Assume that the change in concentration of \(\mathrm{COCl}_{2}\) is small enough to be neglected in the following problem. (a) Calculate the equilibrium concentration of all species in an equilibrium mixture that results from the decomposition of \(\mathrm{COCl}_{2}\) with an initial concentration of \(0.3166 \mathrm{M}\). \(\operatorname{COCl}_{2}(g) \rightleftharpoons \operatorname{CO}(g)+\mathrm{Cl}_{2}(g) \quad K_{c}=2.2 \times 10^{-10}\) (b) Confirm that the change is small enough to be neglected.

Understanding the use of an ICE table is crucial when studying chemical equilibrium. ICE stands for Initial, Change, and Equilibrium, which are the three stages of concentration levels in a chemical reaction reaching equilibrium. To accurately calculate the concentrations of reactants and products at equilibrium, one would start by filling out the ICE table with the known initial concentrations. In our case, the initial concentration of COCl_2 is 0.3166 M while that of CO and Cl_2 is 0 M.

The 'Change' row represents the shift in concentrations as reactants turn into products; these changes are commonly represented by a variable, typically 'x'. For every mole of COCl_2 that decomposes, one mole each of CO and Cl_2 form. Thus, the change for COCl_2 is '-x', and '+x' for both CO and Cl_2.

The 'Equilibrium' row then incorporates these changes to express the concentrations of all species when the reaction has reached equilibrium. In this problem, since it is assumed that x is small, the ICE table simplifies the problem and helps visualize the process of the reaction reaching equilibrium.

The equilibrium constant expression is derived from the law of mass action, which correlates the concentrations of reactants and products of a reaction at equilibrium to a constant, known as the equilibrium constant (Kc for concentration). For the decomposition of COCl_2, the expression is

\[ K_{c} = \frac{[\mathrm{CO}][\mathrm{Cl}_{2}]}{[\mathrm{COCl}_{2}]} \]
where the concentrations are those at equilibrium. Filling in the known values and assuming the change in concentration 'x' is small allows us to solve for 'x'. Doing so tells us the concentration of both CO and Cl_2 at equilibrium. This equation is pivotal as it holds the key to quantifying the extent of a reaction and understanding how different factors affect the equilibrium position.

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time, even though the reactions continue to occur. It does not imply that the reactants and products are present in equal amounts but that their ratio does not change. In our exercise, COCl_2 is in equilibrium with CO and Cl_2. At equilibrium, their concentrations remain constant, dictated by the equilibrium constant, Kc. Understanding equilibrium is essential not only for calculating concentrations, but also for predicting how changes in conditions, like temperature or pressure, will shift the equilibrium, based on Le Châtelier's Principle.

The reaction quotient, Q, plays a pivotal role in determining the direction a reaction will proceed to reach equilibrium. It is calculated using the same formula as the equilibrium constant, Kc, but with the current concentrations of the reactants and products, not those at equilibrium. The formula is:

\[ Q = \frac{[\mathrm{CO}][\mathrm{Cl}_{2}]}{[\mathrm{COCl}_{2}]} \]
By comparing Q to Kc, one can infer if the reaction will proceed forward or in reverse to achieve equilibrium. If Q < Kc, the forward reaction is favored; if Q > Kc, the reverse reaction is favored. In this exercise, Q starts at zero, as initially the product concentrations are zero, and as the reaction proceeds, Q increases until it equals Kc at equilibrium, indicating no further net change in concentrations.