Percentage Composition
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You should be comfortable calculating molar masses from the periodic table and converting between mass and moles using the relationship \( n = \frac{m}{M} \).
Why this matters
Percentage composition allows chemists to determine the elemental makeup of any substance from its chemical formula.
In laboratory practice, we rarely weigh individual atoms. Instead, we measure macroscopic masses and use the mole concept to bridge the gap between the visible world and the atomic scale. Percentage composition provides a standardized way to express the relative amounts of elements in a compound. This information is crucial for identifying unknown substances, verifying purity, and understanding the stoichiometric relationships that govern chemical reactions. By knowing the percentage of each element, we can predict how much of a reactant is available for a reaction or how much product will be formed, making it a fundamental tool in both analytical and synthetic chemistry.
The Concept of Mass Percent
Imagine you have a bag of mixed nuts containing almonds, cashews, and peanuts. If you were to weigh the entire bag and then weigh just the almonds, you could calculate what fraction of the total mass is due to almonds. Percentage composition works on exactly the same principle, but at the atomic level. For any given compound, the total mass is the sum of the masses of all the atoms present. By comparing the mass contributed by a specific element to the total molar mass of the compound, we determine the percentage by mass of that element. This is a purely mathematical relationship derived from the atomic masses listed in the periodic table. It does not depend on the size of the sample; whether you have one mole of water or one kilogram of water, the percentage of hydrogen by mass remains constant. This property makes percentage composition a powerful identifier for pure substances, as each unique chemical formula yields a unique set of mass percentages. It is the first step in determining the empirical formula of an unknown compound, linking the qualitative identity of a substance to its quantitative composition.
Calculating Mass Percent from a Formula
To calculate the mass percent of an element in a compound, we first determine the molar mass of the entire compound. This is done by summing the atomic masses of all atoms in the chemical formula. Next, we calculate the total mass of the specific element in one mole of the compound by multiplying the atomic mass of the element by the number of atoms of that element present in the formula. The mass percent is then found by dividing the mass of the element by the total molar mass of the compound and multiplying by 100. The general formula is expressed as: \[ \text{Mass \% of Element} = \frac{\text{Mass of Element in 1 mol Compound}}{\text{Molar Mass of Compound}} \times 100\% \] For example, consider water, \(\mathrm{H_2O}\). The molar mass is approximately \( 2(1.008) + 16.00 = 18.016\ \mathrm{g\,mol^{-1}} \). The mass of hydrogen is \( 2 \times 1.008 = 2.016\ \mathrm{g\,mol^{-1}} \). The mass percent of hydrogen is \( \frac{2.016}{18.016} \times 100\% \approx 11.19\% \). Similarly, for oxygen, it is \( \frac{16.00}{18.016} \times 100\% \approx 88.81\% \). Note that the sum of the percentages of all elements in a compound must equal 100%. This serves as a useful check for calculation errors. If the sum is not 100%, it indicates a mistake in the molar mass calculation or the arithmetic of the division.
Handling Compounds with Multiple Atoms
When a compound contains multiple atoms of the same element, it is critical to account for all of them in the numerator. A common error is to use the atomic mass of a single atom instead of the total mass of all atoms of that element in the formula unit. For instance, in calcium phosphate, \(\mathrm{Ca_3(PO_4)_2}\), there are three calcium atoms, two phosphorus atoms, and eight oxygen atoms. The molar mass is calculated as \( 3(40.08) + 2(30.97) + 8(16.00) = 120.24 + 61.94 + 128.00 = 310.18\ \mathrm{g\,mol^{-1}} \). The mass percent of calcium is \( \frac{120.24}{310.18} \times 100\% \approx 38.76\% \). The mass percent of phosphorus is \( \frac{61.94}{310.18} \times 100\% \approx 19.97\% \). The mass percent of oxygen is \( \frac{128.00}{310.18} \times 100\% \approx 41.27\% \). Summing these gives \( 38.76 + 19.97 + 41.27 = 100.00\% \), confirming the calculation. This method is essential for determining the composition of complex minerals and biological molecules where the number of atoms of each element can be large and varied.
