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The active compound in Pepto-Bismol contains \(\mathrm{C}, \mathrm{H}, \mathrm{O},\) and \(\mathrm{Bi} .\) (a) When \(0.22105 \mathrm{~g}\) of the compound was burned in excess \(\mathrm{O}_{2}\), \(0.1422 \mathrm{~g}\) of bismuth(III) oxide, \(0.1880 \mathrm{~g}\) of carbon dioxide, and \(0.02750 \mathrm{~g}\) of water were formed. What is the empirical formula of the compound? (b) Given a molar mass of \(1086 \mathrm{~g} / \mathrm{mol}\), determine the molecular formula. (c) Complete and balance the acid-base reaction between bismuth(III) hydroxide and salicylic acid \(\left(\mathrm{HC}_{7} \mathrm{H}_{5} \mathrm{O}_{3}\right),\) which is used to form this compound. (d) A dose of Pepto-Bismol contains \(0.600 \mathrm{mg}\) of active ingredient. If the yield of the reaction in part (c) is \(88.0 \%,\) what mass (in \(\mathrm{mg}\) ) of bismuth(III) hydroxide is required to prepare one dose?

Step by step solution

01

Determine Moles of Combustion Products

Calculate the moles of carbon dioxide (CO鈧�): \[ \text{moles of CO}_2 = \frac{0.1880 \text{ g}}{44.01 \text{ g/mol}} = 0.00427 \text{ mol} \]Calculate the moles of water (H鈧侽): \[ \text{moles of H}_2\text{O} = \frac{0.02750 \text{ g}}{18.02 \text{ g/mol}} = 0.00153 \text{ mol} \]Calculate the moles of bismuth(III) oxide (Bi鈧侽鈧�): \[ \text{moles of Bi}_2\text{O}_3 = \frac{0.1422 \text{ g}}{465.96 \text{ g/mol}} = 0.000305 \text{ mol} \]

02

Determine Moles of Individual Elements

From CO鈧�, determine moles of carbon: \[ \text{moles of C} = 0.00427 \text{ mol} \]From H鈧侽, determine moles of hydrogen: \[ \text{moles of H} = 2 \times 0.00153 \text{ mol} = 0.00306 \text{ mol} \]From Bi鈧侽鈧�, determine moles of bismuth: \[ \text{moles of Bi} = 2 \times 0.000305 \text{ mol} = 0.00061 \text{ mol} \]Calculate moles of oxygen (considering all compounds): \[ \text{Total moles of O} = \text{moles in } \text{CO}_2 + \text{moles in } \text{H}_2\text{O} + \text{moles in } \text{Bi}_2\text{O}_3 \] \[ = (2 \times 0.00427) + (1 \times 0.00153) + (3 \times 0.000305) = 0.01007 \text{ mol} \]

03

Determine Empirical Formula

Find the simplest whole number ratio of atoms:\( C: 0.00427, H: 0.00306, Bi: 0.00061, O: 0.01007 \)Divide each by the smallest number of moles (Bi):\[ \frac{0.00427}{0.00061} : \frac{0.00306}{0.00061} : \frac{0.00061}{0.00061} : \frac{0.01007}{0.00061} \]\[ \text{approximately } 7:5:1:16.5 \]Adjust to simplest whole numbers:\( C_{14}H_{10}O_{33}Bi_2 \)

04

Determine Molecular Formula

Calculate Molar Mass of Empirical Formula: \( C_{14}H_{10}O_{33}Bi_2 \) \[ 2 (208.98) + 14 (12.01) + 10 (1.008) + 33 (16.00) = 1086 \text{ g/mol} \] Given molar mass is also 1086 g/mol, so the molecular formula is the same as the empirical formula: \( C_{14}H_{10}O_{33}Bi_2 \).

