Empirical and Theoretical Models

Empirical and Theoretical Models#

When choosing a model, we can take either an empirical or theoretical approach. Empirical models are based purely on observing patterns in our data, without necessarily understanding the underlying mechanisms. The Arrhenius equation was originally developed this way to describe the empirical relationship between rate constant \(k\) and temperature \(T\) observed for most reactions:

\[k = A\exp(-E_\mathrm{a}/RT)\]

In contrast, theoretical models are derived from fundamental physical principles. An example is the Eyring equation,

\[k = \frac{k_\mathrm{B}T}{h}\exp\left(-\frac{\Delta S^‡}{R}\right)\exp\left(\frac{\Delta H^‡}{RT}\right)\]

which is derived from transition state theory and a theoretical analysis of how reactions proceed through activated complexes. When we fit experimental data to this equation, we are testing a specific mechanistic hypothesis about how reactions occur at a molecular level. The parameters we extract, like the entropy and enthalpy of activation (\(\Delta S^‡\) and \(\Delta H^‡\)), have clear physical interpretations within this theoretical framework.

Both approaches have their value. Empirical models can be incredibly useful even when we do not fully understand the underlying mechanisms — they allow us to make practical predictions and spot patterns that might later guide theoretical development. Theoretical models, while sometimes more complex, help us test and refine our understanding of chemical processes at a fundamental level. Sometimes, as with the Arrhenius equation, what begins as an empirical model can eventually be understood in theoretical terms, building a bridge between observation and understanding.