Empirical formulas

Empirical Formulas

Introduction

Every chemical compound can be described by more than one kind of formula. The molecular formula tells you the actual number and type of every atom in one molecule. The empirical formula is something different and, in many ways, more fundamental: it tells you the simplest whole-number ratio of the elements in the compound, stripped of any redundancy.

Understanding how to determine and use empirical formulas is an essential skill in chemical analysis because experimental techniques frequently provide mass-percent data rather than atom counts.

In this lecture, we will define the empirical formula, contrast it with the molecular formula, and then walk step-by-step through the method for calculating an empirical formula from percent composition data.

Figure 1: Molecular Formula vs. Empirical Formula

Explore the differences between molecular and empirical formulas using glucose as an example, highlighting how the actual atom count compares to the simplest ratio.

What Is an Empirical Formula?

An empirical formula gives the lowest whole-number ratio of atoms of each element in a compound.

The word empirical comes from the Greek word for "experience" or "experiment," reflecting the fact that empirical formulas are determined directly from experimental measurements rather than from a theoretical model of the molecule.

Consider hydrogen peroxide:

H2O2\mathrm{H_2O_2}

The molecular formula tells us that each molecule contains:

• 2 hydrogen atoms

• 2 oxygen atoms

The ratio of hydrogen to oxygen is:

2:22:2

which simplifies to:

1:11:1

Therefore, the empirical formula is:

HO\mathrm{HO}

Notice that HO does not tell us how many atoms are actually present in a molecule. It only tells us the simplest ratio.

Many molecular formulas can share the same empirical formula:

• HO

• H₂O₂

• H₃O₃

all reduce to the same ratio:

1:11:1

For some compounds, the empirical formula and molecular formula are identical.

Water:

H2O\mathrm{H_2O}

already has the simplest possible ratio and therefore cannot be reduced further.

Key Point

The empirical formula is the simplest whole-number ratio of elements in a compound. It may or may not be identical to the molecular formula.

Figure 2: Simplifying Chemical Ratios

This infographic explains how to simplify chemical formulas to their empirical forms by dividing element subscripts by their greatest common factor, using hydrogen peroxide and glucose as examples.

Examples: Molecular vs. Empirical Formulas

The relationship between molecular and empirical formulas is illustrated below.

Table 1. Molecular and Empirical Formula Comparisons

Table showing the molecular formulas, their ratios, and corresponding empirical formulas for various chemical compounds.

To convert a molecular formula into an empirical formula:

1. Identify the subscripts.

2. Find their greatest common factor.

3. Divide all subscripts by that value.

Key Point

To convert a molecular formula into an empirical formula, divide all subscripts by their greatest common factor.

Figure 3: Molecular-to-Empirical Conversion

Explore the process of converting a molecular formula to its empirical form, using the example of C₆H₁₂O₆ to CH₂O, by simplifying subscripts to their smallest whole-number ratio.

From Percent Composition to Empirical Formula

A useful rhyme summarizes the entire empirical formula process:

Percent to Mass

Mass to Mole

Divide by Small

Multiply 'til Whole

Each line corresponds to one step of the calculation.

Step 1: Percent to Mass

Assume a 100 g sample.

This allows each percentage to become a mass directly.

Example:

If carbon is:

40.0%40.0\%

then the sample contains:

40.0 g40.0\ \mathrm{g}

of carbon.

Step 2: Mass to Mole

Convert each mass into moles using molar mass.

The conversion is:

moles=massmolar mass\text{moles}=\frac{\text{mass}}{\text{molar mass}}

Each element is converted separately.

Step 3: Divide by Small

After all mole values have been calculated, divide every value by the smallest mole value obtained.

This converts the mole amounts into a mole ratio.

The resulting values should be close to whole numbers.

Step 4: Multiply 'til Whole

Sometimes the ratios are not whole numbers.

Common fractional values must be removed by multiplying all ratios by the same integer.

Table displaying fractions and their corresponding multiplication factors, illustrating how each fraction can be multiplied by values of 0.5, 2, 0.33, 0.67, 3, 0.25, 0.75, and 4.

The goal is always a whole-number ratio.

Key Point

Percent → Mass → Mole → Divide by Smallest → Whole-Number Ratio

This sequence always leads to the empirical formula.