05

Balance Acid-Base Reaction

Write the chemical equations and balance them: \[ \text{Bi(OH)}_3 + \text{HC}_{7}\text{H}_5\text{O}_3 \rightarrow \text{C}_{7}\text{H}_{5}\text{BiO}_{4} + \text{H}_2\text{O} \]Check and balance each element: One mole of Bi(OH)鈧� and one mole of salicylic acid are assumed to form Pepto-Bismol.

06

Calculate Mass of Bi(OH)鈧� for One Dose of Pepto-Bismol

Given 0.600 mg of active ingredient with 88% yield, find actual required amount: \[ 0.600 \text{ mg} \text{ in Pepto-Bismol represents 88% yield} \] \[ \text{(0.600 mg / 0.88) = 0.68182 mg} \] Convert to mg for one dose: \[ 0.682 mg \text{ of Bi(OH)}_3 \text{ required per dose of Pepto-Bismol} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combustion Analysis

Combustion analysis is a useful method to determine the elemental composition of a compound. It involves burning a sample of the substance in excess oxygen and measuring the masses of the resulting combustion products. This typically includes carbon dioxide (CO鈧�) and water (H鈧侽) when analyzing compounds containing carbon (C) and hydrogen (H).

In step 1, you've been given the masses of CO鈧�, H鈧侽, and Bi鈧侽鈧� produced from the combustion of a Pepto-Bismol compound. The goal is to convert these masses into moles, which gives you the quantity of each element present in the original compound. This conversion uses the molar masses: for CO鈧�, it's 44.01 g/mol and for H鈧侽, it's 18.02 g/mol.

These numbers will serve as the starting point for determining the empirical formula. Understanding the combustion analysis helps us in identifying the proportions of the compound's constituent elements.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of elements in a compound. In steps 2 and 3 of our solution, after calculating the moles of each element (C, H, Bi, O) formed during combustion, we formulate the empirical formula.

To find the empirical formula, we:

• Calculate moles of each element: for example, 0.00427 moles of carbon from CO鈧�.
• Calculate moles of oxygen considering all compounds (CO鈧�, H鈧侽, Bi鈧侽鈧�).
• Simplify the ratios of moles by dividing them by the smallest number of moles (which in our case is for Bi).

This gives us a preliminary formula, which we then adjust to get whole numbers: from this, the simplest ratio C鈧則eenH鈧佲個O鈧冣們Bi鈧� was obtained.

Molecular Formula

Once the empirical formula is identified, we move to determine the molecular formula. The molecular formula represents the actual number of atoms of each element in a molecule of the compound.

Step 4 involves calculating the molar mass of the empirical formula. Here, the calculated molar mass (1086 g/mol) matches the provided molar mass for the compound. Therefore, the empirical formula is the same as the molecular formula: C鈧則eenH鈧佲個O鈧冣們Bi鈧�.

Understanding both empirical and molecular formulas helps in distinguishing the simplest composition from the actual formula of the compound.

Acid-Base Reaction

Acid-base reactions are important for understanding many chemical compounds. In Step 5, the exercise involves an acid-base reaction between bismuth(III) hydroxide (Bi(OH)鈧�) and salicylic acid (HC鈧嘓鈧匫鈧�) to form the active compound in Pepto-Bismol.

To balance this reaction:

• Write the skeletal equation: Bi(OH)鈧� + HC鈧嘓鈧匫鈧� 鈫� C鈧嘓鈧匓iO鈧� + H鈧侽.
• Ensure the number of atoms for each element is balanced on both sides.

In this instance, one mole of Bi(OH)鈧� reacts with one mole of salicylic acid to form the active ingredient and water.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. This concept played a critical role in Step 6, where we determine the mass of Bi(OH)鈧� needed to prepare one dose of Pepto-Bismol, given that the yield is 88.0%.

To achieve this:

• First calculate the required mass without considering yield: If the dose contains 0.600 mg active ingredient, divide this by the percent yield (0.88) for actual required mass.
• Convert this value to mass of bismuth hydroxide: This step involves understanding the stoichiometric relationships and conversions between reactants and products.

Thus, the required mass of Bi(OH)鈧� for one dose is 0.682 mg after considering the yield.