Problem 78 - Chemistry: Principles and Reactions

Question

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 \(\left.n_{A} / V=[\mathrm{A}] .\right)\)

Step-by-Step Solution

Answer

Based on the provided solution, the derived relationship between the equilibrium constant \(K\) and other parameters is given by the equation: $$K = K_c \times (RT)^{\Delta r_B}$$ where \(K_c\) is the equilibrium constant using molarities, \(R\) is the gas constant, \(T\) is the temperature, and \(\Delta r_B\) is the change in the number of moles of gas in the reaction.

Step 1: Start with the expression for the equilibrium constant using partial pressures

First, we will write the expression for the equilibrium constant \(K\) using partial pressures of the species involved in the reaction. For a general reaction of the form: $$\mathsf{aA + bB}\mathop{\longleftrightarrow}[{}][{}]\mathsf{cC + dD}$$ the equilibrium constant is given by: $$K = \frac{P_C^c P_D^d}{P_A^a P_B^b}$$

Step 2: Rewrite the expression for partial pressures using the ideal gas law

In this step, we will rewrite the expression of the partial pressures of each species in terms of their moles and molarities, making use of the ideal gas law, as given by the hint: $$P_{\Lambda} = \frac{n_{\Lambda} RT}{V}$$ So, the partial pressures of all species become: $$P_A = \frac{n_A RT}{V}, \ P_B = \frac{n_B RT}{V}, \ P_C = \frac{n_C RT}{V}, \ P_D = \frac{n_D RT}{V}$$

Step 3: Substitute and cancel terms

Now, we will substitute the expressions for partial pressures from Step 2 into the expression for the equilibrium constant \(K\) from Step 1: $$K = \frac{\left(\frac{n_C RT}{V}\right)^c \left(\frac{n_D RT}{V}\right)^d}{\left(\frac{n_A RT}{V}\right)^a \left(\frac{n_B RT}{V}\right)^b}$$ We can then cancel out common terms in the numerator and the denominator: $$K = \frac{n_C^c n_D^d R^{c + d} T^{c + d}}{n_A^a n_B^b R^{a + b} T^{a + b}}$$

Step 4: Rewrite the expression in terms of molarities

Now, we will rewrite the expression of the moles of each species in terms of their molarities, using the definition of molarity as given in the hint: $$n_\mathrm{A} / V = [\mathrm{A}], \ n_\mathrm{B} / V = [\mathrm{B}], \ n_\mathrm{C} / V = [\mathrm{C}], \ n_\mathrm{D} / V = [\mathrm{D}]$$ So, the moles of all species become: $$n_A = [\mathrm{A}]V, \ n_B = [\mathrm{B}]V, \ n_C = [\mathrm{C}]V, \ n_D = [\mathrm{D}]V$$

Step 5: Substitute and simplify the expression

Finally, we will substitute the expressions for the moles of each species from Step 4 into the expression for the equilibrium constant \(K\) from Step 3: $$K = \frac{([\mathrm{C}]V)^c ([\mathrm{D}]V)^d R^{c + d} T^{c + d}}{([\mathrm{A}]V)^a ([\mathrm{B}]V)^b R^{a + b} T^{a + b}}$$ We can again cancel out common terms: $$K = \frac{[\mathrm{C}]^c [\mathrm{D}]^d}{[\mathrm{A}]^a [\mathrm{B}]^b} \times \frac{R^{c + d} T^{c + d}}{R^{a + b} T^{a + b}}$$ Now, let's define \(K_c\) as the equilibrium constant using molarities: $$K_c = \frac{[\mathrm{C}]^c [\mathrm{D}]^d}{[\mathrm{A}]^a [\mathrm{B}]^b}$$ And let's define \(\Delta r_B\) as the change in the number of moles of gas in the reaction: $$\Delta r_B = (c + d) - (a + b)$$ So we can re-write the expression for \(K\) as: $$K = K_c \times (RT)^{\Delta r_B}$$ This is the desired relationship we set out to derive for this exercise.

Key Concepts

Equilibrium Constant

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 \(K_p\).

Ideal Gas Law

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

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

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.