Question

A 4.00 -g sample of a mixture of \(\mathrm{CaO}\) and \(\mathrm{BaO}\) is placed in a 1.00 -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 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 CaO in the mixture.

Short answer

The number of moles of CO₂ that reacted can be calculated by finding the difference between the initial and final moles of CO₂ using the ideal gas law. Then, the moles of CO₂ that reacted correspond to the moles of CaO and BaO that reacted. By calculating the mass of the reacted CaO and BaO using their molar masses, we can determine the mass percentage of CaO in the mixture as the ratio of the mass of CaO to the total mass of the mixture multiplied by 100%.

Step-by-step solution

Calculate the initial moles of CO₂

We are given the initial pressure, volume, and temperature of the CO₂ gas. We can use the ideal gas law to find the initial moles of CO₂. The ideal gas law is: \[PV = nRT\] where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. We will use R = 62.36 L.torr/mol.K for this problem, and convert the given temperature to Kelvin. \[T(K) = 25+273.15 = 298.15 K\] Now we can solve for the initial moles of CO₂: \[n_{initial} = \frac{PV}{RT} = \frac{730 \, \text{torr} \cdot 1.00 \, \text{L}}{62.36 \, \frac{\text{L}\cdot\text{torr}}{\text{mol}\cdot\text{K}} \cdot 298.15 \, \text{K}}\]

Calculate the final moles of CO₂

We can now use the final pressure of CO₂ to calculate the final moles of CO₂ after the reaction occurred: \[n_{final} = \frac{PV}{RT} = \frac{150 \, \text{torr} \cdot 1.00 \, \text{L}}{62.36 \, \frac{\text{L}\cdot\text{torr}}{\text{mol}\cdot\text{K}} \cdot 298.15 \, \text{K}}\]

Calculate the moles of CO₂ that reacted

The number of moles of CO₂ that reacted can be calculated as the difference between the initial and final moles of CO₂: \[n_{reacted} = n_{initial} - n_{final}\]

Calculate the mass of CaO and BaO that reacted

Both CaO and BaO react with CO₂ in a 1:1 ratio. So, the moles of CO₂ that reacted corresponds to the moles of CaO and BaO that reacted. We can calculate the mass of the reacted CaO and BaO using their molar masses: \[m_{CaO} = n_{CaO}M_{CaO}\] \[m_{BaO} = n_{BaO}M_{BaO}\] where \(n_{CaO}\) and \(n_{BaO}\) are the moles of CaO and BaO, respectively, and \(M_{CaO}\) and \(M_{BaO}\) are their respective molar masses.

Determine mass percentage of CaO in the mixture

To determine the mass percentage of CaO in the mixture, we need to find the ratio of the mass of CaO to the total mass of the mixture and multiply by 100%: \[\% \, \text{CaO} = \frac{m_{CaO}}{m_{CaO} + m_{BaO}} \times 100\%\] By performing calculations in these steps, the mass percentage of CaO in the mixture can be determined.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle used to relate the pressure, volume, temperature, and number of moles of a gas in a closed system. It is represented by the equation:

\[PV = nRT\]
Where P is the pressure of the gas, V is the volume it occupies, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvins. The constant R can vary depending on the units of pressure and volume used, so it's crucial to use consistent units throughout the calculation. For example, in our problem, we used 62.36 L.torr/mol.K for R when the pressure is given in torr and the volume in liters. Understanding this equation allows us to determine unknown quantities of a gas when the other conditions are known.

The ideal gas law also underpins many stoichiometric calculations in chemistry. It helps us determine the amount of a reactant or product in a chemical reaction under certain conditions, as demonstrated in the given exercise. Applying this law faciliates transitions from theoretical knowledge to practical applications, such as predicting behavior of gases during reactions.

Molar Mass

Molar mass is a property that links the mass of a substance to its amount in moles. It is defined as the mass in grams of one mole of a substance. This value can be found on the periodic table for individual elements and is calculated for compounds by adding the molar masses of the respective elements. For example, calcium oxide (CaO) and barium oxide (BaO) have unique molar masses, as noted in the problem. The molar mass is essential for converting between grams and moles, which allows us to quantify how much of a substance is reacting or produced in a chemical reaction.

To find the molar mass of a compound like CaO, you would add the molar mass of calcium to that of oxygen. Similarly, for BaO, you add the molar masses of barium and oxygen. With knowledge of molar masses (M_CaO and M_BaO), one can convert the moles of reactants or products into masses, more practical for laboratory measurements, such as weighing substances.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products through the breaking and forming of chemical bonds. Each reaction is governed by a set of stoichiometric coefficients, which indicate the proportional amounts of each substance involved. For instance, the reaction of CO₂ with CaO and BaO to form CaCO₃ and BaCO₃ occurs in a 1:1 molar ratio, meaning one mole of CO₂ reacts with one mole of either CaO or BaO.

In the context of our exercise, understanding the stoichiometry of the reaction enabled us to deduce the amount of CaO and BaO that reacted by simply looking at how much CO₂ reacted. This is because stoichiometry provides a quantitative relationship between the reactants and products in a balanced chemical equation. Being adept at interpreting these relationships is pivotal for predicting the outcome of chemical reactions and for calculating the quantities of substances that participate in these reactions.

Mass Percentage Calculation

The mass percentage is a way of expressing the concentration of a component in a mixture. It is calculated as the mass of the component divided by the total mass of the mixture, multiplied by 100%. This calculation is particularly useful in chemistry for specifying the composition of mixtures and solutions. In our exercise, for instance, the mass percentage of CaO was determined by first calculating the individual masses of CaO and BaO that reacted and then dividing the mass of CaO by the total mass of the mixture.

This concept not only is important in pure chemistry but also has practical implications in fields like pharmacology, materials science, and nutrition. By mastering the mass percentage calculation, students can better understand and describe mixture compositions, an essential skill for many analytical and synthetic procedures in both laboratory and industrial settings.