Collectively, these conversions are called mole-mass calculations.

As an example, consider the balanced chemical equation

Fe₂O₃ + 3SO₃ → Fe₂(SO₄)₃

If we have 3.59 mol of Fe₂O₃, how many grams of SO₃ can react with it? Using the mole-mass calculation sequence, we can determine the required mass of SO₃ in two steps. First, we construct the appropriate molar ratio, determined from the balanced chemical equation, to calculate the number of moles of SO₃ needed. Then using the molar mass of SO₃ as a conversion factor, we determine the mass that this number of moles of SO₃ has.

The first step resembles the exercises we did in Section 6.4 “Mole-Mole Relationships in Chemical Reactions”. As usual, we start with the quantity we were given:

[latex]3.59\cancel{\text{ mol Fe}_2\text{O}_3}\times\frac{3\text{ mol SO}_3}{1\cancel{\text{ mol Fe}_2\text{O}_3}}[/latex] = 10.8 mol SO₃

The mol Fe₂O₃ units cancel, leaving mol SO₃ unit, and based on 3 sig figs in the 3.59 mol SO₃, the answer was rounded to 3 sig figs.  As you will see soon, such intermediate rounding introduces rounding error in the final answer.

Next convert this answer to grams of SO₃, using the molar mass of SO₃ as the conversion factor.

[latex]10.8\cancel{\text{ mol SO}_3}\times\frac{80.06\text{ g SO}_3}{1\cancel{\text{ mol SO}_3}}[/latex] = 865 g SO₃

The answer to this step is expressed to three significant figures. Thus, in a two-step process, we find that 865 g of SO₃ will react with 3.59 mol of Fe₂O₃. Many problems of this type can be answered in this manner.

To avoid rounding error, it is better to do multi-step problems all at once, keeping intermediate answers in the calculator and rounding only once at the end.:

Notice that, when rounding after each step, the answer was 865 g SO₃.  But when the calculator was used to hold onto all of the decimal places of the intermediate number, the answer came out to 862 g SO₃.  This should call to mind the fact that the last digit of a measured number or of a calculated number based on measurements is an uncertain estimate.  However, the most accurate answer is the one obtained by doing all of the steps at once and rounding for the correct number of sig figs at the very end.  In this case the most accurate answer is 862 g SO₃.

Example 8

How many grams of CO₂ are produced if 2.09 mol of HCl are reacted according to this balanced chemical equation?

CaCO₃ + 2HCl → CaCl₂ + CO₂ + H₂O

Solution

Our strategy will be to convert from moles of HCl to moles of CO₂ and then from moles of CO₂ to grams of CO₂. We will need the molar mass of CO₂, which is 44.01 g/mol. Performing these two conversions in a single-line gives 46.0 g of CO₂:

The molar ratio between CO₂ and HCl comes from the balanced chemical equation.

It is a small step from mole-mass calculations to mass-mass calculations. If we start with a known mass of one substance in a chemical reaction (instead of a known number of moles), we can calculate the corresponding masses of other substances in the reaction. The first step in this case is to convert the known mass into moles, using the substance’s molar mass as the conversion factor. Then—and only then—we use the balanced chemical equation to construct a conversion factor to convert that quantity to moles of another substance, which in turn can be converted to a corresponding mass. Sequentially, the process is as follows:

This three-part process should be combined into a single calculation that contains three conversion factors as shown in the following example.

To Your Health: The Synthesis of Taxol

Taxol is a powerful anticancer drug that was originally extracted from the Pacific yew tree (Taxus brevifolia). As you can see from the accompanying figure, taxol is a very complicated molecule, with a molecular formula of C₄₇H₅₁NO₁₄. Isolating taxol from its natural source presents certain challenges, mainly that the Pacific yew is a slow-growing tree, and the equivalent of six trees must be harvested to provide enough taxol to treat a single patient. Although related species of yew trees also produce taxol in small amounts, there is significant interest in synthesizing this complex molecule in the laboratory.

After a 20-year effort, two research groups announced the complete laboratory synthesis of taxol in 1994. However, each synthesis required over 30 separate chemical reactions, with an overall efficiency of less than 0.05%. To put this in perspective, to obtain a single 300 mg dose of taxol, you would have to begin with 600 g of starting material. To treat the 26,000 women who are diagnosed with ovarian cancer each year with one dose, almost 16,000 kg (over 17 tons) of starting material must be converted to taxol. Taxol is also used to treat breast cancer, with which 200,000 women in the United States are diagnosed every year. This only increases the amount of starting material needed.

Clearly, there is intense interest in increasing the overall efficiency of the taxol synthesis. An improved synthesis not only will be easier but also will produce less waste materials, which will allow more people to take advantage of this potentially life-saving drug.

Figure 6.4 The Structure of the Cancer Drug Taxol. Because of the complexity of the molecule, hydrogen atoms are not shown, but they are present on every atom to give the atom the correct number of covalent bonds (four bonds for each carbon atom).

Key Takeaway

• A balanced chemical equation can be used to relate masses or moles of different substances in a reaction.