Problem 115 - Chemistry: The Central Science

Question

A \(4.00-\mathrm{g}\) 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.

Step-by-Step Solution

Answer

The number of moles of CO₂ that have reacted is 0.03135 mol. The mass percentage of CaO in the mixture is 33.28%.

Step 1: Find the initial number of moles of CO₂

Start by finding the number of moles of CO₂ at the beginning, using the Ideal Gas Law, which states: PV = nRT Where - P is the pressure (converted to atm) - V is the volume - n is the number of moles - R is the gas constant (0.08206 L atm/mol K) - T is the temperature (converted to Kelvin) Initially, the pressure of CO₂ is 730 torr. First, convert this to atm: \( (730\,\text{torr})(\frac{1\,\text{atm}}{760\,\text{torr}}) = 0.9605\,\text{atm} \). The temperature is given as 25°C. Convert this to Kelvin: \( 25 + 273.15 = 298.15\,\text{K} \). The volume of the vessel is 1.00 L. Now, apply the Ideal Gas Law to calculate the number of moles of CO₂: \( n = \frac{PV}{RT} = \frac{(0.9605\,\text{atm})(1.00\,\text{L})}{(0.08206\,\text{L atm/mol K})(298.15\,\text{K})} = 0.03945\,\text{mol} \)

Step 2: Find the final number of moles of CO₂ after the reaction

After the reaction, the pressure of CO₂ is 150 torr, which needs to be converted to atm: \( (150\,\text{torr})(\frac{1\,\text{atm}}{760\,\text{torr}}) = 0.19737\,\text{atm} \). Apply the Ideal Gas Law to calculate the number of moles of CO₂ after the reaction: \( n = \frac{PV}{RT} = \frac{(0.19737\,\text{atm})(1.00\,\text{L})}{(0.08206\,\text{L atm/mol K})(298.15\,\text{K})} = 0.008097\,\text{mol} \)

Step 3: Calculate the number of moles of CO₂ reacted

Subtract the final number of moles of CO₂ from the initial number of moles: \( 0.03945\,\text{mol} - 0.008097\,\text{mol} = 0.03135\,\text{mol CO2 reacted} \)

Step 4: Determine moles of CO₂ reacted with CaO and BaO separately

Let x be the moles of CO₂ reacted with CaO; then, (0.03135 - x) is the moles of CO₂ reacted with BaO. The reaction with CaO is: CaO + CO₂ → CaCO₃, And the reaction with BaO is: BaO + CO₂ → BaCO₃ Now, calculate the mass of CaO and BaO in the sample using their molar masses: - Mass of CaO = mass of the sample - mass of BaO - Mass of CaO = 4.00 g - mass of BaO - Mass of CaO = x * (molar mass of CaO) = x * 56.079 g/mol - Mass of BaO = (0.03135 - x) * (molar mass of BaO) = (0.03135 - x) * 153.33 g/mol Since Mass of CaO + Mass of BaO = 4.00 g, we can create the following equation: \( x(56.079) + (0.03135 - x)(153.33) = 4.00 \)

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

Solve the equation from step 4 for x: \( x = 0.02374\,\text{mol of CO2 reacted with CaO} \) Now, calculate the mass of CaO in the sample: \( \text{mass of CaO} = 0.02374\,\text{mol} * 56.079\,\text{g/mol} = 1.331\,\text{g} \) Finally, calculate the mass percentage of CaO in the mixture: \( \text{mass percentage of CaO} = \frac{1.331\,\text{g}}{4.00\,\text{g}} * 100 = 33.28\% \) Thus, the mass percentage of CaO in the mixture is 33.28%.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry, advanced by combining several empirical gas laws. It is expressed as \( PV = nRT \), where:

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

The Ideal Gas Law ties together pressure, volume, temperature, and the amount of gas into one coherent equation. This equation is incredibly useful for calculating missing variables when you know the other three. In our exercise, we used this formula to determine the moles of \( \mathrm{CO}_{2} \) both initially and after the reaction occurred. Start by converting given units to the ideal units: convert pressure from torr to atm by dividing by 760, and temperature from Celsius to Kelvin by adding 273.15. Once everything is set, plug the values into the equation and solve for the unknown!

Reaction Calculations

Reaction calculations are crucial in stoichiometry as they help determine how reactants convert to products. It involves several steps:
Step 1: Calculate Initial and Remaining Moles
Before and after the reaction, the moles of \( \mathrm{CO}_{2} \) were calculated using the Ideal Gas Law. This gave us an understanding of the amount of \( \mathrm{CO}_{2} \) present initially and after forming calcium carbonate and barium carbonate.
Step 2: Determine Moles Reacted
By taking the difference between initial and remaining moles of \( \mathrm{CO}_{2} \), we derived the moles of \( \mathrm{CO}_{2} \) that reacted with \( \mathrm{CaO} \) and \( \mathrm{BaO} \).
Step 3: Identify Individual Reactions
Knowing the chemical equations \( \mathrm{CaO} + \mathrm{CO}_{2} \rightarrow \mathrm{CaCO}_{3} \) and \( \mathrm{BaO} + \mathrm{CO}_{2} \rightarrow \mathrm{BaCO}_{3} \), helps us understand complete conversion, which is crucial for further stoichiometric calculations like determining masses.

Gas Pressure Conversion

Gas pressure conversion is often necessary in chemical calculations to ensure all units are compatible, particularly when using the Ideal Gas Law. Pressure in atmospheres (atm) is standard for the law, but many situations, such as this exercise, present pressure in torr. Conversion is straightforward: 1 atm is equivalent to 760 torr.
Example Conversion
Given the initial pressure of \( \mathrm{CO}_{2} \) was 730 torr, it needed to be converted to atm to fit the Ideal Gas Law. Use the relation:\[\text{Pressure in atm} = \frac{\text{Pressure in torr}}{760}\]For 730 torr, this comes out to \(0.9605\, \text{atm}\). Similarly, for the final pressure of 150 torr, the conversion yields \(0.19737\, \text{atm}\). These conversions were pivotal in correctly applying the Ideal Gas Law to determine moles before and after the reaction, thus ensuring accuracy in our stoichiometric calculations.