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pH of a mixture of acids and bases

Neutralisation (Protolysis)

According Broenstedt, the neutralization reaction (acid-base reaction; protolysis) between an acid $HB_1$ and a base $B_2$ is the exchange of a proton:

Neutralisation (Protolysis): $HB_1$ $+$ $B_2$ $\rightleftarrows$ $HB_2$ $+$ $B_1$

Examples: $HCl$ $+$ $CH_3COO^-$ $\rightleftarrows$ $CH_3COOH+Cl^-$ $HCO_3^-$ $+$ $OH^-$ $\rightleftarrows$ $CO_3^{2-}$ $+$ $H_2O$

Position of the equilibrium

The equilibrium constant for the protolysis reaction depends upon the acidity constants in question (in the case of strong acids or strong bases, one consideres constants of couples $H_3O^+/H_2O$ of $pK_a=-1,74$ respectively $H_2O/OH^-$ of $pK_a=15,74$ ), indeed: $K$ $=$ $\frac{[HB_2][B_1]}{[HB_1][B_2]}$ = $\frac{[HB_2]}{[H_3O^+][B_2]}\cdot \frac{[H_3O^+][B_1]}{[HB_1]} $ = $\frac{K_{a1}}{K_{a2}}$

Equilibrium constant of a protolysis reaction $K$ $=$ $\frac{K_{a1}}{K_{a2}}$

We can look (see exercises) in which case the protolysis reaction is (almost) complete ( $\rightarrow $ instead of $\rightleftarrows$):

Strong acid 1/ strong base 2

The reaction between a strong acid and a strong base is always complete.

Example: $2\;L\;HCl\; 0.1 M$ are mixed with $1\;L\;NaOH \;0.1 M$: Record of moles :

$H_3O^+$ + $OH^- $$\rightarrow$ $2H_2O$
Before:0.2 0.1
After:0.1 0

Finally ($Na^+$ et $Cl^-$ do not influence the pH): $pH$ $=$ $-log\frac{n_{H_3O^+}}{V}$ = $-log\frac{0,1}{3}$ $=$ $ 0.48$

Weak acid 1/ strong base 2

The reaction between a weak acid 1 and a strong base 2 is complete ($\rightarrow$), if $pK_{a1}\lt 11$

Example: $1\;L\;CH_3COOH\; 0.1 M$ are mixed with $2\;L\;NaOH \;0.1 M$: Record of moles :

$CH_3COOH$ + $OH^- $$\rightarrow$ $CH_3COO^-$+$H_2O$
Before:0.1 0.2 0
After:0 0.1 0.1

Finally ($Na^+$ does not influence the pH), we are in the presence of a mixture of weak base/strong base: $pH$ $=$ $14$ $+$ $log\frac{n_{OH^-}}{V}$ $=$ $14$ $+$ $log\frac{0,1}{3}$ $=$ $ 13,52$

Strong acid 1/ weak base 2

The reaction between a strong acid 1 and a weak base 2 is complete ($\rightarrow$), if $pK_{a2}\gt 3$

Example: $2\;L\;HCl\; 0,1 M$ are mixed with $1\;L\;NH_3 \;0,1 M$: Record of moles :

$H_3O^+$ + $NH_3 $$\rightarrow$ $NH_4^+$+$H_2O$
Before:0.2 0.1 0
After:0.1 0 0.,1

Finally ($Na^+$ does not influence the pH), we are in the presence of a mixture of weak acid/strong acid: $pH$ $=$ $-log\frac{n_{H_3O^+}}{V}$ = $-log\frac{0,1}{3}= 0,48$

Weak acid 1/ weak base 2

(1) The reaction between a weak acid 1 and weak base 2 is complete ($\rightarrow $), if $pK_{a2}$ $-$ $pK_{a1}\gt 3$ (2) The reaction between a weak acid and weak base 1 2 is incomplete ($\rightleftarrows$) if $-3\lt pK_{a2}$ $-$ $pK_{a1}\lt 3$ (3)There is no reaction between a weak acid 1 and weak base 2 ($\leftarrow), if $ pK_{a2}$ $-$ $pK_{a1}\lt -3$

Justifications: (1) $K$ $=$ $\frac{K_{a1}}{K_{a2}}\gt 10^3$ (2) $10^{-3}\lt K$ $=$ $\frac{K_{a1}}{K_{a2}}\lt 10^3$ (3) $ K$ $=$ $\frac{K_{a1}}{K_{a2}}\lt 10^{-3}$

Examples: The reaction between chloric acid ($pK_{a1}$ $=$ $-1$) and ammonia ($pK_{a2}$ $=$ $9,2$) is complete, because $9,2-(-1)\gt3$ $HClO_3$ $+$ $NH_3$ $\rightarrow$ $ NH_4^+$ $+$ $ClO_3^-$ The reaction between acetic acid ($pK_{a1}$ $=$ $4,75$) and methanoate ($pK_{a2}$ $=$ $3,75$) is incomplete, because $3,75-4,75\lt3$ $CH_3COOH$ $+$ $HCOO^-$ $\rightleftarrows$ $ CH_3COO^-$ $+$ $HCOOH$