Figure 4: Empirical Formula Roadmap

The "Empirical Formula Roadmap" guides users through converting a molecular formula to its simplest whole-number ratio, using glucose as an example to illustrate the process.

Worked Example: NutraSweet (Aspartame)

NutraSweet (aspartame) has the following percent composition:

Table showing the percent composition of elements in a chemical compound, indicating the proportions of Carbon (C), Hydrogen (H), Nitrogen (N), and Oxygen (O).

The molar mass is:

294.30 g/mol294.30\ \mathrm{g/mol}

We will first determine the empirical formula.

Step 1: Percent to Mass

Assume a 100 g sample.

Table displaying the mass composition of a chemical element, detailing the percentages of Carbon (C), Hydrogen (H), Nitrogen (N), and Oxygen (O) in grams.

Step 2: Mass to Mole

Convert each mass into moles.

For carbon:

57.1412.011=4.757 mol\frac{57.14}{12.011}=4.757\ \mathrm{mol}

For hydrogen:

6.161.008=6.111 mol\frac{6.16}{1.008}=6.111\ \mathrm{mol}

For nitrogen:

9.5214.007=0.6797 mol\frac{9.52}{14.007}=0.6797\ \mathrm{mol}

For oxygen:

27.1815.999=1.699 mol\frac{27.18}{15.999}=1.699\ \mathrm{mol}

Step 3: Divide by Small

The smallest mole value is:

0.67970.6797

Divide all values by this number.

For carbon:

4.7570.6797=6.998≈7\frac{4.757}{0.6797}=6.998\approx7

For hydrogen:

6.1110.6797=8.99≈9\frac{6.111}{0.6797}=8.99\approx9

For nitrogen:

0.67970.6797=1\frac{0.6797}{0.6797}=1

For oxygen:

1.6990.6797=2.499≈2.5\frac{1.699}{0.6797}=2.499\approx2.5

Step 4: Multiply 'til Whole

The oxygen ratio contains:

2.52.5

Because the ratio ends in 0.5, multiply every value by 2.

Carbon:

7×2=147 \times 2=14

Hydrogen:

9×2=189 \times 2=18

Nitrogen:

1×2=21 \times 2=2

Oxygen:

2.5×2=52.5 \times 2=5

The empirical formula is therefore:

C14H18N2O5\mathrm{C_{14}H_{18}N_2O_5}

Key Point

When a ratio ends in 0.5, multiply every ratio by 2. For 0.33 or 0.67 multiply by 3. For 0.25 or 0.75 multiply by 4.

Figure 5: Eliminating Fractional Ratios

Learn how to convert fractional ratios into whole-number ratios using a step-by-step process involving the least common denominator, enabling easy application in empirical formulas.

Common Rounding Guidelines

After dividing by the smallest mole value, the resulting values may not be exact whole numbers due to rounding in the original experimental data.

Use the following guidelines.

Table 2. Common Fractional Ratios

This table outlines the actions to be taken based on specific value ranges, including rounding, multiplication, and rechecking calculations.

Key Point

Values within approximately 0.1 of a whole number should be rounded. Larger deviations usually require multiplying by an integer.

Figure 6: Fractional Ratio Decision Tree

A guide to simplifying fractional ratios into whole-number ratios, this decision tree helps in determining the correct method—rounding or multiplying by the least common denominator—based on calculated ratios.

Summary

An empirical formula expresses the simplest whole-number ratio of elements in a compound. It may be identical to or different from the molecular formula. Empirical formulas are commonly determined from percent composition data using a four-step process: convert percentages to masses, convert masses to moles, divide by the smallest mole value, and multiply to eliminate fractional ratios. Mastering this process provides a foundation for chemical analysis and molecular formula determination.

Key Points

• The empirical formula gives the simplest whole-number ratio of elements.

• Molecular formulas and empirical formulas are not always identical.

• Convert molecular formulas to empirical formulas by dividing subscripts by their greatest common factor.

• Assume a 100 g sample when converting percent composition data.

• Convert masses to moles using molar masses.

• Divide all mole values by the smallest mole quantity.

• Multiply ratios when fractional values remain.

• 0.5 requires multiplication by 2.

• 0.33 and 0.67 require multiplication by 3.

• 0.25 and 0.75 require multiplication by 4.