Problem 122 - Chemistry: The Central Science

Question

A 4.00-g sample of a mixture of \(\mathrm{CaO}\) and \(\mathrm{BaO}\) is placed in a \(1.00-\mathrm{L}\) vessel containing \(\mathrm{CO}_{2}\) gas at a pressure of 730 torr and a temperature of \(25^{\circ} \mathrm{C}\). The \(\mathrm{CO}_{2}\) reacts with the \(\mathrm{CaO}\) and \(\mathrm{BaO}\), forming \(\mathrm{CaCO}_{3}\) and \(\mathrm{BaCO}_{3}\). When the reaction is complete, the pressure of the remaining \(\mathrm{CO}_{2}\) is 150 torr. (a) Calculate the number of moles of \(\mathrm{CO}_{2}\) that have reacted. (b) Calculate the mass percentage of \(\mathrm{CaO}\) in the mixture.

Step-by-Step Solution

Answer

0.0313 moles of CO₂ have reacted, and the mass percentage of CaO in the mixture is 10.30%.

Step 1: Find the initial moles of CO₂

To find the initial moles of CO₂, we can use the ideal gas law: \(PV = nRT\), where P is pressure, V is volume, n is moles, R is the gas constant, and T is temperature in Kelvin. Rearrange the formula to solve for n: \(n = \frac{PV}{RT}\) The initial pressure of CO₂ is given as 730 torr. To convert this to atmospheres, divide by 760 torr/atm: \(730 \text{ torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 0.9605 \text{ atm}\) Now we can plug in the values: - P = 0.9605 atm - V = 1.00 L - R = 0.0821 \(\frac{\text{L atm}}{\text{mol K}}\) (Ideal gas constant) - T = 25°C + 273.15 = 298.15 K \[n_\text{initial} = \frac{(0.9605 \text{ atm})(1.00 \text{ L})}{(0.0821 \frac{\text{L atm}}{\text{mol K}})(298.15 \text{ K})} = 0.0394 \text{ moles of CO}_{2}\]

Step 2: Find the final moles of CO₂

After the reaction, the pressure of CO₂ is given as 150 torr. Convert to atmospheres: \(150 \text{ torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 0.1974 \text{ atm}\) Using the same method as before, plug in the values to the ideal gas law equation: - P = 0.1974 atm - V = 1.00 L - R = 0.0821 \(\frac{\text{L atm}}{\text{mol K}}\) - T = 298.15 K \[n_\text{final} = \frac{(0.1974 \text{ atm})(1.00 \text{ L})}{(0.0821 \frac{\text{L atm}}{\text{mol K}})(298.15 \text{ K})} = 0.0081 \text{ moles of CO}_{2}\]

Step 3: Calculate the number of moles of CO₂ that have reacted

To find the moles of CO₂ that have reacted, subtract the final moles of CO₂ from the initial moles of CO₂: \[n_\text{reacted} = n_\text{initial} - n_\text{final} = 0.0394 - 0.0081 = 0.0313 \text{ moles of CO}_{2}\] So, 0.0313 moles of CO₂ have reacted.

Step 4: Calculate the moles of CaO and BaO in the mixture

In the reaction, CO₂ reacts with CaO and BaO to form CaCO₃ and BaCO₃. The balanced equations are: \[1)\,\text{CaO} + \text{CO}_{2} → \text{CaCO}_{3}\] \[2)\,\text{BaO} + \text{CO}_{2} → \text{BaCO}_{3}\] Since both reactions consume 1 mole of CO₂ per 1 mole of CaO/BaO, the total moles of CaO and BaO in the mixture are equal to the moles of CO₂ reacted: 0.0313 moles.

Step 5: Calculate the mass percentage of CaO in the mixture

Let x = moles of CaO in the mixture. Then, the moles of BaO in the mixture will be 0.0313 - x. Next, convert moles to mass using molar mass: Mass of CaO (g) = x moles × 56.08 g/mol (Molar mass of CaO = 56.08 g/mol) Mass of BaO (g) = (0.0313 - x) moles × 153.33 g/mol (Molar mass of BaO = 153.33 g/mol) The sum of the masses of CaO and BaO should be equal to the total mass of the mixture (4.00 g). This gives us the equation: \(56.08x + 153.33(0.0313 - x) = 4.00\) Now, solve the equation for x: \[\begin{aligned} 56.08x + 153.33(0.0313 - x) &= 4.00 \\ 56.08x + 4.8003 - 153.33x &= 4.00 \\ -97.25x &= -0.8003 \\ x &= 0.00823 \end{aligned}\] Now we can calculate the mass percentage of CaO: \[\frac{\text{mass of CaO}}{\text{total mass}} = \frac{(0.00823 \text{ moles})(56.08\frac{\text{g}}{\text{mol}})}{4.00\text{ g}}\times 100\%= 10.30\%\] So, the mass percentage of CaO in the mixture is 10.30%.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. Given by the formula \(PV = nRT\), where \(P\) denotes pressure, \(V\) is volume, \(n\) stands for moles, \(R\) represents the ideal gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin.

For solving problems involving the ideal gas law, it's vital to ensure that the pressure is in atmospheres and the temperature is in Kelvin. When given other units, such as torr for pressure, a conversion is necessary (1 atm = 760 torr). In the provided exercise, the initial and final pressures of the \(CO_2\) gas were converted from torr to atmospheres to solve for the moles before and after the reaction.

Moreover, the ideal gas law assumes that gases behave ideally, which means gas particles do not attract or repel each other and don't occupy space. Although real gases do not perfectly comply with these assumptions, the ideal gas law offers a close approximation for many conditions, particularly under standard temperature and pressure.

Mole Concept

The mole concept is central to chemistry as it provides a bridge between the atomic scale and the macroscopic world. One mole is defined as exactly \(6.022 \times 10^{23}\) particles (Avogadro's number) of the substance, which could be atoms, molecules, ions, or other entities depending on the context.

In relation to the exercise, understanding the mole concept is essential for converting between moles and grams. By knowing the molar mass of a substance (in grams per mole), we can calculate how many moles are present in a given mass or conversely, how many grams are represented by a certain number of moles. This is pivotal when computing the amount of \(CO_2\) that has reacted, and subsequently, the individual amounts of \(CaO\) and \(BaO\) present in the initial sample.

Why is understanding the mole concept important?

Knowing the mole concept allows chemists to count particles by weighing, which is practical and facilitates stoichiometric calculations in chemical reactions.

Percent Composition

Percent composition refers to the mass percent of each element in a compound or a mixture. It's calculated by dividing the mass of a particular component by the total mass and then multiplying by 100%. This concept is vital when dealing with mixtures or compounds as it leads to an understanding of the composition and can aid in determining empirical and molecular formulas.

In the problem at hand, after determining the number of moles of \(CaO\) and \(BaO\) that reacted with \(CO_2\), the mass percentage of \(CaO\) in the original mixture could be calculated. You find the mass of \(CaO\) by multiplying the moles of \(CaO\) by its molar mass. The total mass of the mixture is known (4.00 g), allowing the determination of the mass percentage of \(CaO\) after solving the algebraic equations obtained from the stoichiometry of the reactions.

Practical Applications of Percent Composition

Knowledge of percent composition is essential not only for chemical synthesis but also for quality control in various industries, such as pharmaceuticals and food manufacturing, where the components' proportions can affect product efficacy and safety.