In pure water, #["H"^+] = ["OH"^"-"] = 1.00 × 10^"-7" color(white)(l)"mol/L"#.
Imagine that we add the #2.6 × 10^"-9" color(white)(l)"mol/L H"^+#, but the equilibrium hasn't had time to re-establish itself.
We have just set up a new initial condition.
Let's put this into an ICE table.
#color(white)(mmmmm)"H"_2"O" ⇌ color(white)(mmmmmll)"H"^+color(white)(mmmm) +color(white)(mll) "OH"^"-"#
#"I/mol·L"^"-1": color(white)(mmmm)1.00×10^"-7" + 2.6×10^"-9" color(white)(ml)1.00 × 10^"-7"#
#"C/mol·L"^"-":color(white)(mmmmmmmmm) -x color(white)(mmmmmmml)-x#
#"E/mol·L"^"-1":color(white)(mmmmm) 1.026 × 10^"-7" -xcolor(white)(mml) 1.00 × 10^"-7" - x#
#K_w = ["H"^+]["OH"^"-"] = 1.00 × 10^"-14"#
#(1.026 × 10^"-7" -x)(1.00 × 10^"-7" - x) = 1.00 × 10^"-14"#
#x^2 - 2.026 × 10^"-7"x + 1.026 × 10^"-14" = 1.00 × 10^"-14"#
#x^2 - 2.026 × 10^"-7"x + 0.026 × 10^"-14" = 0#
#x = 1.30 × 10^"-9"#
#["H"^+] = 1.026 × 10^"-7" -x = 1.026 × 10^"-7" - 1.03 × 10^"-9" = 1.016 × 10^"-7"#
#"pH" = -log["H"^+] = -log(1.016 × 10^"-7") = 6.99#
