### Limitations of Thermodynamics Analysis:

So far, we have used thermodynamic concepts to describe systems. However, equilibrium, the key requirement for thermodynamics analysis to be applicable, is rarely found in natural systems. Most times, natural systems are *moving* toward equilibrium. However, in many cases, they can only achieve such equilibrium conditions if left undisturbed for eons. In other words, the equilibrium approximations are not even good enough for geologic formations.

An interesting thing to notice is that a system needs to be out of equilibrium for a process to occur spontaneously. In other words, the methods introduced in the two previous chapters cannot be readily applied to most natural systems because they are not in equilibrium.

This difficulty can be at least partially circumvented by two simplifying assumptions - reversibility and local equilibrium.

is a temporal assumption that dictates that in a sufficiently slow, idealized process, changes proceed in sufficiently small steps that the system can be considered to be in equilibrium at any given time.*Reversibility*is a spatial assumption that limits the changing parts of the system to small enough volumes that can be considered to be in equilibrium.*Local Equilibrium*

In some cases, the thermodynamics analysis of a system simply collapses, either as a consequence of a gradual or a drastic change in one or more variables of the system.

The effect of drastic changes on the state of a system is intuitive. For example, we can easily surmise that a large asteroid impact event may result in mass extinctions. However, understanding how gradual changes may also result in mass extinction events is much more difficult to understand, let alone predict. To find and understand the cases in which gradual and progressive changes in the value of an independent variable results in sudden changes in the value of the dependent variable, we can draw from the mathematical foundations of thermodynamics, i.e., a set of differential equations. Without going into the computational side, thermodynamics analysis does not work at points in which the derivative of the function under study does not exist. Those points are called ** Discontinuities**. From drawing the graphs of functions in calculus classes, you probably remember that a function is not continuous at points where you need to lift your pencil.

### Review of Math Concepts:

For a summary of key differential equations, you can visit the thermodynamic equations page on Wikipedia.

For a review of the definitions of function and continuity, Wikipedia pages are linked.