Acids and bases can be thought of as chemical opposites...

Acids and bases can be thought of as chemical opposites (acids are proton donors, and bases are proton acceptors). Therefore, one might think that $K_{\mathrm{a}}=1 / K_{\mathrm{b}} .$ Why isn't this the case? What is the relationship between $K_{\mathrm{a}}$ and $K_{\mathrm{b}}$ ? Prove it with a derivation.

Key Concepts

Base Dissociation Constant (Kb)

Kb quantifies the extent to which a base accepts a proton from water, generating hydroxide ions. For the reaction B + H2O ? BH+ + OH?, it measures the equilibrium position and, thereby, the strength of the base. A higher Kb indicates a stronger base, and it is defined similarly to Ka but for the proton-accepting member of a conjugate acid–base pair.

Relationship Between Ka and Kb

For a conjugate acid–base pair, the product of the acid dissociation constant (Ka) and the base dissociation constant (Kb) is equal to the ion-product constant of water (Kw). This arises from considering the equilibria of the acid dissociation and the base's reaction with water. Multiplying the two equilibrium expressions leads to cancellation of the common species and results in Ka × Kb = Kw. Therefore, rather than being reciprocals, Ka and Kb are related by the water ionization constant, implying that Ka = Kw / Kb and vice versa.

Ion-Product Constant of Water (Kw)

Kw is the equilibrium constant for the self-ionization of water, defined by the reaction H2O ? H+ + OH?, and it is equal to [H+][OH?]. At 25°C, Kw has a constant value of approximately 1.0 × 10?¹?. This constant links the acid and base dissociation constants for conjugate acid–base pairs in aqueous solution.

Acid Dissociation Constant (Ka)

This constant measures the extent to which an acid donates its proton to water, forming hydronium ions. It is defined for the reaction HA + H2O ? H3O+ + A? and reflects the strength of the acid—the higher the value, the stronger the acid. Understanding Ka is crucial in acid–base equilibria as it quantifies the dissociation process in solution.

Transcript

00:01 And this problem, we're asked what the relationship between ka and kb is.

00:05 And so we'll begin solving this problem by writing out ka and kb just by definition.

00:11 And so ka for some reaction, ha plus h2o forming h3o plus and the congia base of ha is equal to a concentration of a minus times a concentration of h3o plus.

00:34 Over the concentration of h .a.

00:39 Kb.

00:41 Or that is associated with the not reverse reaction, but for the reaction of the conjugate base a -minus with water, which forms hydroxide ion and the initial acid h -a, that is the concentration of o .h.

01:01 Minus times the concentration of h -a.

01:04 Over the concentration of a minus.

01:10 And so this can be just written out by definition.

01:13 And now we're going to combine these two expressions in order to gain some understanding of the relationship between ka and kb.

01:21 And so i'm going to just rewrite exactly what we had over here.

01:26 So ka, the mass action expression here.

01:36 And kb is equal to oh minus ha over a minus.

01:48 And what we're going to do here is we're going to rearrange kb.

01:52 We're going to try and get an expression for a minus over ha, so that way we can replace it over here.

02:02 So, rewriting kb, we can say that a minus over the concentration of ha is equal to the concentration of oh minus over kb...