A common interpretation of entropy is that it is somehow a measure of chaos or randomness. There is some utility in that concept. Given that entropy is a measure of the dispersal of energy in a system, the more chaotic a system is, the greater the dispersal of energy will be, and thus the greater the entropy will be. Ludwig Boltzmann (1844 – 1906) (O'Connor & Robertson, 1998) understood this concept well, and used it to derive a statistical approach to calculating entropy. Boltzmann proposed a method for calculating the entropy of a system based on the number of energetically equivalent ways a system can be constructed.

Boltzmann proposed an expression, which in its modern form is:

\[S = k_b \ln(W) \label{Boltz}\]

This rather famous equation is etched on Boltzmann’s grave marker in commemoration of his profound contributions to the science of thermodynamics (Figure \(\PageIndex{1}\)).

Example \(\PageIndex{1}\):

Calculate the entropy of a carbon monoxide crystal, containing 1.00 mol of \(\ce{CO}\), and assuming that the molecules are randomly oriented in one of two equivalent orientations.

Solution:

Using the Boltzmann formula (Equation \ref{Boltz}):

\[S = nK \ln (W) \nonumber\]

And using \(W = 2\), the calculation is straightforward.