Problem 89 - Chemistry The Molecular Nature of Matter

Question

A \(0.200 \mathrm{~mol}\) sample of a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{CO}_{2}\) with a total pressure of 845 torr was exposed to an excess of solid \(\mathrm{CaO},\) which reacts with \(\mathrm{CO}_{2}\) according to the equation $$ \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{CaCO}_{3}(s) $$ After the reaction was complete, the pressure of the gas had dropped to 322 torr. How many moles of \(\mathrm{CO}_{2}\) were in the original mixture? (Assume no change in volume or temperature.)

Step-by-Step Solution

Answer

The number of moles of \(\mathrm{CO}_{2}\) in the original mixture is approximately 0.085 mol.

Step 1: Understand the Chemical Reaction

Know that calcium oxide (CaO) reacts with carbon dioxide (CO2) to form calcium carbonate (CaCO3). This reaction removes CO2 gas from the mixture, and the reaction goes to completion due to excess CaO.

Step 2: Relate Pressure Change to Moles of CO2

Understand that the pressure drop from 845 torr to 322 torr is due to the removal of CO2 gas upon reaction. The pressure of the remaining gas corresponds to the partial pressure of N2 since it does not react with CaO.

Step 3: Calculate Moles of N2 Using Ideal Gas Law

Since there's no volume or temperature change, use the proportional relationship from the ideal gas law: \(P_{1} / n_{1} = P_{2} / n_{2}\). Let \(n_{1}\) be the total initial moles (0.200 mol), \(P_{1}\) be the initial pressure (845 torr), and \(P_{2}\) be the final pressure (322 torr) to solve for \(n_{2}\), the moles of N2.

Step 4: Calculate Initial Moles of CO2

Subtract the moles of N2 (from Step 3) from the total initial moles to find the moles of CO2 initially present in the mixture.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. It is commonly expressed as PV = nRT, where R is the universal gas constant. In practical applications, it helps us to understand and predict how gases will behave under different conditions.

When the volume and temperature of a gas are held constant, as in our exercise, the pressure of the gas is directly proportional to the number of moles. This is due to the fact that more molecules will exert more force on the walls of the container. This relationship allows us to calculate the amount of gas before and after a chemical reaction if we know the corresponding pressures and assume ideal behavior.

Chemical Reactions

Chemical reactions involve breaking and forming of bonds between atoms which results in the transformation of substances. In our exercise, the chemical reaction is between calcium oxide (CaO) and carbon dioxide (CO2) to form calcium carbonate (CaCO3). This is a typical example of a solid-gas reaction where the reactant in gaseous form (CO2) is removed from the gas mixture upon reacting with a solid reactant (CaO).

Understanding the stoichiometry of the reaction is crucial, as it allows us to determine the amount of reactants consumed and products formed. In the given problem, the reaction goes to completion because of the excess CaO, implying that all the CO2 present reacts and is subtracted from the gas mixture.

Partial Pressure of Gases

Partial pressure is the pressure that a gas in a mixture would exert if it alone occupied the entire volume. According to Dalton's Law of Partial Pressures, the total pressure of a mixture of gases is the sum of the partial pressures of the individual gases. In the context of the given problem, before the reaction, the total pressure is the sum of the partial pressures of N2 and CO2.

After the reaction has occurred and CO2 has been removed, the pressure measured is solely due to the N2 since it does not react with CaO. Thus, the final pressure after the reaction gives us the partial pressure of the N2, which we then use in our stoichiometric calculations.

Mole Concept

The mole concept is a fundamental principle that allows chemists to count particles by weighing them. A mole (abbreviated as mol) is defined as the amount of substance that contains the same number of particles as there are atoms in 12 grams of carbon-12. This number is Avogadro's number (\(6.022 \times 10^{23}\) particles/mol).

In stoichiometry, moles provide a bridge between the mass of substances and the number of particles or volume of gases at standard temperature and pressure (STP). In the given exercise, the change in pressure after the completion of the reaction allows us to determine the number of moles of CO2 present initially by using the proportionality of pressure and moles (from the Ideal Gas Law) and subtracting the moles of N2 from the total initial moles.