At a particular temperature, \(K=2.0 \times 10^{-6}\) for the reaction $$ 2 \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) $$ If \(2.0\) moles of \(\mathrm{CO}_{2}\) is initially placed into a \(5.0-\mathrm{L}\) vessel, calculate the equilibrium concentrations of all species.

The concept of chemical equilibrium is foundational to understanding how reactions proceed in chemistry. At a basic level, equilibrium represents a state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentration of reactants and products remains constant over time, although not necessarily equal.

When a reaction reaches this state, it is said to be 'at equilibrium.' Importantly, reaching equilibrium does not mean that the reaction has stopped; rather, reactants continue to form products, and products continue to form reactants at equal rates. Understanding this dynamic allows us to predict how various changes to the system, such as altering concentrations, temperature, or pressure, could shift the equilibrium position, a concept often described by Le Chatelier's Principle.

An ICE Table, standing for Initial, Change, and Equilibrium, is a strategic tool used to track the changes in concentrations of reactants and products during a dynamic chemical process. It is particularly useful in visualizing how equilibrium is established in a reaction system.

The 'Initial' section of the table lists the starting concentrations of reactants and products. Often, as in the original exercise, the initial concentration of products is zero in a reaction that starts only with reactants. The 'Change' section expresses the amounts by which the reactants and products will increase or decrease as the system approaches equilibrium. This is symbolized with a variable, often 'x,' which signifies this shift. Finally, the 'Equilibrium' section reveals the concentrations of reactants and products once the system has reached its equilibrium state. By setting up an ICE table, students can systematically calculate unknown concentrations at equilibrium using the equilibrium constant of the reaction.

The equilibrium constant, represented by K or Kc for concentrations, is a numerical value that conveys the ratio of the concentration of products to the concentration of reactants at equilibrium for a reversible chemical reaction. Its definition is specific to the reaction at hand, derived from the balanced chemical equation.

For the example given, the equilibrium constant expression is \[ K_c = \frac{[\mathrm{CO}]^2[\mathrm{O_2}]}{[\mathrm{CO_2}]^2} \]. Calculating these constants involves algebraic manipulation where we substitute equilibrium concentrations from the ICE table into the Kc expression. Solving for the remaining variables allows us to determine the equilibrium concentrations of all species in the reaction. The difficulty often lies in solving a cubic or a polynomial equation, which may require approximation techniques or numeric methods for solutions. Understanding these calculations helps to forecast how the system would respond under different conditions, which is extremely pertinent in chemical industries and research.