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Chapter 10: Problem 121

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 \(97.33 \mathrm{kPa}\) 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 \(20.0 \mathrm{kPa}\). (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

Expert verified
The number of moles of CO2 that have reacted is ∆n = n_initial - n_final. The mass percentage of CaO in the mixture is calculated as (mass of CaO / 4.00 g) × 100, where mass of CaO is obtained using the moles of CaO (x) found in step 5.

Step by step solution

01

Find the initial moles of CO2

We can find the initial moles of CO2 gas using the Ideal Gas Law, which is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. The initial pressure of CO2 is 97.33 kPa, and the volume is 1.00 L. The temperature, given in Celsius, should be converted to Kelvin: T = 25 + 273.15 = 298.15 K. Using the pressure in atm (1 kPa = 0.00986923 atm), we have, P = 97.33 kPa × 0.00986923 atm/kPa ≈ 0.9606 atm Now we can find the initial moles of CO2: \(n_{initial} = \frac{PV}{RT}\)
02

Find the final moles of CO2

Similar to step 1, we will use the Ideal Gas Law to find the final moles of CO2 after reaction. The final pressure is given as 20.0 kPa. P = 20.0 kPa × 0.00986923 atm/kPa ≈ 0.19738 atm Again, using the Ideal Gas Law, \(n_{final} = \frac{PV}{RT}\)
03

Calculate the moles of CO2 that have reacted

The difference between the initial and final moles of CO2 will give us the moles of CO2 that have reacted: ∆n = n_initial - n_final
04

Determine the stoichiometry of the reaction

Since one mole of CO2 reacts with one mole of CaO and one mole of BaO, we can find the moles of CaO and BaO that reacted with CO2: moles of CaO = x moles of BaO = ∆n - x We also know that the mass of the mixture is 4.00 g, so: mass of CaO + mass of BaO = 4.00 g Using the molar masses of CaO (56.08 g/mol) and BaO (153.33 g/mol), we can write: 56.08x + 153.33(∆n - x) = 4.00
05

Solve for x

Rearrange the equation from step 4 to solve for x, the moles of CaO: x = (4.00 - 153.33∆n) / (56.08 - 153.33) Now substitute the value of ∆n obtained in step 3 and find the moles of CaO, x.
06

Calculate the mass percentage of CaO

Finally, we can find the mass percentage of CaO in the mixture using the mass of CaO and the total mass of the mixture: mass percentage of CaO = (mass of CaO / 4.00 g) × 100 Substitute the value of x found in step 5, and find the mass percentage of CaO.

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