For the reaction below, \(K_{\mathrm{p}}=1.16\) at \(800 .{ }^{\circ} \mathrm{C}\). $$ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) $$ If a 20.0-g sample of \(\mathrm{CaCO}_{3}\) is put into a \(10.0\) - \(\mathrm{L}\) container and heated to \(800 .{ }^{\circ} \mathrm{C}\), what percentage by mass of the \(\mathrm{CaCO}_{3}\) will react to reach equilibrium?

Understanding the equilibrium constant, represented as Kp, is essential for predicting the extent of a reaction when dealing with gases at a constant temperature. In the context of chemical equilibrium, Kp is a numerical value that provides insight into the relative quantities of reactants and products present when a reaction reaches its equilibrium state.

For a given reversible reaction involving gases, the Kp expression is formulated using the partial pressures of the gaseous components. It specifically applies to reactions where temperature remains unchanged because the equilibrium constant itself is sensitive to temperature changes. The key thing to remember is that Kp does not include solids or liquids, only gases.

The Kp for the reaction \[ \mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s) + \mathrm{CO}_{2}(g) \]is given by \[ K_{\mathrm{p}} = \frac{{P_{\mathrm{CO}_2}}}{{1}} \]where \( P_{\mathrm{CO}_2} \) is the partial pressure of \( \mathrm{CO}_2 \) gas. In our problem, we were given that Kp equals 1.16 at 800°C. By using this constant, we were able to solve the problem and determine what percentage of calcium carbonate reacts in a closed system.

Le Chatelier's principle provides a predictive quality to chemical reactions when a system at equilibrium is subjected to changes such as concentration, pressure, volume, or temperature. It states that if an external stress is applied to a system in equilibrium, the system will adjust itself to minimize the impact of that disturbance.

This principle explains how shifts in equilibrium occur. For instance, if we increase the pressure of a system by decreasing the volume, a reaction that produces fewer moles of gas is favored. Oppositely, if the pressure is decreased, the reaction moves toward producing more gas.

Temperature changes are also significant. For exothermic reactions, an increase in temperature shifts the equilibrium towards the reactants, while for endothermic reactions, the equilibrium shifts towards the products. In the solution given, if we were to change the container size or the temperature, Le Chatelier's principle would guide us to anticipate how the equilibrium would shift. However, since we maintained constant conditions, the reaction's behavior was straightforward and predictable using Kp.

Partial pressure is a concept in chemistry that refers to the pressure a single gas in a mixture of gases would exert if it occupied the entire volume of the mixture at the same temperature. It plays a crucial role in calculating Kp since Kp is derived from the partial pressures of the gaseous reactants and products.

In our exercise, the partial pressure of \( \mathrm{CO}_2 \), \( P_{\mathrm{CO}_2} \), is a key element in determining the equilibrium state of the reaction. It is directly proportional to the number of moles of \( \mathrm{CO}_2 \) and can be calculated using the ideal gas law: \[ P_{\mathrm{CO}_2} = \frac{{n_{\mathrm{CO}_2}}}{{V}} \cdot R \cdot T \]With this relationship, we can link moles of gas to pressure, which enables us to use the equilibrium constant effectively. It's also worth noting that the partial pressure is independent of the presence of other gases, which allows for straightforward calculations in a mixture.

The ICE table method is a systematic way to organize and calculate changes in concentrations or pressures of reactants and products throughout a chemical reaction. ICE stands for Initial, Change, and Equilibrium, each representing a state of the reaction components.

Initially, you list the initial concentrations (or pressures) of reactants and products. 'Change' denotes the changes these components undergo as the reaction moves towards equilibrium. Lastly, 'Equilibrium' states the final concentrations once equilibrium is reached. By creating an ICE table, we easily visualize and solve for unknown variables in equilibrium reactions.

In the problem, we created an ICE table that helped us solve for the unknown value of x, which represented the change in moles of \( \mathrm{CaCO}_3 \). After setting up the table, we incorporated the given Kp value and partial pressure calculations to solve for the equilibrium concentrations, and in turn, determine the percentage of \( \mathrm{CaCO}_3 \) that reacted. The ICE table is an invaluable tool in equilibrium problems like these and greatly simplifies complex calculations.