Entropy is inherently a statistical phenomenon. We have seen that entropy describes the relationship between heat and temperature, or more precisely it is defined by the ratio of heat to temperature. In a physical sense this means that entropy depends on the number of energy states that are accessible to the system at a given temperature. The great the number of accessible states, the greater the entropy. However, the dependent is not linear, but rather it is logarithmic. This fact may be hard to grasp interitively. Yet, in general chemistry, we teach that the free energy is related logarithmically to the equilibrium constant and that does not appear to raise any eyebrows. We will first present the statistical definition of entropy and then we will show how this definition is a foundation for the other definitions.
Interpretations based on the factorial and combinatoric
The statistical entropy has several important ramifications. First, we can interpret the microscopic occupation of energy levels in a stastical sense. We should understand the entropy as a measure of how many energy states are accessible to the system at a given temperature. The greater then number of energy states, the greater the entropy. This is easy to see if all of the eneryg states have the same energy. Then we can just count the number of degenerate states and the entropy will be proportional to the logarithm of that number (S = k ln W>. However, if the states do not all have the same energy then we need a way to count the occupations of states that have the same energy. This is where the factorial enters the picture. The factorial is a statistical function that tells us how many different ways we can put M objects into N different boxes. The total number of ways is the combinatoric, W = MN. However, if the number of ways is subject to a constraint (i.e. the constraint of a particular energy), then the factorial must be used to count the number of different ways to populate the boxes.
We can tink of conformational entropy as a special case of statistical entropy, in which the combinatoric of the possible conformations of a polymer gives rise to an entropy contribution. This model can be applied to polymers. Proteins are a special case of polymers and an interesting one since there is a spontaneous process known as protein folding that occurs despite the rathrer large entropy barrier due to conformational entropy
