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I am reading a chemistry book that makes the following point about Sørensen's pH discovery regarding [H+] being in negative powers of 10:

To dilute a solution from a pH 1 to a pH 4 (just 3 pH units) would require dilution by 10 x 10 x 10 = 1000 times.

I don't understand how this calculation is derived, nor is there any proof available in the book.

Can someone explain how this is the case?

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    $\begingroup$ let $c_1$ be concentration 1, $V_1$ be Volume 1 and so on then $c_1= 10^{-pH}$ and $c_1\times V_1 = c_2\times V_2 $ $\endgroup$ – MaxW Mar 10 '17 at 22:16
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    $\begingroup$ There is nothing even remotely resembling a discovery in this. The power of logarithms was known for centuries by then. $\endgroup$ – Ivan Neretin Mar 10 '17 at 22:34
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The confusion here probably comes from your statement that Sørensen discovered the pH scale for expressing [$\ce{H+}$]. The pH scale is something that he created or defined for convenience. The definition of pH is basically:

$$\mathrm{pH} = -\log[\ce{H^+}]$$

Since this is a (negative) log scale, a pH of 4 has 10 times the $[\ce{H+}]$ as does a pH of 5, and so on. In other words each pH unit represents a factor of 10 change in $[\ce{H+}]$.

The purpose behind such a scale is simply to avoid having to write out a bunch of zeros, and having to read a bunch of zeros!

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