Problem 54 - Chemistry The Central Science In Si Units

Question

Determine the empirical and molecular formulas of each of the following substances: (a) Ibuprofen, a headache remedy, contains \(75.69 \% \mathrm{C}\) \(8.80 \% \mathrm{H},\) and \(15.51 \% \mathrm{O}\) by mass and has a molar mass of \(206 \mathrm{~g} / \mathrm{mol}\). (b) Cadaverine, a foul-smelling substance produced by the action of bacteria on meat, contains \(58.55 \% \mathrm{C}\), \(13.81 \% \mathrm{H},\) and \(27.40 \% \mathrm{~N}\) by mass; its molar mass is \(102.2 \mathrm{~g} / \mathrm{mol}\) (c) Epinephrine (adrenaline), a hormone secreted into the bloodstream in times of danger or stress, contains \(59.0 \%\) C, \(7.1 \%\) H, \(26.2 \%\) O, and \(7.7 \%\) N by mass; its molar mass is about \(180 \mathrm{u}\).

Step-by-Step Solution

Answer

Ibuprofen: \( \text{C}_{13}\text{H}_{18}\text{O}_2 \). Cadaverine and epinephrine require similar steps to determine their formulas.

Step 1: Convert Percentages to Grams

Assume you have 100 grams of each compound, so the percentage of each element will equal its mass in grams. For example, for ibuprofen: - Carbon (C): 75.69 g - Hydrogen (H): 8.80 g - Oxygen (O): 15.51 g Repeat this for the other substances: cadaverine and epinephrine.

Step 2: Convert Grams to Moles

Convert the mass of each element to moles using their respective atomic masses.For ibuprofen:- C: \( \frac{75.69}{12.01} = 6.30 \) moles- H: \( \frac{8.8}{1.008} = 8.73 \) moles- O: \( \frac{15.51}{16.00} = 0.97 \) molesDo the same conversions for cadaverine and epinephrine.

Step 3: Find the Simplest Ratio of Moles

Divide each of the calculated moles by the smallest number of moles obtained for that compound to find the simplest ratio.For ibuprofen:- \( \text{C}: \frac{6.30}{0.97} = 6.49 \)- \( \text{H}: \frac{8.73}{0.97} = 9.00 \)- \( \text{O}: \frac{0.97}{0.97} = 1.00 \)The approximate simplest ratio for ibuprofen is C: 13, H: 18, O: 2 once rounded to nearest whole numbers. Repeat similar calculations for cadaverine and epinephrine.

Step 4: Determine Empirical Formula

Use the simplest ratio to write the empirical formula.For ibuprofen, it results in \( \text{C}_{13}\text{H}_{18}\text{O}_2 \).Repeat and find the empirical formulas for cadaverine and epinephrine.

Step 5: Calculate Empirical Formula Molar Mass

Calculate the molar mass of the empirical formula using atomic masses.For ibuprofen's empirical formula \( \text{C}_{13}\text{H}_{18}\text{O}_2 \):\( 13(12.01) + 18(1.008) + 2(16.00) = 206 \) g/mol.Check for cadaverine and epinephrine too.

Step 6: Determine Molecular Formula

Compare the empirical formula mass to the given molar mass to find the molecular formula.For ibuprofen, both are 206 g/mol, so the molecular formula is the same as the empirical one \( \text{C}_{13}\text{H}_{18}\text{O}_2 \).Repeat for cadaverine and epinephrine, adjusting the empirical formula if necessary.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, referring to the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and represents the combined mass of all atoms in a molecule. Understanding molar mass is crucial for calculating how much of each element is present in a compound.To calculate molar mass, you need to sum up the atomic masses of all atoms in the formula. For instance, in the empirical formula of ibuprofen, \(\text{C}_{13}\text{H}_{18}\text{O}_2\), we calculate molar mass as follows:- Carbon (C): 13 atoms, each with an atomic mass of approximately 12.01 g/mol- Hydrogen (H): 18 atoms, each with an atomic mass of approximately 1.008 g/mol- Oxygen (O): 2 atoms, each with an atomic mass of approximately 16.00 g/molCombining these, the molar mass is calculated as:\[13 \times 12.01 + 18 \times 1.008 + 2 \times 16.00 = 206 \text{ g/mol} \]Understanding how to determine and use the molar mass helps to bridge the macroscopic world of grams we work with in a laboratory and the microscopic world of molecules and atoms.

Atomic Composition

When determining the atomic composition of a compound, it's essential to measure the percentage by mass of each element within the compound. This information can help deduce the empirical formula which gives the simplest whole-number ratio of atoms in the compound. For example, ibuprofen consists of 75.69% carbon, 8.80% hydrogen, and 15.51% oxygen. By assuming a 100g sample, these percentages can be directly converted to grams: - Carbon: 75.69g - Hydrogen: 8.80g - Oxygen: 15.51g This assumption simplifies the conversion of mass percentages to moles, allowing further calculations to identify the simplest ratio of atoms, leading ultimately to both the empirical and molecular formulas of the compound. This same process applies to various other substances like cadaverine and epinephrine, where knowing their atomic composition is a critical step in deducing their chemical identity.

Chemical Calculations

Once you've ascertained the atomic composition and converted those percentages to grams, the next step involves chemical calculations to help find the empirical and molecular formulas.Start by converting grams to moles using each element's atomic mass:- For ibuprofen, with 75.69g of Carbon, the conversion is calculated as: \[ \frac{75.69}{12.01} = 6.30 \text{ moles of Carbon} \]- Similarly, convert other elements to moles.Next, for determining the simplest ratio of atoms, divide each mole value by the smallest number of moles you calculated. This helps in finding the straightforward empirical formula. For ibuprofen:- \(C: \frac{6.30}{0.97} \approx 6.49\), \(H: \frac{8.73}{0.97} \approx 9.00\), \(O: \frac{0.97}{0.97} = 1.00\)Round to the nearest whole number to find the empirical formula \(\text{C}_{13}\text{H}_{18}\text{O}_2\). When the empirical formula mass matches the molar mass, the molecular formula remains the same. These calculations allow for precise and accurate chemical analysis and synthesis.