Question

Elixirs such as Alka-Seltzer use the reaction of sodium bicarbonate with citric acid in aqueous solution to produce a fizz: $$ 3 \mathrm{NaHCO}_{3}(a q)+\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}(a q) \longrightarrow $$ $$ 3 \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(a q) $$ a. What mass of \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}\) should be used for every \(1.0 \times 10^{2} \mathrm{mg} \mathrm{NaHCO}_{3} ?\) b. What mass of \(\mathrm{CO}_{2}(g)\) could be produced from such a mixture?

Short answer

a. For every \(1.0 \times 10^{2} \mathrm{mg} \mathrm{NaHCO}_{3}\), \(0.0762 \, \mathrm{g}\) of citric acid (\(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}\)) should be used. b. From such a mixture, \(0.0524 \, \mathrm{g}\) of carbon dioxide (\(\mathrm{CO}_{2}\)) could be produced.

Step-by-step solution

Write the balanced chemical equation

First, let's write down the balanced chemical equation of the reaction given in the exercise: \[3 \mathrm{NaHCO}_{3}(a q)+\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}(a q) \longrightarrow 3 \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(a q)\]

Calculate the molar masses of the compounds

Next, we need to find the molar masses of sodium bicarbonate (\(\mathrm{NaHCO}_{3}\)), citric acid (\(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}\)) and carbon dioxide (\(\mathrm{CO}_{2}\)). Using the periodic table, we can find the molar masses as follows: Molar mass of \(\mathrm{NaHCO}_{3} = 23 + 1 + 12 + (3 \times 16) = 84 \, \mathrm{g/mol}\) Molar mass of \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7} = (6 \times 12) + (8 \times 1) + (7 \times 16) = 192 \,\mathrm{g/mol}\) Molar mass of \(\mathrm{CO}_{2} = 12 + (2\times16) = 44 \, \mathrm{g/mol}\)

Determine the mass of citric acid required

We know the mass of sodium bicarbonate to be used: \(1.0 \times 10^{2} \mathrm{mg} \mathrm{NaHCO}_{3} = 0.100 \,\mathrm{g}\) (converting mg to g). Now, we need to determine the stoichiometric ratio of citric acid to sodium bicarbonate from the balanced equation. The ratio is \(\frac{\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}}{\mathrm{NaHCO}_{3}} = \frac{1}{3}\). Using the given mass of \(\mathrm{NaHCO}_{3}\) and their molar masses, we can calculate the mass of citric acid required: Mass of \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7} = \) (Mass of \(\mathrm{NaHCO}_{3}) \times \frac{\text{Molar mass of}\, \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}}{\text{Molar mass of}\, \mathrm{NaHCO}_{3}} \times\frac{\text{Mole ratio of}\, \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}}{\text{Mole ratio of}\, \mathrm{NaHCO}_{3}}\) Mass of \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7} = 0.100 \,\mathrm{g} \times \frac{192 \,\mathrm{g/mol}}{84 \, \mathrm{g/mol}} \times \frac{1}{3} = 0.0762 \, \mathrm{g}\)

Calculate the mass of CO₂ produced

Using a similar procedure, we'll find the mass of \(\mathrm{CO}_{2}\) produced from the same mass of sodium bicarbonate. The stoichiometric ratio of \(\mathrm{CO}_{2}\) to \(\mathrm{NaHCO}_{3}\) is \(\frac{3}{3} = 1.\) Mass of \(\mathrm{CO}_{2} = \) (Mass of \(\mathrm{NaHCO}_{3}) \times \frac{\text{Molar mass of}\, \mathrm{CO}_{2}}{\text{Molar mass of}\, \mathrm{NaHCO}_{3}} \times\frac{\text{Mole ratio of}\, \mathrm{CO}_{2}}{\text{Mole ratio of}\, \mathrm{NaHCO}_{3}}\) Mass of \(\mathrm{CO}_{2} = 0.100 \,\mathrm{g} \times \frac{44 \,\mathrm{g/mol}}{84 \, \mathrm{g/mol}} \times \frac{3}{3} = 0.0524 \, \mathrm{g}\) Now we have our results: a. For every \(1.0 \times 10^{2} \mathrm{mg} \mathrm{NaHCO}_{3}\), \(0.0762 \, \mathrm{g}\) of citric acid (\(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}\)) should be used. b. From such a mixture, \(0.0524 \, \mathrm{g}\) of carbon dioxide (\(\mathrm{CO}_{2}\)) could be produced.

