Question

Mandelic acid is an organic acid composed of carbon \((63.15 \%),\) hydrogen \((5.30 \%),\) and oxygen \((31.55 \%) .\) Its molar mass is \(152.14 \mathrm{g} / \mathrm{mol} .\) Determine the empirical and molecular formulas of the acid.

Short answer

Empirical and molecular formulas: \(C_8H_8O_3\).

Step-by-step solution

Convert Percentages to Grams

Assume a 100-g sample of mandelic acid. Therefore, it contains 63.15 grams of carbon, 5.30 grams of hydrogen, and 31.55 grams of oxygen.

Convert Grams to Moles

Using the molar masses (C = 12.01 g/mol, H = 1.01 g/mol, O = 16.00 g/mol), convert to moles:- Carbon: \( \frac{63.15 \text{ g}}{12.01 \text{ g/mol}} \approx 5.26 \text{ mol} \)- Hydrogen: \( \frac{5.30 \text{ g}}{1.01 \text{ g/mol}} \approx 5.25 \text{ mol} \)- Oxygen: \( \frac{31.55 \text{ g}}{16.00 \text{ g/mol}} \approx 1.97 \text{ mol} \)

Determine the Mole Ratio

Divide each mole value by the smallest mole quantity (1.97 mol for oxygen):- Carbon: \( \frac{5.26}{1.97} \approx 2.67 \)- Hydrogen: \( \frac{5.25}{1.97} \approx 2.67 \)- Oxygen: \( \frac{1.97}{1.97} = 1 \)

Convert to Whole Numbers

Since the ratios are approximately 2.67 for C and H, multiply all the ratios by 3 to get approximate whole numbers:- Carbon: \(2.67 \times 3 = 8 \)- Hydrogen: \(2.67 \times 3 = 8 \)- Oxygen: \(1 \times 3 = 3 \)Thus, the empirical formula is \(C_8H_8O_3\).

Determine the Molecular Formula

Compute the empirical formula mass of \(C_8H_8O_3\): - \(C: 8 \times 12.01 = 96.08 \)- \(H: 8 \times 1.01 = 8.08 \)- \(O: 3 \times 16.00 = 48.00 \)Total: \(96.08 + 8.08 + 48.00 = 152.16 \text{ g/mol} \)Since the empirical formula mass is approximately equal to the given molar mass (152.14 g/mol), the molecular formula is also \(C_8H_8O_3\).

Key concepts

Empirical Formula

An empirical formula represents the simplest whole number ratio of atoms in a compound. It does not provide information about the actual number of atoms, just the ratio. For instance, in mandelic acid, we start from percentages of elements like carbon, hydrogen, and oxygen to find their ratios.

To find the empirical formula, begin by assuming you have a 100 g sample of the compound. This makes converting percentages directly into grams very convenient, as they remain numerically the same. Next, you convert these masses into moles by using the molar masses of the respective elements. Here is how it unfolds for each element:

• Carbon: 63.15 g (from 63.15% of the sample)
• Hydrogen: 5.30 g
• Oxygen: 31.55 g

After this conversion into moles, you'll divide by the smallest number of moles calculated to get a mole ratio that can be simplified to whole numbers. In this example, we achieve the empirical formula: \(C_8H_8O_3\). This ratio tells us there are 8 carbon atoms, 8 hydrogen atoms for every 3 oxygen atoms in the simplistic structure.

Molecular Formula

The molecular formula indicates the actual number of atoms of each element in a single molecule of a compound. This formula can be the same as the empirical formula or a simple integer multiple of it, depending on the compound.

For mandelic acid, once the empirical formula \(C_8H_8O_3\) was determined, the next step was to verify whether this empirical formula corresponds to the molecular formula by comparing empirical formula mass with the given molar mass of the compound. Here’s how you do it:

• Calculate the empirical formula mass: Add the product of the atomic masses and their respective atoms in the empirical formula.
• Empirical mass of \(C_8H_8O_3\) is 152.16 g/mol.
• Compare this with the compound’s given molar mass, which is 152.14 g/mol.

This close match indicates that the empirical formula is indeed the molecular formula for mandelic acid. In this case, both formulas are identical.

Molar Mass Calculation

Molar mass is the mass of one mole of a given substance. It is expressed in grams per mole (g/mol) and is obtained by summing up the atomic masses of the elements in a compound, considering the number of atoms of each.

To calculate the molar mass of any compound, follow these steps below:

• Identify the number of atoms of each element in the given molecular formula.
• Utilize the periodic table to find the atomic mass of each element (in atomic mass units).
• Multiply the atomic mass of each element by the number of its atoms in the molecular formula.
• Add all these values together to get the molar mass.

For example, for \(C_8H_8O_3\):

• Carbon: 8 atoms \(\times 12.01 \text{ g/mol} = 96.08 \text{ g/mol}\)
• Hydrogen: 8 atoms \(\times 1.01 \text{ g/mol} = 8.08 \text{ g/mol}\)
• Oxygen: 3 atoms \(\times 16.00 \text{ g/mol} = 48.00 \text{ g/mol}\)

The total molar mass equals 152.16 g/mol, which matches or nearly matches the provided molar mass, further confirming the molecular formula.

Mole Ratio

A mole ratio is a proportion that helps chemists understand the quantities of substances involved or produced in a chemical reaction. It is determined by dividing the number of moles of each substance by the smallest number of moles calculated in the solution.

In the context of determining the empirical formula, mole ratios are derived from converting the grams of each element to moles, then simplifying those ratios to the smallest whole numbers. Here’s how:

• Convert grams to moles using molar masses.
• Identify the smallest number of moles in the calculations.
• Divide each mole value by the smallest mole number to find a comparative ratio.
• Adjust these ratios to whole numbers by multiplying all ratios by a common factor, if necessary.

In our example of mandelic acid, the initial mole numbers were approximately 5.26 for carbon, 5.25 for hydrogen, and 1.97 for oxygen. By dividing by the smallest (oxygen), we get ratios of roughly 2.67, which can be rounded to 8:8:3 after multiplying by 3, forming the empirical formula \(C_8H_8O_3\).