(a) Calculate the reaction free energy of \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})\) when the partial pressures of \(\mathrm{N}_{2}, \mathrm{H}_{2}\), and \(\mathrm{NH}_{3}\) are \(1.0,4.2\), and 63 bar, respectively, and the temperarure is \(400 \mathrm{~K}\). For this reaction, \(K=41\) at \(400 \mathrm{~K}\). (b) Indicate whether this reaction mixture is likely to form reactants, is likely to form products, or is at equilibrium.

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. As a result, there's no net change in the concentration of reactants and products over time. Equilibrium does not mean that reactants and products are present in equal amounts – rather, it indicates that their ratios are constant.

When a system reaches equilibrium, it may appear to be static, but in reality, both reactants and products are continually being formed and consumed at equal rates. Understanding equilibrium is crucial in predicting the behavior of chemical reactions and formulating products in various industries.

The reaction quotient, Q, is a mathematical expression used to determine the direction in which a reaction will proceed at any point in time. It takes into account the current concentrations or partial pressures of the reactants and products. The value of Q can be compared with the equilibrium constant, K, to predict the shift that will occur in order to reach equilibrium.

For gaseous reactions, the reaction quotient is calculated using partial pressures. It is given by the partial pressures of the products raised to the power of their stoichiometric coefficients, divided by the partial pressures of the reactants raised to their respective powers. If Q is less than K, the reaction will proceed in the forward direction to form more products. If Q is greater than K, the reaction will shift to form more reactants.

The equilibrium constant, K, is a number that expresses the ratio of the concentration of products to reactants for a reversible reaction at chemical equilibrium and a constant temperature. K is constant for a given reaction at a specific temperature and is independent of the initial concentrations of reactants and products.

In the application, if the reaction quotient Q equals the equilibrium constant K, the reaction is at equilibrium. The value of K can help predict whether a reaction mixture will proceed to form more products or reactants as it attempts to reach equilibrium. K is determined experimentally and varies with temperature, so it’s essential to ensure that K and the reaction conditions (such as temperature) correspond accordingly.

Gibbs free energy, denoted as \(\Delta G\), is a thermodynamic quantity that predicts the direction of a chemical reaction. For reactions occurring at constant temperature and pressure, a negative \(\Delta G\) indicates that a process is spontaneous, meaning it can occur without any added energy. Conversely, a positive \(\Delta G\) implies that the process is non-spontaneous.

\(\Delta G\) also relates closely to the concepts of equilibrium and reaction quotient. It can be calculated from the standard Gibbs free energy change (\(\Delta G^\circ\)), which is based on standard conditions, along with the actual concentration or pressure of reactants and products, encapsulated in the reaction quotient (Q). The relationship between \(\Delta G\), \(\Delta G^\circ\), and Q is used to determine the spontaneity of the reaction under non-standard conditions.

Partial pressures are used in chemistry to describe the pressure exerted by a single gas when it is part of a mixture of gases. Each gas in a mixture exerts pressure as if it were the only gas present, and the total pressure of the mixture is the sum of the partial pressures of each individual gas.

Partial pressures are particularly relevant in the context of gaseous equilibria, where they are used to calculate the reaction quotient (Q) and equilibrium constants (K). When dealing with reactions involving gases, changes in partial pressures can affect both Q and K, thus influencing the direction in which the equilibrium will shift. Furthermore, the concept of partial pressures is also essential in applications ranging from respiratory physiology to the design of industrial chemical processes.