of the first commercial nitrogenous fertilizers, of mineral origin (atomic masses: $\text{Na} = 23\ \text{u}$; $\text{N} = 14\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Calculate its percentage by mass of nitrogen.
2. b) How many kilograms of nitrogen does a $40\ \text{kg}$ bag provide?

A liquid hydrocarbon found in gasoline has a percentage composition of $85.7\ \%$ carbon and $14.3\ \%$ hydrogen, and its molar mass is $70\ \text{g/mol}$ (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$).

1. a) Determine its empirical formula and the molar mass of that empirical formula.
2. b) Determine its molecular formula and check that it is consistent with the given molar mass.
3. c) Explain why percentage analysis alone can never distinguish between two compounds with the same empirical formula.

Vitamin C (ascorbic acid) in dietary supplements has a percentage composition of $40.9\ \%$ carbon, $4.5\ \%$ hydrogen and $54.5\ \%$ oxygen, and its molar mass is $176\ \text{g/mol}$ (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Determine its empirical formula. Hint: if the molar ratios are not integers, multiply them all by the same number until you get simple integers.
2. b) Determine its molecular formula.
3. c) A tablet contains $500\ \text{mg}$ of vitamin C: how many moles of vitamin C is this? Express the result to three significant figures.

A steel mill can buy two iron ores. Ore A is hematite with a purity of $80\ \%$ in $\text{Fe}_2\text{O}_3$; Ore B is magnetite with a purity of $75\ \%$ in $\text{Fe}_3\text{O}_4$ (atomic masses: $\text{Fe} = 56\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Calculate the percentage of iron in each of the two pure oxides.
2. b) Calculate how many kilograms of iron are obtained from one metric ton of each ore.
3. c) If both ores cost the same per metric ton, which one is worth buying? Justify the decision with the previous calculations.

In a combustion analysis, $4.4\ \text{g}$ of a hydrocarbon is completely burned and $13.2\ \text{g}$ of carbon dioxide and $7.2\ \text{g}$ of water are collected (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$; $\text{O} = 16\ \text{u}$; molar masses: $\text{CO}_2 = 44\ \text{g/mol}$; $\text{H}_2\text{O} = 18\ \text{g/mol}$). The diagram summarises the analysis:

1. a) Calculate the mass of carbon contained in the $\text{CO}_2$ collected and the mass of hydrogen contained in the water collected. Check that they add up to $4.4\ \text{g}$: which stoichiometric law justifies this check?
2. b) Determine the empirical formula of the hydrocarbon.
3. c) Knowing that the $4.4\ \text{g}$ burned was $0.1\ \text{mol}$ of substance, determine its molecular formula.

An agricultural technician needs to apply $46\ \text{kg}$ of nitrogen to a field and is deciding between two fertilizers: urea containing $46\ \%$ nitrogen, in $25\ \text{kg}$ bags at $\$25$ per bag, or ammonium nitrate containing $34\ \%$ nitrogen, in $25\ \text{kg}$ bags at $\$20$ per bag.

1. a) Calculate how many kilograms of each fertilizer are needed to provide the $46\ \text{kg}$ of nitrogen.
2. b) Taking into account that only whole bags can be bought, calculate how many bags of each type must be bought.
3. c) Which of the two options is cheaper? Justify this by calculating the total cost of each option.

The analysis of a reddish rock from a quarry shows that the iron oxide it contains has a percent composition of $70.0\ \%$ iron and $30.0\ \%$ oxygen (atomic masses: $\text{Fe} = 56\ \text{u}$; $\text{O} = 16\ \text{u}$).

1. a) Determine the empirical formula of the oxide from the percentages.
2. b) Calculate the percentage of iron in the three known oxides, $\text{FeO}$, $\text{Fe}_2\text{O}_3$ and $\text{Fe}_3\text{O}_4$, and check that only one matches the analysis.
3. c) Why can percent composition analysis not distinguish between pure $\text{Fe}_2\text{O}_3$ and a mixture of oxides that gives the same overall proportion? Comment on what other test you would perform.

A sample of $4.0\ \text{g}$ of a highly corrosive cleaning product contains $2.3\ \text{g}$ sodium, $1.6\ \text{g}$ oxygen and $0.1\ \text{g}$ hydrogen (atomic masses: $\text{Na} = 23\ \text{u}$; $\text{O} = 16\ \text{u}$; $\text{H} = 1\ \text{u}$).

1. a) Check that the three partial masses add up to the total mass of the sample and interpret this check using the law of conservation of mass.
2. b) Determine the empirical formula of the compound.
3. c) Its measured molar mass is $40\ \text{g/mol}$: do the empirical and molecular formulas coincide in this case? What substance is it?

A gas sold in cylinders for cooking has a density of $1.96\ \text{g/L}$ at standard temperature and pressure (molar volume under these conditions: $22.4\ \text{L/mol}$) and a percentage composition of $81.8\ \%$ carbon and $18.2\ \%$ hydrogen (atomic masses: $\text{C} = 12\ \text{u}$; $\text{H} = 1\ \text{u}$).

1. a) Calculate its molar mass from the density and the molar volume.
2. b) Determine its empirical formula.
3. c) Determine its molecular formula and identify which gas it is.

Unit challenge. Does your kitchen obey the laws of chemical combination? You will experimentally test the stoichiometry of the reaction between baking soda ($NaHCO_3$) and store-bought vinegar ($CH_3COOH$, $5\,\%$ m/v acidity).

1. Data collection: Using a precision kitchen scale ($\pm 0.1\text{ g}$), pour $50.0\text{ mL}$ of vinegar into three identical glasses. Add $2.0\text{ g}$, $4.0\text{ g}$, and $6.0\text{ g}$ of baking soda, respectively. Determine the total mass before and after fizzing stops to find the mass of $CO_2$ released ($\pm 0.2\text{ g}$).

2. Processing and plotting: Prepare a data table with your measurements and uncertainties. Plot the mass of $CO_2$ versus the mass of $NaHCO_3$ and check whether Proust's law is supported by the appearance of a limiting reactant.

3. Everyday calculations: Check the price of a pack of baking soda on a supermarket website and calculate the actual cost in dollars per mole of $CO_2$ obtained. Also determine the percent yield relative to the theoretical stoichiometric value.

4. Final report: Write a brief report containing the data table, the graph, the quantitative calculations, and a critical discussion of the mass losses.

Evaluation criteria:

• Rigor in data collection, uncertainty propagation, and graph quality.

• Correctness in stoichiometric relationships and use of the laws of chemical combination.

• Consistency in yield analysis and calculation of the actual economic cost.