6.2: Background

Before proceeding read sections 15.2: Equilibrium Constants and 15.3: Determining an Equilibrium Constant.  The generic expression for an equilibrium constant for the reaction \[aA+bB \rightleftharpoons cC+dD \] is: \[ K =\dfrac{[C]^c[D]^d}{[A]^a[B]^b} \; \\ \; \\ \text{(noting all species are at equilibrium concentrations)}\]

Note: The above equation describes the concentrations after the reaction has completed and the system has entered a state of equilibrium.  You can mix reactants in any proportions, and they will react until the above equation equals the equilibrium constant.  The Rice diagram below is used to predict how a reaction will proceed.

Rice Diagram

We typically use a RICE diagram to describe the change in concentration as a reaction proceeds from initial to final (equilibrium) concentrations.  For the above generic reaction, the RICE diagram is:

Table \(\PageIndex{2}\): Generic RICE Diagram

Reactants	aA             +	bB	⇌	cC             +	dD
Initial	[A]Initial	[B]Initial		[C]Initial	[D]Initial
Change	-ax	-bx		+cx	+dx
Equilibrium	[A]Eq	[B]Eq		[C]Eq	[D]Eq

noting that:

\[ \left [ A\right ]_{Eq}=\left [ A\right ]_{Initial}-ax \\ \left [B\right ]_{Eq}=\left [ B\right ]_{Initial}-bx \\ \left [ C\right ]_{Eq}=\left [C\right ]_{Initial}+cx \\ \left [D\right ]_{Eq}=\left [D\right ]_{Initial}+dx \label{15.3.3} \]

where,

x=extent of reaction

and

\[K=\frac{\left [C\right ]_{eq}^{c}\left [D\right ]_{eq}^{d}}{\left [A\right ]_{eq}^{a}\left [B\right ]_{eq}^{b}}\]

We will use the equilibria of a weak acid as our example because we can measure the hydronium ion concentration with a pH meter.

pH and Concentration

In this lab we will use the pH meter to measure the hydronium ion concentration. From section 4.5 (gen chem 1) we know

pH = -log[H₃O⁺]

[H₃O⁺] = 10^-pH = 1/10^pH

Weak Acid Equilibria

From section 3.5.1.2: Acid-Base Reactions we know that the Bronsted-Lowry definition of an acid is a proton donor, which can generically be represented as HA, and in aqueous solutions the proton is donated to water

\[HA(aq)+H_2O(l)⇌H_3O^+(aq)+A^-(aq)\]

At first glance this gives an equilibrium constant of

\[K=\frac{[H_{3}O^{+}]_{eq}[A^{-}]_{eq}}{[HA]_{eq}[H_{2}O]_{eq}}\]

but water is in excess (section 15.2.5) and so this reduces to

\[K_a=\frac{[H_{3}O^{+}]_{eq}[A^{-}]_{eq}}{[HA]_{eq}}\]

The water is typically removed from the equation and implicitly represented, where hydronium is written as H⁺ (implicitly meaning H₃O⁺ ), giving:

\[HA \rightleftharpoons H^+ + A^- \\ \; \\ \text{which results in the equilibrium expression of)} \\ \; \\  K_a=\frac{[H^{+}][A^{-}]}{[HA]} \]

Which results in the following RICE diagram.

Table \(\PageIndex{3}\): Weak Acid RICE Diagram

Reaction	\(HA\)	\(H^+\)	\(A^-\)
Initial	[HA]i	0	0
Change	-x	+x	+x
Equilibrium	[HA]i-x	x	x

Noting that \(x=10^{-pH}\) (at equilibrium) and substituting, gives\[K_a =\frac{(10^{-pH})^2}{[HA]_i-10^{-pH}}\]

So with a pH meter we can study equilibrium problems involving acids (and bases).  We are going to measure the pH of 5 different solutions and perform the following calculations:

1. Calculate K_a knowing [HA]_i (solutions 1-3)
2. Calculate K_a knowing [HA]_i and [NA]_i (solution 4)
3. Calculate [HA]_i knowing Ka (solution 5 - the unknown)  

 

Common Ion Effect

In solution 4 we are mixing equal volumes of 0.1M acetic acid (weak acid) and 0.1M sodium acetate (salt of weak acid section 3.4.2), which have the common ion acetate. Before mixing we would expect little of the acid to dissociate and all the salt to be dissociated.  So before mixing we expect

\[\begin{align}HC_2H_3O_2  &\rightleftharpoons  H^+ + C_2H_3O_2^- \\  NaC_2H_3O_2  &\rightarrow Na^+ + C_2H_3O_2^- \end{align}\]

According to LeChatlier's principle (section 15.6), a system shifts its equilibrium to consume a chemical species involved in a reaction if it is added to a system at equilibrium. If we add sodium acetate to the weak acidic acid we would expect the weak acid equilibria to shift to the left, and thus the common ion (acetate) inhibits the ionization of the weak acid (acetic acid). 

 

Table \(\PageIndex{4}\): Weak Acid and its salt RICE Diagram

Reaction	\(HA\)	\(H^+\)	\(A^-\)
Initial	[HA]i	0	[NaA]i
Change	-x	+x	+x
Equilibrium	[HA]i-x	x	[NaA]i+x

\[K_a=\frac{x\left ( [NaA]_i+x] \right )}{[HA]_i-x} \; \\  \; \\ \; =\frac{x\left ( [NaA]_i] \right )}{[HA]_i}\]

This is saying that the common ion (A^-) added by the salt pushes the equilibrium of the acid (HA) to the left and inhibits the ionization of the acid, driving the pH up. 

Note, the special case for equimolar mixtures of an acid and its conjugate base (where the acetate ion concentration equals the acetic acid concentration ) gives an easy way to measure the equilibrium constant.

\[K_a=\frac{x\left (\cancel{[NaA]_i]} \right )}{\cancel{[HA]_i}} = x = 10^{-pH} \\ \; \\ \underbrace{K_a = 10^{-pH}}_{when \; [HA] = [A^-]}\]

So we have a very easy way to measure the acid ionization constant of a weak acid.