Problem 16 - General Chemistry: Principles and Modern Applications

Question

Explain which of the following statement(s) is (are) correct for sorbic acid, \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{2},\) an inhibitor of mold and yeast. (a) It has a \(\mathrm{C}: \mathrm{H}:\) O mass ratio of 3: 4: 1 (b) It has the same mass percent composition as the aquatic herbicide, acrolein, \(\mathrm{C}_{3} \mathrm{H}_{4} \mathrm{O}\) (c) It has the same empirical formula as aspidinol, \(\mathrm{C}_{12} \mathrm{H}_{16} \mathrm{O}_{4},\) a drug used to kill parasitic worms. (d) It has four times as many \(\mathrm{H}\) atoms as \(\mathrm{O}\) atoms, but four times as much O as H by mass.

Step-by-Step Solution

Answer

Among the provided statements about sorbic acid, statement (d) is the only correct one.

Step 1: Calculate the mole ratios

To determine if statement (a) is correct, calculate the mole ratio of \(\mathrm{C}:\mathrm{H}:\mathrm{O}\) for sorbic acid. Use the atomic masses of carbon (12.01 g/mol), hydrogen (1.01 g/mol), and oxygen (16.00 g/mol) to calculate the moles of each element in one mole of sorbic acid.For Carbon, \( \frac{6 \times 12.01}{112.12} \) gives 0.643 moles. For Hydrogen, \( \frac{8 \times 1.01}{112.12} \) gives 0.072 moles. For Oxygen, \( \frac{2 \times 16.00}{112.12} \) gives 0.285 moles.

Step 2: Compare with given ratio

The calculated mole ratio is approximately \(\mathrm{C}: \mathrm{H}: \mathrm{O}=0.643: 0.072: 0.285\), which is not the same as the given \(\mathrm{C}: \mathrm{H}: \mathrm{O}=3: 4: 1\). Therefore, statement (a) is incorrect.

Step 3: Compare mass percent compositions

To determine if statement (b) is correct, compare the mass percent composition of sorbic acid and acrolein. The mass percent composition is the mass of each element in a compound divided by the overall mass. The mass percent of carbon in sorbic acid is \(\frac{6 \times 12.01}{112.12}\), and for hydrogen it is \(\frac{8 \times 1.01}{112.12}\), and for oxygen it is \(\frac{2 \times 16.00}{112.12}\).Perform similar calculations for acrolein to find its mass percent composition, and compare. Since the two molecules have different numbers of atoms of each element, they will not have the same mass percent composition. Statement (b) is incorrect.

Step 4: Compare empirical formulas

To determine if statement (c) is correct, one can compare the empirical formulas of sorbic acid and aspidinol. The empirical formula is the lowest whole number ratio of the elements in a compound. For sorbic acid, the empirical formula is \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{2}\), and for aspidinol it is \(\mathrm{C}_{12} \mathrm{H}_{16} \mathrm{O}_{4}\). Taking the ratio \(\frac{6}{8}\) and reducing, one get \(\frac{3}{4}\) as the ratio of C to H, different from the ratio in aspidinol. Therefore, statement (c) is incorrect.

Step 5: Compare Hydrogen and Oxygen numbers and masses

To determine if statement (d) is correct, compare the number of hydrogen atoms to oxygen atoms, and the mass of hydrogen to the mass of oxygen. In this case, the ratio of hydrogen in sorbic acid to oxygen is indeed 4 to 1. The mass ratio is \( \mathrm{H}: \mathrm{O} = 8 \times 1.01 : 2 \times 16.00 \), which simplifies to 4:1, exactly as written in statement (d). Therefore, statement (d) is correct.

Key Concepts

Mole Ratio

In chemistry, the mole ratio is a comparison of the number of moles of one element to the number of moles of another element within a molecule. For sorbic acid \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{2} \), the calculation of the mole ratio involves using the atomic masses of its constituent elements: carbon (C), hydrogen (H), and oxygen (O).
To determine the mole ratio of these elements in sorbic acid:

• Carbon: \(6 \times 12.01\) g/mol
• Hydrogen: \(8 \times 1.01\) g/mol
• Oxygen: \(2 \times 16.00\) g/mol

Each of these products is then divided by the molar mass of the compound (112.12 g/mol) to find the number of moles of each element.
This yields a mole ratio of approximately \( \mathrm{C}: \mathrm{H}: \mathrm{O} = 0.643: 0.072: 0.285 \). Therefore, the proposed \( \mathrm{C}: \mathrm{H}: \mathrm{O} \) ratio of 3:4:1 is incorrect. Calculating the mole ratio helps visualize the proportion of different elements in a compound and ensures chemical reactions can be balanced accurately.

Mass Percent Composition

Mass percent composition is the calculation of the percentage of each element's mass in comparison to the entire compound's mass. This concept is crucial when comparing the composition of different compounds.
For sorbic acid, the mass of each element is:

• Carbon: \( \frac{6 \times 12.01}{112.12} \)
• Hydrogen: \( \frac{8 \times 1.01}{112.12} \)
• Oxygen: \( \frac{2 \times 16.00}{112.12} \)

Calculating these gives the mass percent of each element in sorbic acid. By comparing these percentages to another compound like acrolein, \( \mathrm{C}_{3} \mathrm{H}_{4} \mathrm{O} \), we can determine if two compounds have the same composition.
When you do the math for both compounds, you will quickly see that their compositions differ, thus proving the proposed similarity with acrolein false. Understanding mass percent helps scientists and students predict how substances will react with each other in various conditions.

Empirical Formula

The empirical formula reflects the simplest whole-number ratio of the elements in a compound. It doesn't necessarily reflect the actual number of atoms in the molecules of the substance but offers a simplified version for analysis.
For sorbic acid, with its molecular formula \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{2} \), the empirical formula happens to be the same as the molecular since it's already expressed in the simplest form.
By comparison, let's look at aspidinol, \( \mathrm{C}_{12} \mathrm{H}_{16} \mathrm{O}_{4} \). When you divide each subscript by the greatest common factor, the empirical formula remains unchanged. However, comparing this with the empirical formula of sorbic acid results in different element ratios: \( \frac{6}{8} = \frac{3}{4} \) does not equate to aspindinol's ratio, confirming that the two compounds have different empirical formulas.
Empirical formulas are significant for identifying substances in chemical studies and ensuring correct expression in chemical equations.

Hydrogen to Oxygen Ratio

The hydrogen to oxygen ratio in a compound indicates how many times hydrogen atoms are present compared to oxygen atoms, and it is foundational in determining the character and properties of a compound.
In sorbic acid, \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{2} \), the count of hydrogen atoms (H) to oxygen atoms (O) is 8:2, which simplifies to 4:1, meaning there are four hydrogen atoms for every oxygen atom. This was a point of importance in verifying statement (d).
Notably, while the quantity of H atoms is indeed quadruple that of O atoms, when you consider mass, oxygen still weighs more because an oxygen atom has a greater mass than a hydrogen atom. This insight aids in understanding the impact of atomic masses on the properties of substances, emphasizing the rule of thumb that more atoms don't always mean more mass.