Problem 109 - Chemistry An Atoms First Approach

Question

Elixirs such as Alka-Seltzer use the reaction of sodium bicarbonate with citric acid in aqueous solution to produce a fizz: $$\begin{aligned}3 \mathrm{NaHCO}_{3}(a q)+\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}(a q) & \longrightarrow \\\3 \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l) &+\mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(a q)\end{aligned}$$ 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?

Step-by-Step Solution

Answer

a. The mass of \(\mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}\) that should be used for every \(1.0 \times 10^{2} \, \mathrm{mg} \, \mathrm{NaHCO}_{3}\) is: $$\rm Mass \thinspace of \thinspace C_{6}H_{8}O_{7} = (moles \thinspace of \thinspace C_{6}H_{8}O_{7})\times (192\,g/mol)$$ b. The mass of \(\mathrm{CO}_{2}(g)\) that could be produced from such a mixture is: $$\rm Mass \thinspace of \thinspace CO_2 = (moles \thinspace of \thinspace CO_2)\times (44\,g/mol)$$

Step 1: Identify the balanced chemical equation

The balanced chemical equation for the reaction is given by: $$3 \mathrm{NaHCO}_{3}(a \mathrm{q})+\mathrm{C}_{6} \mathrm{H}_{8}\mathrm{O}_{7}(a \mathrm{q}) \longrightarrow 3 \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2}\mathrm{O}(l)+\mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5}\mathrm{O}_{7}(a\mathrm{q})$$

Step 2: Determine the stoichiometric ratios

From the balanced chemical equation, we can see that: - 3 moles of \(\mathrm{NaHCO}_{3}\) reacts with 1 mole of \(\mathrm{C}_{6}\mathrm{H}_{8}\mathrm{O}_{7}\) to produce 3 moles of \(\mathrm{CO}_2\). - Hence, the stoichiometric ratio between \(\mathrm{NaHCO}_{3}\) and \(\mathrm{C}_{6}\mathrm{H}_{8}\mathrm{O}_{7}\) is 3:1, and between \(\mathrm{NaHCO}_{3}\) and \(\mathrm{CO}_2\) is 1:1.

Step 3: Convert mass to moles

Given that the mass of \(\mathrm{NaHCO}_{3}\) is \(1.0 \times 10^2 \, \mathrm{mg}\), we can convert this into moles using the molar mass of \(\mathrm{NaHCO}_{3}\) (84 g/mol): $$\rm moles \thinspace of \thinspace NaHCO_3= \frac{1.0 \times 10^2\,mg \times \frac{1\,g}{10^3\,mg}}{84\,g/mol}$$

Step 4: Calculate the required mass of citric acid

From the stoichiometric ratio, we know that for every 3 moles of \(\mathrm{NaHCO}_3\), 1 mole of \(\mathrm{C}_{6}\mathrm{H}_{8}\mathrm{O}_{7}\) is required. Hence, $$\rm moles \thinspace of \thinspace C_{6}H_{8}O_{7} = \frac{1}{3}\times moles \thinspace of \thinspace NaHCO_3$$ Next, convert moles of \(\mathrm{C}_{6}\mathrm{H}_{8}\mathrm{O}_{7}\) to mass using its molar mass (192 g/mol): $$\rm Mass \thinspace of \thinspace C_{6}H_{8}O_{7} = (moles \thinspace of \thinspace C_{6}H_{8}O_{7})\times (192\,g/mol)$$

Step 5: Calculate the mass of carbon dioxide produced

From the stoichiometric ratio, we know that for every mole of \(\mathrm{NaHCO}_3\), 1 mole of \(\mathrm{CO}_2\) is produced. Hence, $$\rm moles \thinspace of \thinspace CO_2 = moles \thinspace of \thinspace NaHCO_3$$ Next, convert moles of \(\mathrm{CO}_2\) to mass using its molar mass (44 g/mol): $$\rm Mass \thinspace of \thinspace CO_2 = (moles \thinspace of \thinspace CO_2)\times (44\,g/mol)$$

Key Concepts

Balanced Chemical Equation

When it comes to understanding a chemical reaction, the first step is to recognize the balanced chemical equation. This is essential because it illustrates the relationship and ratios between reactants and products. A balanced chemical equation ensures that the number of atoms for each element is the same on both sides of the equation. This is in line with the law of conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction. For instance, in the reaction between sodium bicarbonate and citric acid, the balanced equation is: \[3 \mathrm{NaHCO}_{3}(a q) + \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7}(a q) \rightarrow 3 \mathrm{CO}_{2}(g) + 3 \mathrm{H}_{2} \mathrm{O}(l) + \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}(a q)\]. This tells us that 3 moles of sodium bicarbonate react with 1 mole of citric acid to produce 3 moles of carbon dioxide, 3 moles of water, and 1 mole of sodium citrate. Balancing a chemical equation is crucial for determining the correct stoichiometric ratios, guiding subsequent calculations.

Molar Mass Calculation

Molar mass is a vital concept in stoichiometry. It refers to the mass of one mole of a given substance. This value is typically expressed in grams per mole (g/mol). Determining the molar mass of a compound involves summing up the atomic masses of all the atoms in its molecular formula. For example:

• The molar mass of sodium bicarbonate, \( \mathrm{NaHCO}_{3} \), is calculated from: sodium (Na), hydrogen (H), carbon (C), and three oxygens (O). Summing these gives: 23 (Na) + 1 (H) + 12 (C) + 16*3 (O) = 84 g/mol.
• The molar mass of citric acid, \( \mathrm{C}_{6} \mathrm{H}_{8} \mathrm{O}_{7} \), is calculated from: six carbons (C), eight hydrogens (H), and seven oxygens (O). Summing these gives: 6*12 (C) + 8*1 (H) + 7*16 (O) = 192 g/mol.

Accurate molar mass calculations are crucial for converting between mass and moles, allowing us to explore quantitative relationships in chemical reactions.

Mass to Moles Conversion

Converting from mass to moles is a foundational step in solving stoichiometry problems. This conversion uses the formula:\[\text{Moles of a substance} = \frac{\text{Mass of the substance (g)}}{\text{Molar mass of the substance (g/mol)}}\].For example, if we have 100 mg of sodium bicarbonate, we first need to convert this into grams:

• 100 mg = 0.1 g.

Then, using the molar mass of sodium bicarbonate (84 g/mol), we can find the moles:\[\text{Moles of } \mathrm{NaHCO}_{3} = \frac{0.1}{84} \approx 0.00119 \text{ moles}\].This calculation is integral to determining how much of the other reactants and products are involved using the stoichiometric ratios from the balanced chemical equation.

Chemical Reaction

Now, let's dive into what a chemical reaction truly means. A chemical reaction involves the transformation of reactants into products. During this process, the molecules of different substances interact and bonds are broken and formed, leading to new chemical arrangements. These reactions can lead to various changes, such as energy release in the form of heat or light, color changes, gas formation, or a precipitate forming. In the given reaction between sodium bicarbonate and citric acid, the fizz produced is carbon dioxide gas. This is a common example where gas evolution signifies a reaction is occurring. Reactants and products are expressed using chemical formulas, and their quantities are determined by a balanced chemical equation. Understanding the nature of chemical reactions helps in predicting the outcomes of mixing various chemical substances, which is not only crucial in labs but also in everyday applications like baking and pharmaceuticals.