# Why are many chemical relationships logarithmic?

I've noticed that in many relationships in chemistry, one variable is linearly related to the logarithm of another. Why is this the case?

For example, I carried out an experiment where the dependent variable, the amount of solute adsorbed to a surface, was almost perfectly linearly correlated (Pearson correlation coefficient = 0.93) with the pOH, which is a logarithmic quantity.

Edit: Would this perhaps be better in stats.stackexchange.com?

• I don't think it's better suited in stats. However, what I would say is that this is perhaps a very... general question and I am not sure if there is one single answer. Some logarithmic relationships (e.g. in thermodynamics) arise from first-order ODEs of the form $\mathrm{d}x/x = \mathrm{d}y$. Some arise from the fact that the reaction quotient $Q$ is a product of activities, so $\log Q$ is a sum of logarithms of activities. And so on and so forth... – orthocresol Oct 16 '16 at 10:24
• I see your point - this is too general. This specific case involves the adsorption of fluoride to activated alumina, where there was competitive adsorption between $\ce{F^-}$ and $\ce{OH^-}$; in other words, hydroxide ions occupied the surface sites, and the addition of hydroxide meant that fewer sites were available for fluoride to adsorb to. – Marcel Oct 16 '16 at 11:16
• Regarding your point about reaction quotients, will that mean that I could get a prediction for the relationship by creating some kind of combined equilibrium expression for the two competitive equilibria (fluoride/alumina and hydroxide/alumina)? If that is the case, does it matter that these are heterogeneous equilibria? – Marcel Oct 16 '16 at 11:34
• Sorry, I'm not sure about absorption, don't really know much about the topic. – orthocresol Oct 16 '16 at 11:51

Some logarithmic behaviour is built into the way the world works and some is a consequence of how we perceive things which span very large ranges.

The simplest chemical and nuclear reactions (first order ones) are, for example, inherently logarithmic. Take any first order process (like radioactive decay). The process has a fixed probability of happening (a radioactive atom or an unstable molecule might have a 10% probability of decaying in a minute). What you observe in the bulk with processes like this is that the rate of the reaction decays exponentially. You start with a rate of say 12 units per minute. After a certain time half the initial material has gone and the rate is now 6 after the same time again the rate is now 3 and so on. This is exponential decay emerging from a very simple process with a fixed probability. A logarithm emerging from a probability, if you prefer. This sort of thing is very common in chemistry and physics.

The other cause of logarithms is human perception. Our perception of light and sound are both inherently logarithmic. This is partially because there is no sensible way to map the range of magnitudes of many natural phenomena to a linear scale. The intensity of sound and light vary over huge orders of magnitude. The Sun is about 15 trillion time brighter than the faintest visible star and it is fairly inconvenient to divide this scale into equal bands. So perception tends to divide the scale logarithmically with each perceived difference representing a fixed ratio not a linear difference (each star magnitude, for example is about 2.5 times brighter than the next lower one, but represents a just-about discernible difference).

The logarithmic patterns are more a function of math than physical properties. The logarithmic identity: $\log(x^5) = 5\log(x)$ is responsible for most of your observations. If I have a property $y$ that is dependent on $x^a$ where $a$ is a constant, I can log both sides to get a relation of: $\log(y) = \log(x^a) = a\log(x)$. This means that for every $1$ increment increase in $\log(x)$, $\log(y)$ will increase by $a$ increments (a linear relation).

On reason is that probability plays an important role in physics, especially at the microscopic scale.

When you combine two systems to a new system some physical quantities simply add up while probabilities multiply.

The arising functional equations $$f(x \cdot y) = f(x) + f(y)$$ and $$f(x + y) = f(x) \cdot f(y)$$ are satisfied by logarithmic or exponential functions.