Statistical and thermodynamic defintion of entropy

[jd12345]

How are both statistical and thermodynamic defintion of entropy equivalent?
THe statistical definition i.e. S = k ln ω amkes sense to me. It is the number of mcirostates an atom/moelcule can take over but how is the thermodynamic definition i.e. ΔS = q/T equivalent to it?

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/entropy.html

this site explains entropy but i don't understand the last paragraph about it. Is temperature defined in such a way that the thermodynamic definition of entropy becomes correct?
it makes me feel as if temperature is specifically defined in such a way that the equation ΔS = q/T becomes correct

 

[eljose79]

.

Thank you for your question. The relationship between the statistical and thermodynamic definitions of entropy is a fundamental concept in thermodynamics and statistical mechanics. Both definitions are equivalent, meaning that they describe the same physical phenomenon in different ways.

To understand this equivalence, it is important to first understand the basic principles of thermodynamics and statistical mechanics. Thermodynamics is the study of the relationships between heat, work, and energy in a system. It is based on a set of laws, including the second law of thermodynamics, which states that the total entropy of a closed system can never decrease.

On the other hand, statistical mechanics is based on the behavior of individual particles in a system, and uses statistical methods to describe the overall behavior of the system. It is based on the concept of microstates, which are the different ways that the particles in a system can be arranged. The statistical definition of entropy (S = k ln ω) is based on the number of microstates (ω) that are available to a system at a given energy level.

Now, to understand how these two definitions are equivalent, let's consider a simple example. Imagine a box filled with gas molecules. According to the statistical definition, the entropy of the system would increase if the molecules were allowed to spread out and occupy a larger volume. This is because there are more microstates available to the system in this scenario.

On the other hand, from a thermodynamic perspective, this increase in volume would result in an increase in the system's internal energy. This increase in energy would lead to an increase in temperature, as defined by the equation q = mcΔT. Therefore, the thermodynamic definition of entropy (ΔS = q/T) takes into account the relationship between energy, temperature, and heat transfer.

In summary, the thermodynamic definition of entropy is based on changes in energy and temperature, while the statistical definition is based on the number of microstates available to a system. However, these two definitions are equivalent and describe the same physical phenomenon. Temperature is not specifically defined to make the thermodynamic definition of entropy correct, but rather the two definitions are mutually consistent and can be used interchangeably to describe the behavior of a system. I hope this helps clarify the relationship between the two definitions of entropy.