Question

A \(19.9-\mathrm{mL}\) volume of a hydrochloric acid solution reacts completely with a solid sample of magnesium carbonate, producing \(183 \mathrm{~mL}\) of \(\mathrm{CO}_{2}\) that is collected over water at \(24.0^{\circ} \mathrm{C}\) and 738 torr total pressure. The reaction is \(2 \mathrm{HCl}(a q)+\mathrm{MgCO}_{3}(s) \longrightarrow\) \(\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{MgCl}_{2}(a q)\) What is the molarity of the HCl solution?

Short answer

The molarity of the HCl solution is 0.711 M.

Step-by-step solution

Calculate the Moles of CO2 Produced

To find the moles of \( \text{CO}_2 \) produced, we need to use the ideal gas law:\[PV = nRT\]Where:- \( P \) is the pressure of \( \text{CO}_2 \) in atm- \( V \) is the volume of \( \text{CO}_2 \) in liters- \( n \) is the number of moles- \( R \) is the ideal gas constant \( 0.0821 \text{ L atm mol}^{-1} \text{K}^{-1} \)- \( T \) is the temperature in Kelvin.First, we need to determine the partial pressure of \( \text{CO}_2 \) by subtracting the vapor pressure of water at \( 24.0^\circ \text{C} \), which is approximately 22.4 torr, from the total pressure.The pressure of \( \text{CO}_2 \) (in atm) is:\[P_{\text{CO}_2} = \left(738 - 22.4\right) \text{torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 0.9421 \text{ atm}\]Convert the volume of \( \text{CO}_2 \) to liters:\[V = \frac{183 \text{ mL}}{1000 \text{ mL/L}} = 0.183 \text{ L}\]Convert temperature to Kelvin:\[T = 24.0 + 273.15 = 297.15 \text{ K}\]Now, solve for \( n \):\[n = \frac{{P V}}{{R T}} = \frac{{(0.9421 \text{ atm})(0.183 \text{ L})}}{{(0.0821 \text{ L atm mol}^{-1} \text{K}^{-1})(297.15 \text{ K})}} = 0.00708 \text{ moles of } \text{CO}_2\]

Determine Moles of HCl

According to the balanced chemical equation, 1 mole of \( \text{MgCO}_3 \) produces 1 mole of \( \text{CO}_2 \), and 2 moles of \( \text{HCl} \) react with 1 mole of \( \text{MgCO}_3 \).Since we have 0.00708 moles of \( \text{CO}_2 \), we can infer that 0.00708 moles of \( \text{MgCO}_3 \) reacted, requiring:\[2 \times 0.00708 = 0.01416 \text{ moles of } \text{HCl}\] to react completely with the \( \text{MgCO}_3 \).

Calculate the Molarity of the HCl Solution

Molarity (\( M \)) is calculated using the formula:\[M = \frac{{\text{moles of solute}}}{{\text{liters of solution}}}\]The volume of the HCl solution is \( 19.9 \text{ mL} \), or \( 0.0199 \text{ L} \). Therefore, the molarity of the HCl solution is:\[M = \frac{{0.01416 \text{ moles}}}{{0.0199 \text{ L}}} = 0.711 \text{ M}\]

Final Answer

The molarity of the HCl solution is \( 0.711 \text{ M} \).

Key concepts

Ideal Gas Law

When studying gases, the Ideal Gas Law is a fundamental concept that helps us understand how pressure, volume, temperature, and number of moles are interrelated. This law is represented by the equation \(PV = nRT\). Here, \(P\) stands for the pressure of the gas, \(V\) is the volume of the gas, \(n\) is the number of moles, \(R\) is the ideal gas constant (0.0821 \(\text{L atm mol}^{-1} \text{K}^{-1}\)), and \(T\) is the temperature in Kelvin.

By rearranging this equation, we can solve for the unknown variable, often the number of moles \(n\). When applied to a scenario involving a chemical reaction, like the one between magnesium carbonate and hydrochloric acid, it allows us to calculate the amount of gas produced under specific conditions.

Understanding each variable and its units is essential in applying the Ideal Gas Law accurately. For instance, pressure must be in atmospheres, temperature in Kelvin, and volume in liters. These consistent units ensure accurate calculation and understanding of gas behavior in various reactions.

Hydrochloric Acid

Hydrochloric acid, known as \(\text{HCl}\), is a strong acid commonly used in various chemical reactions. It is a colorless solution that plays a crucial role in processes like digestion and industrial applications. In the chemical reaction given, hydrochloric acid reacts with magnesium carbonate (\(\text{MgCO}_3\)).

As a strong acid, \(\text{HCl}\) disassociates completely in water, releasing hydrogen ions. This characteristic makes it effective in reacting with bases and forming salts and water. Its reaction with magnesium carbonate leads to the production of carbon dioxide \(\text{CO}_2\), water, and magnesium chloride \(\text{MgCl}_2\).
When calculating the molarity of an \(\text{HCl}\) solution, understanding these reactions and the amounts involved is key. Molarity reflects the concentration of \(\text{HCl}\) in solution, helping determine the reactivity and strength of the acid in different chemical contexts.

Magnesium Carbonate

Magnesium carbonate, \(\text{MgCO}_3\), is a white solid compound often used in antacids and various industrial products. In chemical reactions, it behaves as a base or carbonate, reacting with acids to release carbon dioxide. In the provided reaction with hydrochloric acid, \(\text{MgCO}_3\) plays a pivotal role.

When \(\text{MgCO}_3\) reacts with \(\text{HCl}\), it produces \(\text{CO}_2\), water, and \(\text{MgCl}_2\).

• \(\text{MgCO}_3\) as a reactant leads to gas production, useful in calculations involving the Ideal Gas Law.
• In stoichiometry, it's crucial to recognize that one mole of \(\text{MgCO}_3\) produces one mole of \(\text{CO}_2\).

These reactions demonstrate basic principles where carbonates react with acids, releasing gas. This type of reaction is foundational in understanding how compounds transform in chemical contexts.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry involves using balanced equations to calculate the amounts of reactants and products in a chemical reaction. It's like a recipe in chemistry, ensuring precise transformations between substances. In the reaction \(2\text{HCl} + \text{MgCO}_3 \rightarrow \text{CO}_2 + \text{H}_2\text{O} + \text{MgCl}_2\), stoichiometry helps us determine how much of each reactant is required or how much product is produced.

Using stoichiometry, you can relate quantities of reactants like \(\text{HCl}\) and \(\text{MgCO}_3\) to the produced \(\text{CO}_2\). In this case:

• 2 moles of \(\text{HCl}\) react with 1 mole of \(\text{MgCO}_3\).
• These reactions yield 1 mole of \(\text{CO}_2\).

Understanding these relationships enables us to convert between grams, moles, and molecules, applying it to calculate things like the molarity of solutions. It's an essential tool in chemistry, providing insights into how chemical processes operate at the molecular level.