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Chapter 10: Problem 121

A \(4.00-g\) sample of a mixture of \(C a O\) and \(B a O\) 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 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 moles of CO2 reacted were approximately 0.0312. The mass percentage of CaO in the mixture is approximately 11.37%.

Step by step solution

01

Determine Initial Moles of CO2

Use the ideal gas law to find the initial moles of \(\mathrm{CO}_2\). At a pressure of \(97.33 \text{ kPa}\), volume of \(1.00 \text{ L}\), and temperature \(25^{\circ} \text{C}\) (which is \(298 \text{ K}\)), the initial moles can be calculated using \(PV = nRT\). Assuming \(R = 8.314 \text{ J/mol}\cdot\text{K}\), convert the pressure from kPa to Pa: \(97.33 \times 1000 \text{ Pa}\). Solve for \(n\): \[ n = \frac{PV}{RT} = \frac{97,330 \times 1.00}{8.314 \times 298} = \frac{97,330}{2475.172} \approx 0.0393 \text{ moles}\]
02

Determine Final Moles of CO2

Repeat the ideal gas law calculation for the final pressure of \(20.0 \text{ kPa}\). Convert \(20.0 \text{ kPa}\) to \(20,000 \text{ Pa}\). Using the same equation, \[ n_{\text{final}} = \frac{20,000 \times 1.00}{8.314 \times 298} = \frac{20,000}{2475.172} \approx 0.00808 \text{ moles}\]
03

Calculate Reacted Moles of CO2

Subtract the final moles of \(\mathrm{CO}_2\) from the initial moles to find the moles that have reacted: \[ n_{\text{reacted}} = 0.0393 - 0.00808 \approx 0.0312 \text{ moles}\]
04

Setup Equations for Masses

The reaction stoichiometry shows \(1 \text{ mole of CO}_2\) reacts with either \(1 \text{ mole of CaO}\) or \(1 \text{ mole of BaO}\). Suppose \(x\) is the moles of \(\text{CaO}\) and \(y\) is the moles of \(\text{BaO}\), then:\[x + y = 0.0312\]Also, calculate the mass of the sample assuming it is entirely \(\text{CaO}\) and \(\text{BaO}\): \[56.08x + 153.33y = 4.00\]
05

Solve the System of Equations

You have two equations:\[1. x + y = 0.0312\]\[2. 56.08x + 153.33y = 4.00\]Solving these equations, substitute for \(y = 0.0312 - x\) in the second equation:\[56.08x + 153.33(0.0312 - x) = 4.00\]\[56.08x + 4.7878 - 153.33x = 4.00\]Simplifying, solve for \(x\):\[-97.25x = -0.7878\]\[x \approx 0.0081 \text{ moles of CaO}\]
06

Calculate Mass and Percentage of CaO

Convert \(x = 0.0081 \text{ moles}\) of \(\text{CaO}\) to grams using its molar mass (56.08 g/mol):\[\text{mass of } \text{CaO} = 0.0081 \times 56.08 \approx 0.4549 \text{ g}\]The percentage of \(\text{CaO}\) in the mixture is:\[\frac{0.4549}{4.00} \times 100\% \approx 11.37\%\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law
The Ideal Gas Law is fundamental in understanding the behavior of gases. It relates the pressure, volume, temperature, and number of moles of a gas using the equation: \( PV = nRT \). Here, \( P \) is the pressure in pascals, \( V \) is the volume in liters, \( n \) is the number of moles, \( R \) is the universal gas constant (8.314 \( ext{J/mol} \cdot ext{K} \)), and \( T \) is the temperature in Kelvin.

To solve problems involving the Ideal Gas Law, remember to:
  • Convert all units to their respective SI units - pressure in pascals, volume in liters, and temperature in Kelvin.
  • Use the formula \( n = \frac{PV}{RT} \) to find the number of moles when pressure, volume, and temperature are known. This calculation is crucial in determining the amount of gas participating in a reaction.
For example, in the exercise, to find the initial moles of \( ext{CO}_2 \), you would use \( P = 97,330 \) Pa, \( V = 1.00 \) L, and \( T = 298 \) K. Inserting these into the equation yields the initial moles of \( ext{CO}_2 \) before the reaction begins.
Chemical Reactions
Chemical reactions describe the process where substances, known as reactants, transform into different substances, called products. In the exercise, carbon dioxide \( ext{CO}_2 \) reacts with calcium oxide \( ext{CaO} \) and barium oxide \( ext{BaO} \) to form calcium carbonate \( ext{CaCO}_3 \) and barium carbonate \( ext{BaCO}_3 \).