Key concepts

Chemical Reactions

Chemical reactions are at the heart of chemistry, where substances interact to form new compounds. Each reaction involves breaking chemical bonds in the reactants and forming new bonds to create the products. This process is governed by the law of conservation of mass, meaning the mass of the reactants must equal the mass of the products.

In the given exercise, the reaction involves sodium bicarbonate (NaHCO₃) reacting with citric acid (C₆H₈O₇) to form carbon dioxide (CO₂), water (H₂O), and sodium citrate (Na₃C₆H₅O₇). This is an example of an acid-base reaction where the basic compound sodium bicarbonate reacts with the acidic compound citric acid, producing bubbles of carbon dioxide gas, often responsible for the 'fizz' in antacid tablets.

Molar Mass

Understanding molar mass is crucial for stoichiometric calculations in chemistry. Molar mass is the mass of one mole of a given substance (atoms, molecules, etc.) and it is expressed in grams per mole (g/mol).

For the compounds in the exercise:

• Sodium bicarbonate (\(\mathrm{NaHCO}_{3}\)) has a molar mass of 84 g/mol. This is obtained by adding up atomic masses of all atoms in the compound: 23 (Na) + 1 (H) + 12 (C) + (3 x 16) (O).
• Citric acid (\(\mathrm{C}_{6}\mathrm{H}_{8}\mathrm{O}_{7}\)) has a molar mass of 192 g/mol.
• Carbon dioxide (\(\mathrm{CO}_{2}\)) has a molar mass of 44 g/mol.

These values come from summing the atomic masses of each element present in one mole of their respective molecules, allowing us to compare and convert between mass and moles in the reaction.

Balanced Chemical Equation

A balanced chemical equation is essential for accurately representing a chemical reaction. Balancing ensures that the number of atoms for each element is the same on both sides of the equation, respecting the law of conservation of mass.

In our exercise, the balanced equation is given as:\[3 \mathrm{NaHCO}_{3}(aq) + \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}(aq) \longrightarrow 3 \mathrm{CO}_{2}(g) + 3 \mathrm{H}_{2} \mathrm{O}(l) + \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(aq)\]Here, each compound's coefficients are adjusted so that there are equal numbers of each type of atom on both sides of the equation.

• 3 sodium bicarbonate react with 1 citric acid to produce 3 carbon dioxide molecules, 3 water molecules, and 1 sodium citrate.
• This ensures the same number of sodium (Na), hydrogen (H), carbon (C), and oxygen (O) atoms are present on both sides of the reaction.

Balancing chemical equations is a fundamental skill in chemistry to predict the outcomes and measure the relationships between different substances.

Stoichiometric Calculations

Stoichiometry is the calculation of reactants and products in chemical reactions. It involves using the balanced chemical equation to determine the relationships between the quantities of reactants and products.

In the given problem:

• We started by converting the mass of sodium bicarbonate from milligrams to grams for easier calculations.
• Then, using the balanced equation, the mole ratios between the reactants and products were identified.
• The ratio between citric acid and sodium bicarbonate is 1:3. Thus, we used this ratio alongside the molar masses to calculate that for 0.100 g of NaHCO₃, 0.0762 g of C₆H₈O₇ is required.
• For the carbon dioxide produced, a 1:1 ratio exists between CO₂ and NaHCO₃, leading to the production of 0.0524 g of CO₂.

Stoichiometric calculations allow chemists to predict how much product a reaction will produce based on given quantities of reactants, making them essential tools in both laboratory and industrial chemistry. They ensure efficient use of reactants and help in planning experiments or production processes.