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Chapter 19: Problem 84

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 \(\left[\mathrm{CH}_{3} \mathrm{NH}_{2}\right]=0.098 \mathrm{M} ?\)

Short Answer

Expert verified
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 螖G掳 = -13.78 kJ/mol. At equilibrium, 螖G = 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 螖G formula, we find that 螖G = -11.25 kJ/mol.

Step by step solution

01

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-). \( CH_{3}NH_{2} + H_{2}O \rightleftharpoons CH_{3}NH_{3}^{+} + OH^{-}\)
02

Calculate 螖G掳 using Kb

The equilibrium constant for the base dissociation reaction, Kb, is related to the standard Gibbs free energy change (螖G掳) by the following equation: \( 螖G掳 = -RT \ln{K_b}\) , 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掳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 螖G掳.
03

Calculate the value of 螖G at equilibrium

At equilibrium, the value of 螖G equals 0. This means the system is in a state of minimum energy. Hence, 螖G = 0
04

Calculate 螖G using given concentrations

The relationship between 螖G, 螖G掳, and the reaction quotient (Q) is given by the formula: \( 螖G = 螖G掳 + RT \ln{Q}\) 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掳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 螖G using the expression: \( 螖G = 螖G掳 + RT \ln{Q}\) Plug in the calculated value of 螖G掳, and the reaction quotient, Q, to determine the final value of 螖G. Now you have successfully calculated the 螖G value for the given concentrations of H+, CH3NH3+, and CH3NH2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳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掳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.

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