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 \(\mathrm{CaO}\) 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.

Short answer

The mass percentage of CaO in the mixture is \(54.5\%\).

Step-by-step solution

Determine initial and final moles of CO2 in the vessel

To do this, we use the Ideal Gas Law: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. It is important to use consistent units, so we'll use P in atm, V in L, n in moles, R = 0.0821 Latm/molK, and T in K. First, convert the pressures from torr to atm: - Initial pressure: \(730 \, \frac{torr}{1} = 730 \, \frac{torr}{1} \cdot \frac{1 \, atm}{760 \, torr} = 0.9605 \, atm\) - Final pressure: \(150 \, \frac{torr}{1} = 150 \, \frac{torr}{1} \cdot \frac{1 \, atm}{760 \, torr} = 0.1974 \, atm\) Next, convert the temperature from Celsius to Kelvin: - Temperature: \(25^{\circ}C + 273.15 = 298.15 K\) Now, using the Ideal Gas Law, we can find the initial moles (n_initial) and final moles (n_final) of CO2: Initial moles of CO₂ (\(n_{initial}\)): \(n_{initial} = \frac{P_{initial} V}{RT} = \frac{0.9605 \, atm \cdot 1.00 \, L}{0.0821 \frac{L \, atm}{mol \, K} \cdot 298.15 \, K} = 0.0394 \, mol\) Final moles of CO₂ (\(n_{final}\)): \(n_{final} = \frac{P_{final} V}{RT} = \frac{0.1974 \, atm \cdot 1.00 \, L}{0.0821 \frac{L \, atm}{mol \, K} \cdot 298.15 \, K} = 0.00807 \, mol\)

Determine the moles of CO2 that have reacted

We can now find the moles of CO2 that have reacted by taking the difference between the initial and final moles of CO2: \(\Delta n = n_{initial} - n_{final} = 0.0394 \, mol - 0.00807 \, mol = 0.0313 \, mol\)

Determine mass of reacted CaO and BaO

One mole of CO2 reacts with one mole of CaO or one mole of BaO, resulting in one mole of CaCO3 or BaCO3, respectively. We can use stoichiometry to find the mass of reacted CaO and BaO. Let x be the mass of reacted CaO and (4.00 - x) be the mass of reacted BaO. \(\frac{x}{56.08} + \frac{4.00 - x}{153.33} = 0.0313 \, mol\) where 56.08 g/mol is the molar mass of CaO, and 153.33 g/mol is the molar mass of BaO.

Calculate the mass of reacted CaO (x)

Now, we solve the equation in Step 3 for x (mass of CaO): \(x = 2.18 \, g\)

Calculate the mass percentage of CaO in the mixture

Finally, we can find the mass percentage of CaO in the mixture: Mass percentage of CaO = \(\frac{2.18 \, g}{4.00 \, g} \times 100 \% = 54.5\%\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that helps us relate four properties of gases: pressure (P), volume (V), temperature (T), and amount of gas in moles (n). This relationship is expressed as \[ PV = nRT \],where \( R \) is the universal gas constant, with a value of \( 0.0821 \, \text{L atm} / \text{mol K} \). It allows us to predict the behavior of gases in different conditions by understanding how one variable affects the others.

• **Pressure (P)** is often measured in atmospheres (atm) or torr (760 torr = 1 atm).
• **Volume (V)** is the space occupied by the gas, usually measured in liters.
• **Temperature (T)** must always be in Kelvin for the equation to work, so convert from Celsius by adding 273.15.
• **Number of moles (n)** is how we quantify the amount of gas.

By rearranging the Ideal Gas Law, we can solve for any one of these variables if the others are known. For example, to find the number of moles when pressure, volume, and temperature are given, use \[ n = \frac{PV}{RT} \].Understanding this law is crucial when dealing with gases in reactions, as it helps us determine how much of a reactant or product we have in gaseous form.

Mole Calculations

Mole calculations are essential in stoichiometry, the area of chemistry focused on quantitative relationships in chemical reactions. A mole is a unit that measures the amount of substance, based on Avogadro's number, \(6.022 \times 10^{23}\) entities (atoms, molecules, etc.). This allows chemists to relate mass and quantity directly to the substance's chemical make-up. A key step in mole calculations is converting between mass and moles using the molar mass, which is the mass of one mole of a substance:

• **Molar Mass** is expressed in grams per mole (g/mol) and is unique to each compound.
• For example, the molar mass of \(\text{CaO}\) is 56.08 g/mol, meaning one mole of \(\text{CaO}\) weighs 56.08 grams.

To convert from mass to moles, use the formula \( n = \frac{mass}{\text{molar mass}} \).In the given exercise, mole calculations help us determine how much \(\text{CO}_{2}\) reacted by knowing its initial and final moles, and then using stoichiometry to relate these moles to the reactants \(\text{CaO}\) and \(\text{BaO}\).

Chemical Reactions

Chemical reactions involve the transformation of reactants into products. Understanding the stoichiometry of a reaction helps us predict the quantities of reactants consumed and products formed. For gas reactions, like the reaction of \(\text{CO}_{2}\) with \(\text{CaO}\) and \(\text{BaO}\), gases can be involved as products or reactants, making the Ideal Gas Law and mole calculations key.The reaction in the exercise forms calcium carbonate, \(\text{CaCO}_{3}\), and barium carbonate, \(\text{BaCO}_{3}\), when \(\text{CO}_{2}\) reacts with \(\text{CaO}\) and \(\text{BaO}\). This balanced chemical equation shows us the stoichiometry, where one mole of \(\text{CO}_{2}\) reacts with one mole of \(\text{CaO}\) or \(\text{BaO}\) to form one mole of carbonate.

• **Balancing Equations**: Ensure the same number of atoms for each element exists on both sides of the reaction.
• **Reaction Yield**: Knowing the stoichiometry allows determining maximum product yield from given reactants.

Through these concepts, chemical reactions can be effectively analyzed and understood, allowing prediction and manipulation of reactants and products.