Question

Estrone, which contains only carbon, hydrogen, and oxygen, is a female sexual hormone that occurs in the urine of pregnant women. Combustion analysis of a 1.893 -g sample of estrone produces 5.545 \(\mathrm{g}\) of \(\mathrm{CO}_{2}\) and 1.388 \(\mathrm{g} \mathrm{H}_{2} \mathrm{O}\) . The molar mass of estrone is 270.36 \(\mathrm{g} / \mathrm{mol}\) . Find its molecular formula.

Short answer

The molecular formula of estrone is determined using combustion analysis and its molar mass; estrone has a molecular formula of C18H22O2.

Step-by-step solution

Calculate the moles of carbon from CO2

The mass of CO2 produced in the combustion analysis can be used to calculate the moles of carbon. Each mole of CO2 contains one mole of carbon atoms. Use the molar mass of CO2 to convert the mass of CO2 to moles of CO2, and hence to moles of carbon.

Calculate the moles of hydrogen from H2O

Similarly, the mass of H2O produced can be converted to moles of H2O. Since each mole of H2O contains two moles of hydrogen atoms, multiplying the moles of H2O by 2 gives the moles of hydrogen.

Calculate the mass of oxygen in the sample

By subtracting the mass of carbon and hydrogen (calculated from their respective moles and molar masses) from the total mass of the estrone sample, the mass of oxygen can be determined.

Calculate the moles of oxygen

Using the mass of oxygen found in Step 3, convert it to moles of oxygen using the molar mass of oxygen.

Determine the empirical formula

Divide the moles of carbon, hydrogen, and oxygen by the smallest number of moles among them to find the simplest whole number ratio of the elements in the compound. This gives the empirical formula.

Determine the molecular formula

Divide the molar mass of estrone by the molar mass of the empirical formula to find the multiple that relates the empirical formula to the molecular formula. Multiply the subscripts in the empirical formula by this multiple to obtain the molecular formula.

Key concepts

Combustion Analysis

Combustion analysis is a type of chemical analysis used to determine the elemental composition of a substance, particularly for substances containing carbon and hydrogen. It involves burning a known mass of a compound to produce carbon dioxide ((CO_2)) and water ((H_2O)), which are then measured. From the masses of (CO_2) and (H_2O) produced, the amounts of carbon and hydrogen in the original compound can be calculated.

This analytical technique is particularly useful for organic compounds, and in the context of our exercise, it was used to analyze estrone, a female sexual hormone. By knowing the mass of (CO_2) and (H_2O) yielded from combusting the estrone sample, we can backtrack the quantities of carbon and hydrogen in the original sample. The following steps usually encompass a complete combustion analysis:

• Burn the sample and measure the masses of (CO_2) and (H_2O) produced.
• Calculate moles of carbon from the mass of (CO_2).
• Compute moles of hydrogen from the mass of (H_2O).
• Infer the mass of other elements (like oxygen) by difference if needed.

Empirical Formula

The empirical formula of a compound provides the simplest whole number ratio of its constituent elements. It doesn't necessarily indicate the actual number of atoms in a molecule but rather the ratio between them. For instance, the empirical formula for glucose, (C_6 H_12 O_6), is (CH_2 O), as the atoms can be simplified to a 1:2:1 ratio.

After performing a combustion analysis, the empirical formula is found by the following steps:

• Determining the moles of each element present.
• Dividing each by the smallest number of moles to get the simplest ratio.

If the ratios aren't whole numbers, they can be multiplied by the smallest common factor to achieve whole numbers. This is because empirical formulas are represented by whole numbers. In our example, the empirical formula is a key step in establishing the actual molecular formula of estrone.

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole ((g/mol)). It can be found by summing the atomic masses of all the atoms in a formula, which are given on the periodic table. Molar mass is crucial both for converting between grams and moles and for relating the empirical formula to the molecular formula.

When the molar mass is given, as in the case of estrone, it can be used in tandem with the empirical formula to determine the molecular formula. To do this:

• Calculate the molar mass of the empirical formula.
• Divide the compound's given molar mass by the empirical formula's molar mass.
• Use the factor from this division to multiply the subscripts in the empirical formula, deriving the molecular formula.

The exercise provides the molar mass of estrone, which is a vital piece of data for finding the molecular formula.

Moles Calculation

Moles calculation refers to determining the number of moles in a given quantity of a substance. The mole is a unit of measurement for amount of substance in the International System of Units (SI). One mole contains exactly 6.02214076 x 10^23 (Avogadro's number) of particles, atoms, ions, or molecules, whichever is appropriate for the substance in question.

To calculate moles from a known mass, use the formula:\[ \text{moles} = \frac{{\text{mass}}}{{\text{molar mass}}} \]
Specific steps can be outlined as follows:

• Measure the mass of the substance.
• Use the molar mass to convert this mass to moles.

In the context of our estrone analysis:

• The mass of (CO_2) and (H_2O) indicates the moles of carbon and hydrogen respectively.
• The mass of estrone minus the masses of carbon and hydrogen gives the mass of oxygen, which can then be converted to moles of oxygen.

Calculating moles forms the foundation for deriving both empirical and molecular formulas.