# What is 'power law' kinetics?

I've come across a lot of instances where 'power law' description of the kinetics was mentioned, but I cannot think of any integrated rate law that follows something like:

$$A(t)=bt^n+c$$

I'm pretty sure the 'power law' refers to an integrated rate law instead of rate equations like:

$$r = k\prod_i [A_i]^i$$

So what is power law kinetics anyway? Does this 'power law' have a physical meaning or just a polynomial approximation that tells nothing about the mechanism?

Usually these type of complex kinetics arise when there is an inhomogeneous distribution of properties in a system. For example reaction over barriers which have a range of sizes, this can occur in proteins or in quantum dots but not in 'normal' kinetics. Fluorescence quenching in micelles can follow non-single exponential decays because there is usually a distribution of quenchers solubilised in micelles. The fluorescence has the form $\ln(I)\sim -At+B\exp(-Ct)+D$ where , $t$ is time and $A,B,C,D$ constants.

In diffusion controlled quenching or energy transfer quenching in solution the rate constant will depend on time as $k=k_0+k_1/\sqrt{At}$. So there are many situations in which 'strange' kinetics can be observed.

However, vey often 'power law' kinetics is just a name given to an empirical method of fitting data to a function, say of the form $\exp(-kt^n)$ where $k, n$ are variables used to fit the data. In this form the decay function has no physical meaning.

Addendum. The first equation above was calculated as shown below.

As a model we suppose that the fluorophore and quencher are both in a micelle, as example of a restricted geometry, and that effectively no fluorophore exists outside. There is always an excess of quencher over fluorophore whose concentration is small so that no more than one will exist in any micelle. It is possible that zero, one or several quenchers can be in any micelle.

We suppose that the rate constant for quenching is $k_i=k_0+ik_q$ where $k_0$ is the fluorescence decay rate constant value without quencher and $i$ the number of quenchers in any micelle, $i=0,\,1,\,2\cdots$ and $k_q$ the constant for quenching of the fuorophore.

The measured fluorescence signal is the sum of contributions from all micelles and is therefore

$$I(t)=I_0\sum_{i=0}^\infty p_i\mathrm e^{-k_it}$$

where $p_i$ is the probability of there being $i$ quenchers in a micelle. It seems reasonable to suppose that this has a Poisson distribution, and in that case $\displaystyle p_i= \frac{m^i\mathrm e^{-m}}{i!}$ where $m$ is the mean number of quenchers in a micelle. Substituting and simplifying the intensity then becomes

$$I(t)=I_0\mathrm e^{-k_0t-m}\sum_{i=0}\frac{(m\mathrm e^{-k_qt})^i}{i!}$$

The summation term has the form $\sum x^i/i!$ which is the expansion of the exponential thus

$$I(t)=I_0\mathrm e^{-k_0t-m}e^{m\mathrm e^{-k_qt}} = I_0\mathrm e^{-k_0t +m(\mathrm e^{-k_qt} -1) }$$

which when logs are taken is the expression above.

• very well explained ! could you give me a link that would lead to the derivation of your first equation ? Sep 2, 2018 at 15:51
• I have added the derivation as I could not find the reference. Sep 3, 2018 at 8:45