Problem 30 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A student was investigating the following equilibrium reaction Which has an equilibrium constant of 0.220 at \(800^{\circ} \mathrm{C}\) \\[ \mathrm{CaCO}_{3}(\mathrm{s}) \rightleftharpoons \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) \\] and did four experiments. (a) \(0.2 g\) of \(\mathrm{CaCO}_{3}(\mathrm{s})\) was heated to \(800^{\circ} \mathrm{C}\) in a \(1.0 \mathrm{dm}^{3}\) container (b) \(2.0 \mathrm{g}\) of \(\mathrm{CaCO}_{3}(\mathrm{s})\) was heated to \(800^{\circ} \mathrm{C}\) in a \(1.0 \mathrm{dm}^{3}\) container (c) \(0.2 g\) of \(\mathrm{CaCO}_{3}\) (s) was heated to \(800^{\circ} \mathrm{C}\) in a \(500 \mathrm{cm}^{3}\) container (d) \(2.0 g\) of \(\mathrm{CaCO}_{3}(\mathrm{s})\) was heated to \(800^{\circ} \mathrm{C}\) in a \(500 \mathrm{cm}^{3}\) container The pressure of \(\mathrm{CO}_{2}\) (g) measured in each case was ( 1 ) 0.18 bar, (ii) 0.22 bar, (iii) 0.22 bar, \((\text { iv) } 0.22\) bar. Explain these observations.

Step-by-Step Solution

Answer

Experiments (ii), (iii), and (iv) reached equilibrium with \( P_{\text{CO}_2} = 0.22 \) bar. Experiment (i) did not fully reach equilibrium; pressure was \( 0.18 \) bar.

Step 1: Understand the Reaction

The equilibrium reaction given is \( \text{CaCO}_3 (s) \rightleftharpoons \text{CaO} (s) + \text{CO}_2 (g) \). It involves solids \( \text{CaCO}_3 \) and \( \text{CaO} \), and a gaseous product \( \text{CO}_2 \). The system reaches equilibrium with a constant \( K = 0.220 \) at \( 800^{\circ} \text{C} \).

Step 2: Equilibrium Constant Expression

The equilibrium constant for this reaction involves only the concentration of \( \text{CO}_2 \), since the concentration of solids is not included in the equilibrium expression. Therefore, the expression is \( K = P_{\text{CO}_2} = 0.220 \), where \( P_{\text{CO}_2} \) is the partial pressure of carbon dioxide.

Step 3: Analyze Each Experiment

The experiments differ by the mass of \( \text{CaCO}_3 \) and the volume of the container. However, since \( K = P_{\text{CO}_2} \) for equilibrium and \( K = 0.220 \), adjustments in the amounts or volumes will not influence the equilibrium pressure of \( \text{CO}_2 \) as long as the reaction reaches equilibrium.

Step 4: Examine Pressure Measurements

Experiments (ii), (iii), and (iv) all yielded a \( \text{CO}_2 \) pressure of 0.22 bar, matching the equilibrium constant, indicating these systems have reached equilibrium. Experiment (i) had a pressure of 0.18 bar, indicating it did not reach equilibrium fully, likely due to a limiting amount of \( \text{CaCO}_3 \) or insufficient reaction time.

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K \), is a fundamental concept in chemical equilibrium. It represents the ratio of the concentration of the products to the concentration of the reactants, each raised to the power of their respective coefficients in the balanced equation. For the reaction \( \text{CaCO}_3 (s) \rightleftharpoons \text{CaO} (s) + \text{CO}_2 (g) \), the expression simplifies significantly because the reactants and some products are solids.
Solids do not appear in the equilibrium expression because their concentration is constant, making the equilibrium constant solely dependent on the gaseous component, \( \text{CO}_2 \).

Since only \( \text{CO}_2 \) is gaseous, the equilibrium constant \( K \) becomes equated to its partial pressure, given as \( P_{\text{CO}_2} = 0.220 \). This means that at equilibrium and at 800°C, the partial pressure of \( \text{CO}_2 \) is 0.220 bar. The determination of \( K \) helps predict the direction of the reaction and assess whether a given system has reached equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle is a great tool for predicting how a change in conditions can affect a system at equilibrium. This principle states that if an external change is applied to a system at equilibrium, the system will adjust in such a way as to counteract the change and re-establish a new equilibrium.

In the context of our experiments, you may wonder whether changes in mass or volume would affect equilibrium. However, a key insight from Le Chatelier's Principle is that, in this case involving a solid-gas equilibrium with an invariant gas concentration, changing the mass of a solid does not disturb the equilibrium since solids don't appear in the equilibrium expression.
Similarly, according to this principle, increasing the container volume would decrease the system pressure, seeking to produce more \( \text{CO}_2 \) to regain equilibrium pressure, yet due to constant \( K \), the partial pressure will adjust back to match \( K \). Thus, in our given experiments, different mass or volume variations (aside from reaction time factors) mostly do not impact the final pressure detected if equilibrium was reached.

Partial Pressure

The concept of partial pressure is crucial in understanding gas behavior in mixtures and chemical reactions involving gases. In the context of our equilibrium reaction, partial pressure represents the pressure exerted by \( \text{CO}_2 \) exclusively, in the presence of any other gases or solids in the container. Partial pressures help define real behaviors of gases as predicted by the ideal gas law.

For this system, shown in the experiments, the partial pressure answers the ultimate question of what the state of the gas (\( \text{CO}_2 \)) is in equilibrium. The equilibrium constant \( K = 0.220 \) is proven using an ideal representation where \( P_{\text{CO}_2} \approx K \).

• It helps measure the progress of the reaction: if the partial pressure of \( \text{CO}_2 \) is less than 0.220, the reaction needs to progress to reach equilibrium.
• If more, it achieved equilibrium or no forward reaction potential remains.

Therefore, detecting and understanding partial pressure is pivotal in examining reactions involving gases,

Gas Laws

Gas Laws play a critical role in deciphering the behavior of gases in chemical reactions, including equilibrium reactions. Central to our discussion is the Ideal Gas Law, which is articulated as \( PV = nRT \), where \( P \) is the pressure, \( V \) the volume, \( n \) the moles of gas, \( R \) the ideal gas constant, and \( T \) the temperature in Kelvin.
The Ideal Gas Law allows us to interrelate different variables impacting gases and gives a framework for understanding what happens to partial pressures within this equilibrium.

When applied to the reaction of calcium carbonate, the law assists in delineating how changing the volume of the container could potentially impact pressure readings due to the relationship between \( P \), \( V \), and \( n \):

• By decreasing the container's volume by half, in theory, the pressure should increase, but the system will adjust to bring the \( P_{\text{CO}_2} \) back to equilibrium (as projects in experiment \( iii \) and \( iv \)).
• In maintaining a constant volume, regardless of changing \( n \), the pressure remains constant when more reactants transform.

Hence, gas laws provide the tools necessary to anticipate and verify the behavior observed in equilibrium experiments.