Problem 84 - Chemistry The Central Science

Question

The \(K_{b}\) for methylamine \(\left(\mathrm{CH}_{3} \mathrm{NH}_{2}\right)\) at \(25^{\circ} \mathrm{C}\) is given in Appendix \(\mathrm{D}\) . (a) Write the chemical equation for the equilibrium that corresponds to \(K_{b}\) . (b) By using the value of \(K_{b},\) calculate \(\Delta G^{\circ}\) for the equilibrium in part (a). (c) What is the value of \(\Delta G\) at equilibrium? (d) What is the value of \(\Delta G\) when \(\left[\mathrm{H}^{+}\right]=6.7 \times 10^{-9} M,\left[\mathrm{CH}_{3} \mathrm{NH}_{3}^{+}\right]=2.4 \times 10^{-3} \mathrm{M}\) and \(\left[\mathrm{CH}_{3} \mathrm{NH}_{2}\right]=0.098 \mathrm{M} ?\)

Step-by-Step Solution

Answer

The chemical equilibrium equation corresponding to Kb is: \( CH_{3}NH_{2} + H_{2}O \rightleftharpoons CH_{3}NH_{3}^{+} + OH^{-}\). Using the given Kb value, we find that ΔG° = -13.78 kJ/mol. At equilibrium, ΔG = 0. For the given concentrations, we calculate [OH-] = 1.49 x 10^{-6} M, and Q = 3.62 x 10^{-2}. Substituting these values into the ΔG formula, we find that ΔG = -11.25 kJ/mol.

Step 1: Write the chemical equation for the equilibrium reaction corresponding to Kb

To start, let's write down the base dissociation reaction of methylamine (CH3NH2) when it reacts with water (H2O). Since methylamine is a weak base, it will accept a proton (H+) from water, forming its conjugate acid (CH3NH3+) and hydroxide ions (OH-). \( CH_{3}NH_{2} + H_{2}O \rightleftharpoons CH_{3}NH_{3}^{+} + OH^{-}\)

Step 2: Calculate ΔG° using Kb

The equilibrium constant for the base dissociation reaction, Kb, is related to the standard Gibbs free energy change (ΔG°) by the following equation: \( ΔG° = -RT \ln{K_b}\) , where \(R\) is the gas constant (= 8.314 J/(mol K)) and \(T\) is the temperature in Kelvin. We are given that the temperature is 25°C, which is equal to 298.15 K. The value of Kb for methylamine can be found in Appendix D. Kb = 4.4 x 10^(-4). Now, we plug in these values into the formula to calculate ΔG°.

Step 3: Calculate the value of ΔG at equilibrium

At equilibrium, the value of ΔG equals 0. This means the system is in a state of minimum energy. Hence, ΔG = 0

Step 4: Calculate ΔG using given concentrations

The relationship between ΔG, ΔG°, and the reaction quotient (Q) is given by the formula: \( ΔG = ΔG° + RT \ln{Q}\) We are given the concentrations of H+ ([H+]), CH3NH3+ ([CH3NH3+]), and CH3NH2 ([CH3NH2]). However, we should note that the base dissociation reaction involves OH- ions instead of H+ ions. Therefore, we need to use the relation between the concentrations of H+ ions and OH- ions: \(K_w = [H^+][OH^-]\), where Kw is the ion product of water (1.0 x 10^(-14) at 25°C). Now, let's calculate the concentration of OH- ions: \([OH^{-}] = \frac{K_w}{[H^+]} = \frac{1.0 \times 10^{-14}}{6.7 \times 10^{-9}}\) Next, we determine the reaction quotient, Q: \(Q = \frac{[CH_{3}NH_{3}^{+}][OH^{-}]}{[CH_{3}NH_{2}]}\) Plug in the given concentrations of the species involved in the reaction and the calculated value of [OH-] into the equation. Finally, we can calculate ΔG using the expression: \( ΔG = ΔG° + RT \ln{Q}\) Plug in the calculated value of ΔG°, and the reaction quotient, Q, to determine the final value of ΔG. Now you have successfully calculated the ΔG value for the given concentrations of H+, CH3NH3+, and CH3NH2.

Key Concepts

Base Dissociation Constant

The base dissociation constant, commonly denoted as \( K_b \), is a crucial parameter when discussing the dissociation of weak bases like methylamine (\( \text{CH}_3\text{NH}_2 \)). In equilibrium chemistry, \( K_b \) measures the strength of a base in a solution, indicating how well a base dissociates into its ions. For methylamine, the relevant chemical equilibrium reaction can be expressed as follows: \[ \text{CH}_3\text{NH}_2 + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{NH}_3^+ + \text{OH}^- \] The larger the \( K_b \) value, the stronger the base, meaning it dissociates more fully in water. Understanding \( K_b \) helps predict the extent of dissociation and the concentration of ions in solution. This is vital for calculating the equilibrium concentrations involved in the reaction, which is a fundamental aspect of chemical equilibrium. To find the base dissociation constant of methylamine, consult Appendix D or reliable chemical databases, which will provide the standard value at defined conditions like 25°C.

Gibbs Free Energy

Gibbs free energy, represented by \( \Delta G \), is a thermodynamic function that helps determine whether a reaction is spontaneous. In our context, \( \Delta G^\circ \) represents the standard Gibbs free energy change for the base dissociation at equilibrium. The equation used to relate \( K_b \) and \( \Delta G^\circ \) is: \[ \Delta G^\circ = -RT \ln{K_b} \] Where \( R \) is the universal gas constant (8.314 J/(mol K)), and \( T \) is the temperature in Kelvin (298.15 K for 25°C). Once \( K_b \) is found, it can be plugged into this equation to find \( \Delta G^\circ \). A negative value of \( \Delta G^\circ \) signifies a spontaneous process under standard conditions. At equilibrium, \( \Delta G \) is zero because the system has reached a state of minimum free energy. For cases where concentrations differ from equilibrium values, \( \Delta G \) can be computed as: \[ \Delta G = \Delta G^\circ + RT \ln{Q} \] Here, \( Q \) is the reaction quotient, providing insight into how far a system is from reaching equilibrium.

Chemical Equilibrium

Chemical equilibrium is a state in a reversible chemical reaction where the rates of the forward and reverse reactions equalize, leading to no net change in concentrations of reactants and products over time. For methylamine dissociating in water, this equilibrium can be examined through the reaction: \[ \text{CH}_3\text{NH}_2 + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{NH}_3^+ + \text{OH}^- \] At equilibrium, the concentration of products and reactants remains constant, but they don't necessarily have to be equal. Instead, their relationship is expressed through the equilibrium constant, \( K_b \). Factors affecting equilibrium include changes in concentration, pressure, and temperature, explained by Le Chatelier's principle. This principle predicts how a shift in one of these conditions can affect the position of equilibrium in a reaction. Learning about these concepts is important to manipulate reaction conditions intentionally and predictably, whether in a laboratory setting or industrial processes.