Classical Entropy Expression,

In classical thermodynamics, the entropy expression depends on the macroscopic variables of the system, such as volume, temperature, and pressure. The classical entropy expression is derived from empirical observations and does not consider the microscopic details of the system.

For a simple system, the classical entropy expression is given by:

S = k_B * ln(W)

where S is the entropy, k_B is the Boltzmann constant, and W is the number of accessible microstates corresponding to the macroscopic state of the system.

The expression S = k_B * ln(W) states that the entropy is proportional to the natural logarithm of the number of microstates. The Boltzmann constant, k_B, acts as a proportionality constant, relating the macroscopic concept of entropy to the statistical properties of the microscopic states.

It's important to note that this classical entropy expression is a simplification that neglects quantum effects and assumes that the system is in thermodynamic equilibrium. In more complex systems or at low temperatures, quantum mechanical effects become significant, and a more refined treatment, such as statistical mechanics, is required.

Overall, the classical entropy expression provides a useful approximation for describing the behavior of macroscopic systems in thermodynamics based on the concept of microstates, although it does not capture the full microscopic details of the system

derive S = k_B * ln(W)

To derive the expression S = k_B * ln(W) for the entropy of a system, we need to consider the concept of statistical mechanics and the fundamental principles involved. Here's a brief derivation:

Start with the statistical definition of entropy. In statistical mechanics, entropy is related to the number of accessible microstates corresponding to a given macrostate. Let's denote the number of microstates as Ω.


The entropy S is defined as the logarithm of the number of microstates:

S = k_B * ln(Ω)

where k_B is the Boltzmann constant.


The number of microstates, Ω, can be thought of as the ways in which the system's constituents (atoms, molecules, particles) can be arranged or distributed within the constraints of the macroscopic variables, such as energy, volume, and particle number.


By the principles of statistical mechanics, it is assumed that all accessible microstates are equally probable in equilibrium. This is known as the principle of equal a priori probabilities.


Under this assumption, the probability of finding the system in a particular microstate is given by:

P = 1 / Ω

Each microstate has the same probability of occurring.


Now, consider the entropy change when the system transitions from an initial macrostate to a final macrostate. The entropy change is given by:

ΔS = -k_B * Σ[P_i * ln(P_i)]

where P_i is the probability of microstate i in the initial macrostate.


Substitute the probability expression P = 1 / Ω into the entropy change equation:

ΔS = -k_B * Σ[(1 / Ω_i) * ln(1 / Ω_i)]


Simplify the expression using logarithmic properties:

ΔS = k_B * Σ[(1 / Ω_i) * ln(Ω_i)]


Notice that the sum is over all initial microstates. Rearrange the terms to express the sum over all microstates:

ΔS = k_B * Σ[(ln(Ω_i) / Ω_i)]


By recognizing that the sum is essentially an average over all microstates, we can write:

ΔS = k_B * ⟨ln(Ω_i)⟩

where ⟨⟩ denotes the average over all microstates.

In the thermodynamic limit (a large number of particles and high degrees of freedom), the average can be approximated by the ensemble average, denoted by ⟨⟩.


In the ensemble average, the logarithm of the number of microstates can be related to the logarithm of the probability distribution function, denoted as ln(W_i). This approximation is based on the principle of maximum entropy.

⟨ln(Ω_i)⟩ ≈ ⟨ln(W_i)⟩Substitute the ensemble average into the entropy change equation:

ΔS = k_B * ⟨ln(W_i)⟩Finally, in equilibrium, the entropy change is zero (ΔS = 0), and we obtain the entropy expression:

S = k_B * ln(W)

where W represents the total number of microstates accessible to the system.

This derivation shows how the entropy expression S = k_B * ln(W) arises from the principles of statistical mechanics, assuming equal a priori probabilities and the thermodynamic limit. It highlights the connection between the macroscopic concept of entropy and the statistical properties of the microscopic states.