Question

Carbon dioxide gas, saturated with water vapor, can be produced by the addition of aqueous acid to calcium carbonate. $$ \mathrm{CaCO}_{3}(s)+2 \mathrm{H}^{+}(a q) \rightarrow \mathrm{Ca}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{CO}_{2}(g) $$ How many moles of \(\mathrm{CO}_{2}(g),\) collected at \(60 .^{\circ} \mathrm{C}\) and 774 torr total pressure, are produced by the complete reaction of \(10.0 \mathrm{~g}\) of \(\mathrm{CaCO}_{3}\) with acid? What volume does this wet \(\mathrm{CO}_{2}\) occupy? What volume would the \(\mathrm{CO}_{2}\) occupy at 774 torr if a desiccant (a chemical drying agent) were added to remove the water? (The vapor pressure of water at \(60 .\) C is \(149.4 \mathrm{~mm} \mathrm{Hg}\).)

Short answer

In summary, 0.1 moles of CO$_2$ are produced by the complete reaction of 10.0 g of CaCO$_3$ with acid. The wet CO$_2$ occupies a volume of 3.35 L. After adding a desiccant to remove the water, the CO$_2$ occupies a volume of 2.69 L at 774 torr.

Step-by-step solution

Calculate moles of CaCO3

First, we need to determine the moles of CaCO3 in 10.0 g. To do this, we will use the molar mass of CaCO3 that is 100.087 g/mol: \(n_{CaCO_3} = \frac{mass}{molar\,mass}\) \(n_{CaCO_3} = \frac{10.0\,g}{100.087\,g/mol} = 0.1\,mol\)

Moles of CO2 produced

The balanced equation of the reaction given is: \[CaCO_{3}(s)+2H^{+}(aq)\rightarrow Ca^{2+}(aq)+H_{2}O(l)+CO_{2}(g)\] Since 1 mole of CaCO3 produces 1 mole of CO2, the moles of CO2 produced are equal to the moles of CaCO3. \(n_{CO_2} = 0.1\,mol\)

Calculate the partial pressure of CO2

We have to find the partial pressure of CO2 in the gas mixture. Total pressure is given as 774 torr, and the vapor pressure of water at 60°C is 149.4 mm Hg. Let's convert the vapor pressure of water to torr (1 torr = 1 mm Hg): \(P_{H_2O} = 149.4\,mm\,Hg = 149.4\,torr\) Now, using Dalton's law, we can find the partial pressure of CO2: \(P_{total} = P_{CO_2} + P_{H_2O}\) \(P_{CO_2} = P_{total} - P_{H_2O} = 774\,torr - 149.4\,torr = 624.6\,torr\)

Calculate the volume of wet CO2

Now, we can use the ideal gas law to find the volume of wet CO2: \(PV = nRT\) Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin. First, let's convert the temperature to Kelvin: \(T = 60°C + 273.15 = 333.15\,K\) Next, convert the pressure to atm (1 atm = 760 torr): \(P_{CO_2} = \frac{624.6\,torr}{760\,torr/atm} = 0.8216\,atm\) Now, we can find the volume: \(V = \frac{nRT}{P} = \frac{0.1\,mol \times 0.0821\,L\,atm/mol\,K \times 333.15\,K}{0.8216\,atm} = 3.35\,L\) So, the volume of wet CO2 is 3.35 L.

Calculate the volume of CO2 after removing water

To find the volume of CO2 after removing the water with a desiccant, we can use the ideal gas law again. This time, we use only the pressure of CO2 without the pressure of water vapor: Since temperature and moles of CO_2 remain the same, we can directly calculate the volume: \(V = \frac{nRT}{P} = \frac{0.1\,mol \times 0.0821\,L\,atm/mol\,K \times 333.15\,K}{1.0216\,atm} = 2.69\,L\) Thus, the volume of CO2 after removing the water is 2.69 L.

Key concepts

Chemical Reactions

Chemical reactions are fundamental processes in chemistry where substances, known as reactants, transform into new substances called products. In any given reaction, the law of conservation of mass dictates that matter cannot be created or destroyed. This principle is crucially important when balancing chemical equations, ensuring that the number of atoms for each element is equal on both sides of the equation.

For example, the reaction between calcium carbonate (CaCO3) and aqueous acid to form calcium ions (Ca2+), water (H2O), and carbon dioxide (CO2) gas, can be depicted as follows:\[\mathrm{CaCO}_{3}(s)+2\mathrm{H}^{+}(aq) \rightarrow \mathrm{Ca}^{2+}(aq)+\mathrm{H}_{2}\mathrm{O}(l)+\mathrm{CO}_{2}(g)\]

This balanced equation shows a one-to-one stoichiometric relationship between CaCO3 and CO2. Hence, for each mole of CaCO3 that reacts, one mole of CO2 is produced. Understanding this stoichiometric relationship is key to solving problems involving chemical reactions.

Ideal Gas Law

The ideal gas law is a crucial equation in chemistry that relates four physical properties of a gas: pressure (P), volume (V), temperature (T), and the amount of substance in moles (n). It can be expressed as:\[PV = nRT\]

where R is the ideal gas constant, which has different values depending on the units of pressure, volume, and temperature. In the context of the carbon dioxide production reaction, the ideal gas law helps us calculate the volume occupied by CO2 gas at certain conditions of temperature and pressure.

When dealing with gases, it's important to account for the presence of other gases or vapors, such as water vapor, as they contribute to the total pressure (Dalton's law of partial pressures). By using the ideal gas law, we can isolate the behavior of the desired gas and compute its volume, as seen in the given exercise. To perform accurate calculations, it's necessary to convert all measurements to compatible units, like converting Celsius to Kelvin for temperature and torr to atmospheres for pressure.

Molar Mass

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It serves as a bridge between the microscopic scale (atoms or molecules) and the macroscopic scale (grams).

To determine the amount of substance in moles, one must divide the given mass by the substance's molar mass:\[n = \frac{\text{mass}}{\text{molar mass}}\]

In our case, the molar mass of CaCO3 is 100.087 g/mol. By using this value, we can calculate how many moles are in a given mass of CaCO3. For a 10.0 g sample, there are 0.1 moles, which directly correlates to the amount of CO2 produced, thanks to the stoichiometry of the reaction. Understanding molar mass is necessary for solving many problems in chemistry, especially those involving conversions between mass and moles in the context of chemical reactions.