Why is water sometimes omitted from the equilibrium constant?

Why is water sometimes omitted from the equilibrium constant? To answer this equation, we begin by noting that the equilibrium constants of

$K_c=\frac{[C]^p[D]^q...}{[A]^m[B]^n...}\; \; or\; \; K_p=\frac{p_C^{\; \;\; p}p_D^{\; \; \; q}...}{p_A^{\; \;\; m}p_B^{\; \;\; n}...}\; \; \; \; \; \; \; \; 3$

are approximations of the thermodynamic definition of the equilibrium constant, which is:

$K=\frac{a_C^{\; \;\; p}a_D^{\; \; \; q}...}{a_A^{\; \;\; m}a_B^{\; \;\; n}...}\; \; \; \; \; \; \; \; 4$

where a_i is the activity of species i.

For a dilute solution,

$a_i=\frac{[i]}{[i]^o}\; \; \; \; \; \; \; \; 5$

where [i]^o is the concentration of the pure species at standard conditions of 1 bar and 298.15K.

For an ideal gas,

$a_i=\frac{p_i}{p_i^{\: o}}\; \; \; \; \; \; \; \; 6$

where p_i^o is the pressure of the pure species at standard conditions of 1 bar and 298.15K.

Combining eq3 through eq6,

$K_c=\frac{ \left (\frac{[C]}{[C] ^o} \right )^p\left (\frac{[D]}{[D] ^o} \right )^q...}{\left ( \frac{[A]}{[A]^o} \right )^m\left ( \frac{[B]}{[B]^o} \right )^n...}\; \; and\; \; K_p=\frac{ \left (\frac{P_C}{P_C^{\: o}} \right )^p\left (\frac{P_D}{P_D^{\: o}} \right )^q...}{\left ( \frac{P_A}{P_A^{\: o} }\right )^m\left ( \frac{P_B}{P_B^{\: o}} \right )^n...}\; \; \; \; \; \; \; \; 7$

Consider the following reversible reaction:

$CH_3COOH(l)+H_2O(l)\rightleftharpoons CH_3COO^-(aq)+H_3O^+(aq)$

$K_c=\frac{ \left (\frac{[CH_3COO^-]}{[CH_3COO^-] ^o} \right )\left (\frac{[H_3O^+]}{[H_3O^+] ^o} \right )}{\left ( \frac{[CH_3COOH]}{[CH_3COOH]^o} \right )\left ( \frac{[H_2O]}{[H_2O]^o} \right )}\; \; \; \; \; \; \; \; 8$

Water, being in excess, has a concentration that is assumed to be constant throughout the reaction and approximately equals to that of its pure state. Hence $\frac{\left [ H_2O \right ]}{\left [ H_2O \right ]^o}\approx1$ and eq8 becomes

$K_c=\frac{ \left (\frac{[CH_3COO^-]}{[CH_3COO^-] ^o} \right )\left (\frac{[H_3O^+]}{[H_3O^+] ^o} \right )}{\left ( \frac{[CH_3COOH]}{[CH_3COOH]^o} \right )}\; \; \; \; \; \; \; \; 9$

Since the standard state of a solute is defined as 1 mol dm^-3, eq9 approximates to

$K_c=\frac{\left [ CH_3COO^- \right ]\left [ H_3O^+ \right ]}{\left [ CH_3COOH \right ]}\; \; \; \; \; \; \; \; 10$

Therefore, water is excluded from the equilibrium constant if it is a solvent. However, for the reactions

$CH_3COOH(l)+C_2H_5OH(l )\rightleftharpoons CH_3COOC_2H_5(l)+H_2O(l)$

$4HCl(g)+O_2(g)\rightleftharpoons 2H_2O(g)+2Cl_2(g)$

water is not a solvent but reactant, and the respective equilibrium constants are

$K_c=\frac{\left [ CH_3COOC_2H_5 \right ]\left [ H_2O \right ]}{\left [ CH_3COOH \right ]\left [ C_2H_5OH \right ]}$

$K_c=\frac{\left [ H_2O \right ]^2\left [Cl_2 \right ]^2}{\left [ HCl \right ]^4\left [O_2 \right ]}$

As for a reversible reaction containing one or more solid species, e.g.

$HCl(g)+LiH(s)\rightleftharpoons LICl(s)+H_2(g)$

the concentration of the solid species are assumed to be the same as that of their respective pure states and hence

$K_c=\frac{\left [ H_2 \right ]}{\left [HCl\right ]}\; \; or\; \; K_p=\frac{p_{H_2}}{p_{HCl}}$

Question

Write the equilibrium constants for the following reactions:

a) $CaCO_3(s)\rightleftharpoons CaO(s)+CO_2(g)$

b) $CH_3COOC_2H_5(l)+H_2O(l)\rightleftharpoons CH_3COOH(l)+C_2H_5OH(l)$

c) $Co(H_2O)_6^{\; 2+}(aq)+4Cl^-(aq)\rightleftharpoons CoCl_4^{\; 2-}(aq)+6H_2O(l)$

Answer

a) $K_p=p_{CO_2}$

b) $K_c=\frac{\left [ CH_3COOH \right ]\left [ C_2H_5OH \right ]}{\left [ CH_3COOC_2H_5 \right ]\left [ H_2O \right ]}$

Note that H₂O is not a solvent in the hydrolysis reaction but a reactant. A typical ester hydrolysis reaction involves adding, for example, 0.10 M of CH₃COOC₂H₅, 0.10 M of H₂O and a catalyst in an inert organic solvent.

c) $K_c=\frac{\left [ CoCl_4^{\; \; 2-} \right ]}{\left [ Co(H_2O)_6^{\; \; 2+} \right ]\left [ Cl^- \right ]^4}$

Note that the reaction occurs in an aqueous solution, i.e. the solvent is water.

Why is water sometimes omitted from the equilibrium constant?

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