What is the hydrogen ion concentration in a urine specimen that registers a pH of 4 on a strip of pH paper?

And I didn't even need a calculator.........

By definition, #pH=-log_10[H_3O^+]#.

And if #pH=4#, then...........

#[H_3O^+]=10^-4*mol*L^-1=0.0001*mol*L^-1.#

And we know that hydroxide/hydronium ions obey the following equilibrium in aqueous solution.......

#H_3O^+ + HO^(-) rarr2H_2O#; #K_w=[H_3O^+][HO^-]=10^(-14)#. Of course, this value HAS to be measured, and here #K_w# is measured under standard conditions of #298*K# and (almost) #1*atm#.

Why do we use such an absurd definition? Well, back in the day, approx. 30-40 years ago BEFORE the proliferation of hand-held electronic calculators, students and engineers would routinely use log tables for lengthy calculation involving multiplication and division. because the product #axxb=10^(log_10a+log_10b)#, and it was easier to do additions than multiplications, even if we had to take antilogs to get the final product.

These days with a simple electronic calculator (and I bought one for a quid in a discounter's shop a couple of weeks ago) we have access to a computational power that would have astonished both Newton and Gauss (they say that Gauss in particular, a mathematical prodigy, had memorized the log tables so that he could do his calculations).