Question

What is the difference between ${\mathbf{K}}_{\mathbf{c}}$and${\mathbf{K}}_{\mathbf{w}}$ for the autoionization of water?

Short answer

The difference is:

${\mathrm{K}}_{\mathrm{c}}=\frac{\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]}{{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2}\mathrm{and}{\mathrm{K}}_{\mathrm{w}}=\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]$

Step-by-step solution

Definition of concept

The autoionization of water refers to the reaction of water molecules to form two ions, the hydronium ion, which is $\mathbf{H}\mathbf{3}{\mathbf{O}}^{\mathbf{+}}$, and the hydroxide ion, which is ${\mathbf{OH}}^{\mathbf{-}}$.

Explanation

The equilibrium constant of water is Kc

$$\begin{gathered} {\mathrm{H}}_2\mathrm{O}+{\mathrm{H}}_2\mathrm{O}\mathrm{֏}{\mathrm{H}}_3{\mathrm{O}}^{+}+{\mathrm{OH}}^{-} \\ {\mathrm{K}}_{\mathrm{c}}=\frac{\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]}{{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2} \end{gathered}$$

We move it to the other side since the concentration ${\mathrm{H}}_2\mathrm{O}$is constant.

${\mathrm{K}}_{\mathrm{c}}{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2=\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]$

After that, we create a new variable called${\mathrm{K}}_{\mathrm{c}}{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2$, which is the${\mathrm{K}}_{\mathrm{w}}$. So there you have it.

$$\begin{gathered} {\mathrm{K}}_{\mathrm{w}}={\mathrm{K}}_{\mathrm{c}}{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2 \\ =\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right] \end{gathered}$$

Therefore, The difference between ${\mathrm{K}}_{\mathrm{c}}$and ${\mathrm{K}}_{\mathrm{w}}$is that ${\mathrm{K}}_{\mathrm{c}}\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]$

${\mathrm{K}}_{\mathrm{c}}=\frac{\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]}{{\left[{\mathrm{H}}_2\mathrm{O}\right]}^2}$ and ${\mathrm{K}}_{\mathrm{w}}=\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]$