Question

\begin{aligned} &\text { In your own words, define or explain (a) } K_{w} \text { , (b) } K_{a r}\\\ &\text { (d) } \mathrm{pK}_{\mathrm{b}}\\\ &\text { (c) } \mathrm{pOH} \text { , } \end{aligned}

Short answer

In summary: (a) $K_w$ refers to the ion-product constant for water, representing the equilibrium between the self-ionization of water into hydronium and hydroxide ions. It is useful in determining if a solution is acidic, neutral, or basic. (b) $K_{ar}$, the acid dissociation constant, measures the degree of dissociation of an acid, with larger values indicating stronger acids. (c) pOH is the negative logarithm of hydroxide ion concentration and is useful in comparing the basicity of different solutions. It is related to pH through the equation pH + pOH = 14. (d) pKb is a measure of the strength of the conjugate base of a weak acid, defined as the negative logarithm of the base dissociation constant (Kb). Lower pKb values indicate stronger bases.

Step-by-step solution

(a) Definition of Kw

The ion-product constant for water (Kw) refers to the equilibrium constant for the self-ionization of water. In the self-ionization process, one water molecule donates a proton (H+) to another water molecule, forming a hydronium ion (H3O+) and a hydroxide ion (OH-). Mathematically, it can be represented as: \[ K_w = [\mathrm{H_3O^+}][\mathrm{OH^-}] \] At 25°C, the value of Kw is \(1.0 \times 10^{-14}\). This constant is significant in determining if a solution is acidic, neutral, or basic based on the concentrations of hydrogen and hydroxide ions.

(b) Definition of Kar

Kar, or the acid dissociation constant, is an equilibrium constant that gives a measure of the degree of dissociation of an acid. The larger the Kar value, the stronger the acid as it will dissociate more readily into its constituent ions. In a general acid-base reaction, a weak acid (HA) donates a proton to a water molecule, forming a hydronium ion (H3O+) and its conjugate base (A-). The acid dissociation constant for this process can be expressed as: \[ K_{a} = \frac{[\mathrm{H_3O^+}][\mathrm{A^-}]}{[\mathrm{HA}]} \] The Kar value can be used to determine the extent of dissociation of a weak acid and also to predict the direction of the acid-base equilibrium.

(c) Definition of pOH

The pOH of a solution is defined as the negative logarithm (base 10) of the concentration of hydroxide ions (OH-) in the solution. Mathematically, it can be represented as: \[ \mathrm{pOH} = -\log_{10} [\mathrm{OH^-}] \] The pOH value allows us to easily compare the relative basicity or alkalinity of various solutions. Additionally, pOH and pH are related as follows: pH + pOH = pKw . At 25°C, pH + pOH = 14, which means if one knows the value of pOH, they can easily determine the pH of the solution, and vice versa.

(d) Definition of pKb

The pKb is a measure of the strength of the conjugate base of a weak acid, which typically acts as a weak base. It is defined as the negative logarithm (base 10) of the base dissociation constant (Kb) for the base dissociation reaction: \[ \mathrm{pK_b} = -\log_{10} K_{b} \] The Kb value represents the equilibrium constant for the base dissociation reaction, where the weak base (B) reacts with a water molecule to form a hydroxide ion (OH-) and its conjugate acid (HB+). The base dissociation constant can be expressed as: \[ K_{b} = \frac{[\mathrm{HB^+}][\mathrm{OH^-}]}{[\mathrm{B}]} \] A lower pKb value corresponds to a stronger base, as its base dissociation constant (Kb) is higher, indicating a greater tendency for the base to dissociate and form hydroxide ions.

Key concepts

Kw (ion-product constant for water)

Understanding the ion-product constant for water, denoted as Kw, is fundamental in the study of acid-base chemistry. It represents the equilibrium constant for the self-ionization of water, a process wherein water molecules split into ions. The equation for this phenomenon is H2O —→ H+ + OH-. This dissociation is extremely important because it's the basis on which pH is defined. At standard temperature (25°C), Kw has a constant value of 1.0 × 10-14.

This constant helps us find out whether a solution is acidic, neutral, or basic. In pure water, [H⁺] equals [OH^-], each with a concentration of 1.0 × 10-7 mol/L. But when an acid or base is added, the balance shifts, changing these concentrations and consequently affecting the pH or pOH of the solution.

Ka (acid dissociation constant)

The acidity of a solution is characterized by the acid dissociation constant, Ka, which quantifies the extent to which an acid can donate protons to the solvent, typically water, producing hydronium ions (H3O+) and its conjugate base. The general reaction for this process is HA + H2O —→ H3O+ + A-. The value of Ka points to the strength of an acid; a high Ka means the acid readily loses its proton and is thus considered strong.

The dissociation constant provides insight into the acid's behavior in water. Weak acids, which have a small Ka, only partially dissociate, whereas strong acids fully dissociate in solution. It can be mathematically expressed as Ka = [H3O+][A-]/[HA]. This constant is used extensively in pH calculations and is a key factor in predicting the outcome of acid-base reactions.

pOH (negative logarithm of hydroxide ion concentration)

In basic solutions, the hydroxide ion concentration is a critical factor, and pOH provides a simplified method of expressing its level. Just as pH relates to the acidity of a solution, pOH reflects its basicity by being the negative logarithm of the hydroxide ion concentration. The formula used to calculate it is pOH = -log10 [OH-], making it easier to compare the alkalinity of different solutions.

The intrinsic relationship between pOH and pH is described by pH + pOH = pKw, which equals 14 at 25°C. This equation essentially tells us that knowing one of these values will automatically determine the other, a vital aspect in the analysis of a solution's properties. Lower pOH values indicate higher basicity, just as lower pH signifies stronger acidity.

pKb (negative logarithm of base dissociation constant)

Bases have their own dissociation constant, similar to acids, denoted as Kb, and pKb is the negative logarithm that quantifies the strength of weak bases. When a base accepts a proton from water, it increases the water's OH- concentration, and the equilibrium of this reaction determines the base's dissociation constant. The mathematical expression for the equilibrium constant is Kb = [HB+][OH-]/[B], where B signifies the base, and HB+ is the conjugate acid formed.

The pKb value, calculated by the formula pKb = -log10 Kb, inversely indicates the base strength: a lower pKb means a stronger base, implying a greater tendency of the base to dissociate and produce OH- ions. Understanding the pKb aids in predicting the pH of the solution when a particular base is dissolved in water.