For the reaction $$\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g)$$ at \(700^{\circ} \mathrm{C}, K_{\mathrm{c}}=0.534 .\) Calculate the number of moles of \(\mathrm{H}_{2}\) that are present at equilibrium if a mixture of 0.300 mole of \(\mathrm{CO}\) and \(0.300 \mathrm{~mole}\) of \(\mathrm{H}_{2} \mathrm{O}\) is heated to \(700^{\circ} \mathrm{C}\) in a 10.0 -L container.

The equilibrium constant, represented as Kc, quantifies the relative concentrations of reactants and products at equilibrium in a chemical reaction. It is a pivotal concept in chemical equilibrium calculations because it signifies the point at which the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.

In mathematical terms, Kc is expressed as the ratio of the product of the concentrations of the reaction products to the product of the concentrations of the reactants, each raised to the power of their respective coefficients in the balanced equation. For example, for a general reaction \[aA + bB \rightleftharpoons cC + dD,\]

the equilibrium constant Kc would be written as \[Kc = \frac{[C]^{c} [D]^{d}}{[A]^{a} [B]^{b}}.\]

Where the square brackets denote the concentration of the compounds in mol/L, and the letters with subscripts represent the coefficients from the balanced chemical equation. It's crucial to remember that Kc is constant only for a given temperature; changes in temperature can result in a different Kc value.

The reaction quotient, Q, is a measure that tells us how far a reaction has to go to reach equilibrium. It considers the current concentrations or partial pressures of the reactants and products, much like the equilibrium constant, but for conditions that are not necessarily at equilibrium.

For the same general reaction \[aA + bB \rightleftharpoons cC + dD,\]

the reaction quotient Q is given by \[Q = \frac{[C]^{c} [D]^{d}}{[A]^{a} [B]^{b}}.\]

If we compare Q to the known Kc for the reaction at a specific temperature, we can predict which way the reaction will proceed to reach equilibrium. If Q < Kc, the reaction will move forward, producing more products. Conversely, if Q > Kc, the reaction will shift to produce more reactants. When Q = Kc, the reaction is at equilibrium, and no net change in the concentrations of reactants and products will occur.

Le Chatelier's principle is a qualitative tool used to predict how a system at equilibrium reacts to changes in concentration, pressure, or temperature. The principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change.

Here's how Le Chatelier's principle applies to different changes in conditions:

• Concentration: If the concentration of a reactant is increased, the equilibrium will shift to produce more product, and vice versa.
• Pressure: Increasing the pressure by decreasing the volume of a gaseous system will shift the equilibrium toward the side with fewer moles of gas. Similarly, when pressure is decreased, the equilibrium shifts to the side with more moles of gas.
• Temperature: For an exothermic reaction, increasing the temperature causes the equilibrium to shift towards the reactants, while for an endothermic reaction, the equilibrium shifts towards the products.

Understanding this principle aids in controlling chemical reactions, allowing chemists to obtain desired products efficiently.

When dealing with gas-phase reaction equilibrium, it's essential to recognize that it is governed by the same principles as other types of chemical equilibria, but with additional considerations due to the gaseous state of the reactants and products.

In a gas-phase equilibrium, the equilibrium constant may be expressed in terms of partial pressures (Kp) instead of concentrations (Kc). However, the exercise provides Kc, so we'll focus on that representation. For gases, the ideal gas law (\[PV = nRT\]) can be applied to calculate concentrations from moles and volume, where P is pressure, V is volume, n is moles, R is the gas constant, and T is temperature.

In our example problem, when the gaseous mixture of CO and H2O is heated, the number of moles at equilibrium can be found by considering the change in concentrations (\[x\]) resulting from the shifting equilibrium position. By calculating these moles in a gas-phase reaction, we can connect the conceptual understanding of Kc and Le Chatelier's principle to solve practical, numerical problems in chemistry.