Key aspects to consider during a chemical reaction, particularly for stoichiometric calculations, include:
  • The balanced chemical equation, which ensures the same number of each type of atom on both sides of the equation.
  • Mole-to-mole conversions, which help in translating moles of one substance into moles of another using the coefficients from the balanced equation.
  • Conservation of mass, which states that matter is neither created nor destroyed in a chemical reaction, allowing you to compute total reactants' mass equals total products' mass.
In the exercise, the stoichiometry tells us that each mole of \( ext{CO}_2 \) can react with one mole of either \( ext{CaO} \) or \( ext{BaO} \), allowing for calculations of reacted moles to determine how much of each compound is present.
Mole Calculations
Mole calculations are a core part of stoichiometry, allowing for the conversion between mass, moles, and numbers of particles. Understanding and applying mole concepts helps in quantifying reactants and products in chemical reactions.

Here's a brief on important mole calculations:
  • To convert from moles to grams, use the formula: \( ext{mass} = ext{moles} imes ext{molar mass} \).
  • To find the percentage composition of a compound in a mixture, use: \( rac{ ext{mass of component}}{ ext{total mass of mixture}} imes 100 \% \).
  • Use known quantities (like total mass or total moles) to deduce unknowns through system of equations, especially useful in mixtures.
In the given problem, after determining moles of \( ext{CaO} \), you convert it to grams using its molar mass 56.08 g/mol, thereby allowing the computation of the mass percentage of \( ext{CaO} \) in the mixture. This step helps link the chemical calculations to real-world quantities, enabling practical applications of theoretical chemistry knowledge.

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Most popular questions from this chapter

Consider the sample of gas depicted here. What would the drawing look like if the volume and temperature remained constant while you removed enough of the gas to decrease the pressure by a factor of \(2 ?\) [Section 10.3\(]\) (a) It would contain the same number of molecules. (b) It would contain half as many molecules. (c) It would contain twice as many molecules. (d) There is insufficient data to say.

Indicate which of the following statements regarding the kinetic-molecular theory of gases are correct. (a) The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1 / 2} \cdot(\mathbf{b})\) The gas molecules are assumed to exert no forces on each other. (c) All the molecules of a gas at a given temperature have the same kinetic energy. (d) The volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. (e) All gas molecules move with the same speed if they are at the same temperature.

Calcium hydride, \(\mathrm{CaH}_{2}\), reacts with water to form hydrogen gas: $$ \mathrm{CaH}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+2 \mathrm{H}_{2}(g) $$ This reaction is sometimes used to inflate life rafts, weather balloons, and the like, when a simple, compact means of generating \(\mathrm{H}_{2}\) is desired. How many grams of \(\mathrm{CaH}_{2}\) are needed to generate \(145 \mathrm{~L}\) of \(\mathrm{H}_{2}\) gas if the pressure of \(\mathrm{H}_{2}\) is \(110 \mathrm{kPa}\) at \(21^{\circ} \mathrm{C} ?\)

Hurricane Wilma of 2005 is the most intense hurricane on record in the Atlantic basin, with a low-pressure reading of 882 mbar (millibars). Convert this reading into (a) atmospheres, \((\mathbf{b})\) torr, and \((\mathbf{c})\) inches of \(\mathrm{Hg}\).

(a) Place the following gases in order of increasing average molecular speed at \(300 \mathrm{~K}: \mathrm{CO}, \mathrm{SF}_{6}, \mathrm{H}_{2} \mathrm{~S}, \mathrm{Cl}_{2}, \mathrm{HBr}\) (b) Calculate the rms speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\). (c) Calculate the most probable speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\).
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