Problem 127 - Chemistry

Question

Lysine, an essential amino acid in the human body, contains \(\mathrm{C}, \mathrm{H}, \mathrm{O},\) and \(\mathrm{N} .\) In one experiment, the complete combustion of \(2.175 \mathrm{~g}\) of lysine gave \(3.94 \mathrm{~g}\) \(\mathrm{CO}_{2}\) and \(1.89 \mathrm{~g} \mathrm{H}_{2} \mathrm{O} .\) In a separate experiment, \(1.873 \mathrm{~g}\) of lysine gave \(0.436 \mathrm{~g} \mathrm{NH}_{3}\). (a) Calculate the empirical formula of lysine. (b) The approximate molar mass of lysine is \(150 \mathrm{~g}\). What is the molecular formula of the compound?

Step-by-Step Solution

Answer

The empirical formula of Lysine is \( \mathrm{C}_3\mathrm{H}_7\mathrm{N}_1\mathrm{O}_1 \). With the given molar mass, the molecular formula of Lysine is \( \mathrm{C}_6\mathrm{H}_{14}\mathrm{N}_2\mathrm{O}_2 \).

Step 1: Calculate the Moles of Carbon and Hydrogen

From the given data, we know that 2.175 g of Lysine yields 3.94 g of \( \mathrm{CO}_2 \) and 1.89 g of \( \mathrm{H}_2\mathrm{O} \). We know that one mole of \( \mathrm{CO}_2 \) contains 1 mole of Carbon and one mole of \( \mathrm{H}_2\mathrm{O} \) contains 2 moles of Hydrogen. By using the molar masses of \( \mathrm{CO}_2 \) (44.01 g/mol) and \( \mathrm{H}_2\mathrm{O} \) (18.02 g/mol), we can find the moles of Carbon and Hydrogen.

Step 2: Calculate the Moles of Nitrogen

From the second experiment data, we know that 1.873 g Lysine yields 0.436 g of \(\mathrm{NH}_3\). 1 mole of \(\mathrm{NH}_3\) contains 1 mole of Nitrogen. By using the molar mass of \(\mathrm{NH}_3\) (17.03 g/mol), we can find the moles of Nitrogen.

Step 3: Calculate the Moles of Oxygen

To calculate the moles of Oxygen, subtract the mass of Carbon, Hydrogen, and Nitrogen from the total mass of Lysine. Convert the obtained mass of Oxygen to moles by using the molar mass of Oxygen (16.00 g/mol).

Step 4: Calculate the Ratios

For finding the empirical formula, we should express these moles in a simple ratio to each other. We divide each mole value by the smallest mole value found.

Step 5: Calculate the Molecular Formula

Divide the molar mass of Lysine (150 g) by the molar mass of the empirical formula to find the ratio. Multiply the subscripts of the empirical formula by this ratio to obtain the molecular formula of lysine.

Key Concepts

Molecular formula

The molecular formula of a compound reveals the actual number of atoms of each element in a molecule of that compound. It can be seen as a multiple of the empirical formula, which represents the simplest whole-number ratio of atoms in the compound.
To determine the molecular formula, we start from the empirical formula. Then, we use the molar mass of the compound to calculate how many times the empirical formula must be multiplied to result in the correct molecular formula.

• Calculate the molar mass of the empirical formula.
• Divide the compound's known molar mass by the empirical formula's molar mass to find the multiplication factor.
• Apply this factor to derive the molecular formula.

In our example, if the empirical formula's molar mass turns out to be less than the approximate molar mass (150 g/mol) of lysine, we then assess how many times the empirical formula applies to reach the desired mass. Thus, the molecular formula reveals the composition of a substance in its true quantitative form.

Combustion analysis

Combustion analysis is a method used to determine the elemental composition of a substance, particularly organic compounds. In this method, the compound is burned in the presence of excess oxygen. The products typically include carbon dioxide (\( \mathrm{CO}_2 \)) and water (\( \mathrm{H}_2\mathrm{O} \)), and sometimes ammonia (\( \mathrm{NH}_3 \)) for nitrogen-containing compounds. By measuring the amounts of these products, we can determine the quantities of each element in the original compound.
- For carbon, find the amount by noting the mass of \( \mathrm{CO}_2 \) produced.- Hydrogen is determined by the mass of the water produced. - Nitrogen content may be measured if \( \mathrm{NH}_3 \) is formed in a further experiment.- Finally, measure the oxygen content by subtracting the other elements' masses from the total mass of the compound.
Using combustion analysis, we create a clearer picture of the compound's elemental composition, as seen in the experiments with lysine where carbon, hydrogen, and nitrogen amounts are calculated from combusted products.

Molar mass calculation

Molar mass calculation involves determining the mass of one mole of a given substance. This is a critical step in linking empirical and molecular formulas to the actual mass of a substance.

• To calculate molar mass, sum up the atomic masses of all atoms present in one molecule.
• Find atomic masses on the periodic table and add them according to the molecular composition.

In the case of empirical formula calculations, molar mass helps us verify the formula based on the compound's approximate molar mass. By comparing the molar mass of the empirical formula to the known molar mass of the substance, we adjust the formula to more precisely represent the molecule's structure. For instance, if the empirical formula suggested a mass significantly different from lysine's known mass of 150 g/mol, we would adjust it through division or multiplication of subscripts. Thus, molar mass calculation is key in refining empirical formulas into molecular formulas.