Q20P - Chemistry: Molecular Nature Of Matter And Change

Question

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

Step-by-Step Solution

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 1: 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 ${\mathbf{OH}}^{\mathbf{-}}$.

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