Applying Percentage Composition to Samples
Once the mass percent is known, it can be used to determine the mass of a specific element in a given sample of the compound. This is particularly useful in analytical chemistry when determining the amount of a particular element in a mixture or a natural sample. The calculation is straightforward: multiply the total mass of the sample by the mass fraction (percentage divided by 100) of the element. For example, if you have a 50.0 g sample of calcium carbonate, \(\mathrm{CaCO_3}\), and you want to know how much calcium is present, you first calculate the mass percent of calcium. The molar mass of \(\mathrm{CaCO_3}\) is \( 40.08 + 12.01 + 3(16.00) = 100.09\ \mathrm{g\,mol^{-1}} \). The mass percent of calcium is \( \frac{40.08}{100.09} \times 100\% \approx 40.04\% \). The mass of calcium in the 50.0 g sample is \( 50.0\ \mathrm{g} \times 0.4004 = 20.02\ \mathrm{g} \). This application bridges the gap between the theoretical composition of a pure substance and the practical measurement of elements in real-world samples, such as limestone or dietary supplements.
A common misconception is that percentage composition is the same as mole percent. Mole percent refers to the ratio of the number of moles of one component to the total moles of all components in a mixture, whereas mass percent refers to the ratio of mass. In a pure compound, the mole ratio is fixed by the subscripts in the chemical formula, but the mass percent varies depending on the atomic masses of the elements. For example, in water, the mole ratio of hydrogen to oxygen is 2:1, but the mass ratio is approximately 1:8. Confusing these two concepts can lead to significant errors in stoichiometric calculations, particularly when dealing with mixtures or solutions where the composition is not fixed by a chemical formula. C
Worked example
Problem. Determine the mass percent of each element in sucrose, \(\mathrm{C_{12}H_{22}O_{11}}\), and calculate the mass of carbon in a 35.0 g sample of sucrose. Use the following atomic masses: \(\mathrm{C} = 12.01\ \mathrm{g\,mol^{-1}}\), \(\mathrm{H} = 1.008\ \mathrm{g\,mol^{-1}}\), \(\mathrm{O} = 16.00\ \mathrm{g\,mol^{-1}}\).
- Calculate the molar mass of sucrose: \( M = 12(12.01) + 22(1.008) + 11(16.00) = 144.12 + 22.176 + 176.00 = 342.296\ \mathrm{g\,mol^{-1}} \).
- Calculate the mass of carbon in one mole of sucrose: \( m_{\mathrm{C}} = 12 \times 12.01 = 144.12\ \mathrm{g\,mol^{-1}} \).
- Calculate the mass percent of carbon: \( \text{Mass \% C} = \frac{144.12}{342.296} \times 100\% \approx 42.10\% \).
- Calculate the mass of carbon in the 35.0 g sample: \( m_{\mathrm{C, sample}} = 35.0\ \mathrm{g} \times 0.4210 = 14.735\ \mathrm{g} \).
- Round the final answer to three significant figures, consistent with the given sample mass: \( 14.7\ \mathrm{g} \).
Units check. The calculation involves multiplying a mass in grams by a dimensionless fraction, resulting in a mass in grams, which is consistent with the question asking for the mass of carbon.
Three quick tests
- What is the mass percent of oxygen in \(\mathrm{CO_2}\)?
Answer
Approximately 72.7%. Molar mass of \(\mathrm{CO_2}\) is \( 12.01 + 2(16.00) = 44.01\ \mathrm{g\,mol^{-1}} \). Mass of oxygen is \( 32.00\ \mathrm{g\,mol^{-1}} \). Percent is \( \frac{32.00}{44.01} \times 100\% \approx 72.71\% \). - If a compound has a mass percent of 40.0% nitrogen, what is the mass of nitrogen in a 25.0 g sample?
Answer
10.0 g. Calculation: \( 25.0\ \mathrm{g} \times 0.400 = 10.0\ \mathrm{g} \). - Why must the sum of the mass percentages of all elements in a compound equal 100%?
Answer
Because the total mass of the compound is the sum of the masses of all its constituent elements. By definition, the percentage is a part-to-whole ratio, so the sum of all parts must equal the whole.
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