Question

A 6.53 -g sample of a mixture of magnesium carbonate and calcium carbonate is treated with excess hydrochloric acid. The resulting reaction produces \(1.72 \mathrm{~L}\) of carbon dioxide gas at \(28^{\circ} \mathrm{C}\) and 743 torr pressure. (a) Write balanced chemical equations for the reactions that occur between hydrochloric acid and each component of the mixture. (b) Calculate the total number of moles of carbon dioxide that forms from these reactions. (c) Assuming that the reactions are complete, calculate the percentage by mass of magnesium carbonate in the mixture.

Short answer

The balanced chemical equations for the reactions between hydrochloric acid and magnesium carbonate, and hydrochloric acid and calcium carbonate are: \[MgCO_3(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + CO_2(g) + H_2O(l)\] \[CaCO_3(s) + 2HCl(aq) \rightarrow CaCl_2(aq) + CO_2(g) + H_2O(l)\] Using the Ideal Gas Law, we find that the total moles of carbon dioxide formed is \(0.0732\,\text{mol}\). By setting up a mass balance equation and solving for the mass of magnesium carbonate, we find that it is \(3.67\, \text{g}\). Therefore, the percentage by mass of magnesium carbonate in the mixture is \(56.2\%\).

Step-by-step solution

a) Balanced Chemical Equations

For the reaction of hydrochloric acid (HCl) with magnesium carbonate (MgCO3): \[MgCO_3(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + CO_2(g) + H_2O(l)\] For the reaction of hydrochloric acid (HCl) with calcium carbonate (CaCO3): \[CaCO_3(s) + 2HCl(aq) \rightarrow CaCl_2(aq) + CO_2(g) + H_2O(l)\]

b) Calculate Total Moles of Carbon Dioxide

We can use the Ideal Gas Law formula, which is: \(PV = nRT\) Where: - P = pressure (in atm) - V = volume (in L) - n = moles (in mol) - R = ideal gas constant (0.0821 L⋅atm/mol⋅K) - T = temperature (in K) First, we convert the given pressure from torr to atm: \[743 \,\text{torr} \times \frac{1 \,\text{atm}}{760\, \text{torr}} = 0.976 \,\text{atm}\] Next, we convert the given temperature from Celsius to Kelvin: \[(28^\circ \text{C}) + 273.15 = 301.15 \,\text{K}\] Now, we can use the Ideal Gas Law formula to find the total moles (n) of carbon dioxide (CO2) formed: \[n_\text{CO2} = \frac{PV}{RT} = \frac{(0.976 \,\text{atm})(1.72\, \text{L})}{(0.0821 \, \text{L⋅atm/mol⋅K})(301.15 \,\text{K})} = 0.0732\, \text{mol}\]

c) Calculate Percentage by Mass of Magnesium Carbonate

Let x be the mass of magnesium carbonate (MgCO3) and (6.53 - x) be the mass of calcium carbonate (CaCO3) in the mixture. Then, the moles of CO2 produced from each component can be calculated as: For MgCO3: \(n_\text{CO2(MgCO3)}=\frac{x}{M_\text{MgCO3}}\) For CaCO3: \(n_\text{CO2(CaCO3)}=\frac{6.53-x}{M_\text{CaCO3}}\) The total moles of CO2 formed from both components is 0.0732 mol (from part b), so: \(\frac{x}{M_\text{MgCO3}} + \frac{6.53 - x}{M_\text{CaCO3}} = 0.0732\) Where: - \(M_\text{MgCO3}\) is the molar mass of magnesium carbonate, which is 84.31 g/mol. - \(M_\text{CaCO3}\) is the molar mass of calcium carbonate, which is 100.09 g/mol. Solve the equation for x: \(x = 3.67 \,\text{g}\) Finally, we can calculate the percentage by mass of magnesium carbonate in the mixture: \[\text{Percentage of MgCO3} = \frac{3.67 \,\text{g}}{6.53 \,\text{g}} \times 100\% = 56.2\%\]

Key concepts

Balanced chemical equations

In the context of chemical reactions, a balanced chemical equation is essential. It shows the relationship between the reactants and products in a chemical process. When balancing chemical equations, you ensure that the number of atoms of each element is the same on both sides of the equation. This is important because of the Law of Conservation of Mass, which states that matter cannot be created or destroyed in a closed system.

For example, when hydrochloric acid (HCl) reacts with either magnesium carbonate (MgCO_3) or calcium carbonate (CaCO_3), it produces magnesium chloride (MgCl_2) or calcium chloride (CaCl_2), along with carbon dioxide (CO_2) and water (H_2O).

• For the reaction between HCl and MgCO_3: \[ MgCO_3(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + CO_2(g) + H_2O(l) \]


• For the reaction between HCl and CaCO_3: \[ CaCO_3(s) + 2HCl(aq) \rightarrow CaCl_2(aq) + CO_2(g) + H_2O(l) \]

Balancing these equations involves ensuring that there are equal numbers of each type of atom on both sides of the equation. This way, the calculated results reflect the true chemical behavior in the reaction.

Ideal Gas Law

The Ideal Gas Law is a crucial tool in chemistry for relating the physical properties of gases. It combines several gas laws into a single equation to describe the state of an ideal gas:

\[ PV = nRT \]

Where:

• P is the pressure of the gas (in atmospheres).
• V is the volume of the gas (in liters).
• n is the number of moles of gas.
• R is the ideal gas constant (0.0821 L⋅atm/mol⋅K).
• T is the temperature of the gas (in Kelvin).

To apply the Ideal Gas Law, one needs to ensure that all quantities are in the appropriate units. For instance, temperature must be converted to Kelvin by adding 273.15 to the Celsius measurement. Pressure must often be converted from units like Torr to atmospheres (using a conversion factor of 1 atm = 760 Torr).

By using the Ideal Gas Law, the number of moles of carbon dioxide produced can be calculated accurately from volume, temperature, and pressure data. In the given exercise, the calculation involves converting 1.72 L of CO_2 at 28°C and 743 torr into the number of moles of CO_2, providing vital information needed for further stoichiometric calculations.

Percentage composition by mass

Percentage composition by mass is a way of expressing the concentration of an element or compound in a mixture. It is calculated by taking the mass of the individual component, divided by the total mass of the mixture, and then multiplying by 100% to get the percentage.

In the context of the given problem, the percentage composition by mass of magnesium carbonate in the sample is determined after establishing the number of moles of CO_2 produced. For each component (MgCO_3 and CaCO_3), the number of moles of CO_2 they produce can be calculated using their respective balanced reactions and molar masses.

Once the moles are known, you can calculate the mass of each carbonate, and hence, use the formula:

\[ \text{Percentage by mass of component} = \left( \frac{\text{mass of component}}{\text{total mass of mixture}} \right) \times 100 \% \]

This calculation is crucial, as it gives insight into the composition of the sample, which is particularly useful in analytical chemistry for understanding the make-up of complex mixtures. In this case, solving the relevant equation for the mass of MgCO_3 provides its percentage in the original sample, resulting in a calculated percentage of 56.2%.