# Can a negative pH exist? [duplicate]

I am programming a small model to simulate diffusion of $\ce{H+}$ ions in human tissue. It's quite a simplistic model based on a naive algorithm, and I'm mostly doing it to familiarize with the concept of agent-based models.

My space is an $n\times n$ grid where each position $(x;y)$ hosts $h$ $\ce{H+}$ ions.

I understand that $\ce{pH}$ can be calculated based on $h$ as $-\log(h)$. As I want my initial $\ce{pH}$ to be stable at average human $\ce{pH}$, I'd set $h$ across the matrix to be $10^{7.5}$. (In fact, as I don't want to simulate $10^{7.5}$ protons I decided each particle will represent a bulk of $10^5$ protons and made my equation $-\log(10^5\cdot h)$. )

However, I do not understand the equation, wouldn't it at some point yield a negative $\ce{pH}$ as $-\log(x) < 0$ for $x > 1$?

Yes, pH can be negative.

The official definition of pH is: negative log of H+ activity, not negative log of H+ concentration, however. See pH Paradoxes: Demonstrating That It Is Not True That pH ≡ -log[H+] J. Chem. Educ., 2006, 83 (5), p 752.

By either definition pH can be negative in concentrated solutions of strong acids. For example 5M HCl would have a negative pH.

Like you said the $\ce{pH}$ of blood is approximately $7.5$. So since $$\ce{pH} = -\log[\ce{H+}] \\ \text{or}\\ [\ce{H+}] = 10^{-\ce{pH}}$$ this means that the proton or hydrogen ion (same thing) has a concentration of $10^{-7.5}$ not $10^{7.5}$ (notice the negative). If you're dealing with blood you'll never have to worry about negative $\ce{pH}$ values. Like DavePhD stated, negative $\ce{pH}$ values only occur with high concentrations of strong acids.

In short, I think where you're going wrong is with $-\log(10^{7.5})$. It should be $-\log(10^{-7.5})$.

As a side note, you are correct. $\log(x)\geq0$ for $x \geq 1$. So $-\log(x) \leq0$ for $x\geq1$, but when do you ever have proton concentrations above 1 in human biology? Practically never.

$\text{pH}$ can definitely be negative.
It is defined as $$\text{pH}=-\log[\text{H}^{+}]$$
Note that if the $[\text{H}^{+}]\geq 1\text{M}$, the $\text{pH}$ will become negative.
Practically, this is only possible when you have an ultrapure solution of a strong acid.
Hope this helps.