Derive the relationship $$K=K_{\mathrm{c}} \times(R T)^{\Delta r_{\mathrm{B}}}$$ where \(K_{\mathrm{c}}\) is the equilibrium constant using molarities and \(\Delta n_{\mathrm{g}}\) is the change in the number of moles of gas in the reaction (see page 326). (Hint: Recall that \(P_{\Lambda}=n_{\Lambda} R T / V\) and \(\left.n_{A} / V=[\mathrm{A}] .\right)\)

The equilibrium constant (\( K \) is a fundamental concept in chemistry that quantifies the relative concentrations of products and reactants in a chemical reaction at equilibrium. It represents the point at which the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentration of substances over time. This constant is crucial because it helps predict the direction of the reaction and the extent to which reactants are converted into products.

For a general reaction represented as \(aA + bB \leftrightarrow cC + dD\), where \(a\), \(b\), \(c\), and \(d\) are the stoichiometric coefficients, the equilibrium constant \(K_c\) is defined using molarities (\(M\)) as follows:
\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

The square brackets denote the molarity, which is the concentration of the substance in moles per liter (mol/L). If the reaction involves gases, and the concentration is expressed in terms of partial pressure, the equilibrium constant is represented by \(K_p\).

The ideal gas law is an equation of state that describes the behavior of an ideal gas. It is given by the formula:
\[ PV = nRT \]
where \(P\) represents the pressure, \(V\) is the volume, \(n\) denotes the number of moles of gas, \(R\) is the universal gas constant, and \(T\) stands for the absolute temperature in Kelvin. This law helps in understanding how changes in temperature, pressure, and volume affect the state of an ideal gas.

In the context of equilibrium, the ideal gas law enables us to relate molar concentrations to partial pressures for reactions involving gases. By manipulating the ideal gas equation, one can connect the macroscopic properties of gases with microscopic details, such as the number of molecules and their average kinetic energy.

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution (\(mol/L\) or \(M\)). The formula to calculate molarity is given as:
\[ Molarity \left( M \right) = \frac{moles \; of \; solute}{liters \; of \; solution} \]

Understanding molarity is vital for solving problems that involve chemical reactions in solution, as it links the volume of the solution with the amount of substance dissolved in it. Particularly, when dealing with reaction equilibrium in solutions, molarity provides a convenient means to express concentrations that are needed when calculating the equilibrium constant.

Chemical reaction stoichiometry deals with the quantitative relationships between the reactants and products in a chemical reaction. It is the foundation for determining the proportions of chemicals to be combined when reacting and predicting the amount of products formed. Stoichiometry takes into account the conservation of mass and the coefficients in a balanced chemical equation, which represent the mole ratios of the substances involved.

When applied to equilibrium, stoichiometry helps us understand how shifts in the reaction conditions affect the composition of the equilibrium mixture. For instance, changes in the number of moles of reactants or products can alter the value of the equilibrium constant over temperature, as seen in the derivation of the relationship \(K = K_c \times(RT)^{\Delta r_B}\), where \(\Delta r_B\) signifies the change in the number of moles of gases during the reaction.