5.1.3 (d,e) pH & [H+]

Syllabus

(d) use of the expression for pH as:

pH = –log[H⁺]

[H⁺] = 10^–pH

(e) use of the expression for the ionic product of water, K_w

_{What does this mean?}

What is pH?

If you've read the page on K_a and pK_a this will seem quite familiar.

If not, it's easier than it looks at first sight.

[H⁺] is shorthand notation for the concentration of H⁺ ions in water - which is what makes acids acidic.

In reality it is really the concentration of H₃O⁺ ions, since almost no actual H⁺ ions exist in water.

But since H₃O⁺ ions behave in exactly the same way as H⁺ ions we don't usually bother to distinguish between them.

Concentrations of H⁺ ions are very small even in fairly concentrated weak acids and especially in strong bases.

A solution with a pH of 5.8 has [H⁺] = 0.00000158 or 1.58 x 10^-6

A solution with a pH of 13.5 has [H⁺] = 0.000000000000316 or 3.16 x 10^-13

The majority of people aren't familiar with Standard Form notation.

Very small numbers with lots of zeroes are unwieldy and it's easy to lose track of exactly how many zeroes you need to write down.

Logarithms (no matter how much people hate them) reduce very large or very small numbers to much simpler ones.

For instance, a weak-ish alkali may have [H⁺] = 0.000000055 - which is an awkward number.

Taking the Log makes it somewhat less awkward.

Log₁₀(0.000055) = -7.26

And since everyone is happier with positive rather than negative numbers we could ignore the negative sign and simply use 7.26.

Therefore, the definition of pH is: pH = - Log₁₀([H⁺])

Asked for a definition in an exam simply write this and don't bother trying to explain it in English.

Converting [H⁺] to pH.

We've already seen that converting Hydrogen ion concentrations to pH is easy.

pH = - Log₁₀([H⁺])

This is a job for a calculator but you should always look at your [H⁺] and make a quick estimate of what the pH will be before converting just in case you press the buttons incorrectly.

For instance, [H⁺]= 1x10^-5 would give you pH = 5 because you simply take the power and make it positive.

So anything similar will give you a pH close to 5.

eg [H⁺]= 2.5x10^-5 =4.6 and [H⁺]= 8.5x10^-5 =4.1

Both have slightly higher [H⁺] than pH 5 so they are both a little stronger, and hence have slightly lower pH.

In the same way a [H⁺] = 1.25 x10^-11 must give us a pH close to but slightly above 11 - in this case 10.9.

Converting pH to [H⁺]

If you are unfamiliar with Logs then this is probably a bit daunting, but it shouldn't be.

We've already seen that the pH calculated from a [H⁺] is almost exactly the power of ten (see above)

So converting a pH to a [H⁺] also involves the power.

Calculators write the opposite function to Log as invLog or antilog or 10^x.

But it is always the second function of the log button.

Textbooks tend to use 10^x.

pH = - Log₁₀[H⁺]

So, -pH = Log₁₀[H⁺]

anti-log (-pH) = anti-log (Log₁₀[H⁺])

An anti-log and a log function cancel.

So, anti-log (-pH) = [H⁺]

In textbook notation this is: [H⁺] = 10^(-pH)

In practice a pH of 5.2 must give me a [H⁺] close to 1x10^-5 but a bit lower since the H+ concentration is lower. So the power is likely to be -6

[H⁺] = 10^(-pH) = gives us [H⁺] = 10^(-5.2) 6.3 x 10 ^-6