We can use the formula for pH to calculate the concentration of hydrogen ions in each solution:

For the first solution, we have:

Solving for $[\mathrm{H}^+]$, we get:

For the second solution, we want:

Solving for $[\mathrm{H}^+]$, we get:

To dilute the first solution to the desired concentration, we can use the dilution equation:

where $C_1$ and $V_1$ are the initial concentration and volume, and $C_2$ and $V_2$ are the final concentration and volume.

We know $C_1 = 0.1 \mathrm{~M}$, $C_2 = 0.01 \mathrm{~M}$, and $V_1 = 1 \mathrm{~L}$. Solving for $V_2$, we get:

Therefore, we need to add $10-1=9$ liters of water to the initial solution to obtain the desired pH. Converting liters to milliliters, we get:

So the volume of water needed is 9000 mL or 9,000 mL (nearest integer).