${\mathrm{p}}^{\mathrm{H}}$of 10 ml of weak monoprotic acid is 4. To increase ${\mathrm{p}}^{\mathrm{H}}$ of this solution by one unit the volume of water that must be added is 

$$\begin{gathered} \mathrm{For} \mathrm{a} \mathrm{weak} \mathrm{acid}; \\ {\mathrm{p}}^{\mathrm{H}}=\frac{1}{2}{\mathrm{p}}^{{\mathrm{K}}_{\mathrm{a}}}-\frac{1}{2}\log \mathrm{C}; \\ \mathrm{For} \mathrm{a} \mathrm{weak} \mathrm{acid}; \\ {\mathrm{p}}^{\mathrm{H}} \mathrm{after} \mathrm{dilution} = {\mathrm{p}}^{\mathrm{H}} \mathrm{before} \mathrm{dilution} +\frac{1}{2}\log \frac{{\mathrm{V}}_{\mathrm{final}}}{{\mathrm{V}}_{\mathrm{initial}}} \\ 5=4+\frac{1}{2}\log \frac{{\mathrm{V}}_{\mathrm{final}}}{{\mathrm{V}}_{\mathrm{initial}}} \\ {\mathrm{V}}_{\mathrm{final}}=1000\mathrm{ml} \\ {\mathrm{V}}_{{\mathrm{H}}_2\mathrm{O}}=1000\left({\mathrm{V}}_{\mathrm{final}}\right)-10\left({\mathrm{V}}_{\mathrm{initial}}\right)=990 \mathrm{ml} \end{gathered}$$