Statistical interpretation of entropy links microscopic particle behavior to macroscopic thermodynamic properties. It explains why entropy increases in natural processes, as systems tend towards more probable states with more microstates.

The Boltzmann entropy formula, S = k ln W, connects entropy to the number of microstates. This relationship explains the second law of thermodynamics and provides a bridge between microscopic and macroscopic views of systems.

Entropy and Microstates

Statistical Interpretation of Entropy

• The statistical interpretation of entropy relates the macroscopic thermodynamic property of entropy to the microscopic behavior of the particles in a system
• Entropy measures the disorder or randomness of a system, which is directly related to the number of possible microstates (arrangements) of the particles in the system
• A microstate refers to a specific configuration of the particles in a system, while a macrostate describes the system's macroscopic properties (temperature, pressure, volume)
• Macrostates with a larger number of corresponding microstates have higher entropy

Entropy and Natural Processes

• The statistical interpretation of entropy explains why entropy tends to increase in natural processes
• Systems tend to evolve towards macrostates with a larger number of corresponding microstates, as these are more probable
• The tendency of entropy to increase is a consequence of the system moving towards the most likely macrostate
• Examples of entropy increasing in natural processes include:
  • The mixing of two gases (nitrogen and oxygen in the atmosphere)
  • The diffusion of a drop of ink in water
  • The cooling of a hot object to room temperature

Boltzmann Entropy Formula

Derivation of the Boltzmann Entropy Formula

• The Boltzmann entropy formula, $S = k \ln W$, relates entropy ($S$) to the number of microstates ($W$) and the Boltzmann constant ($k$)
• The derivation assumes that entropy is proportional to the logarithm of the number of microstates: $S = k \ln W$
  • This assumption is based on the idea that entropy should be an extensive property (scales with the size of the system) and should increase with the number of microstates
• The Boltzmann constant, $k = 1.38 \times 10^{-23}$ J/K, is a proportionality factor that ensures consistency between the microscopic and macroscopic definitions of entropy

Implications of the Boltzmann Entropy Formula

• The Boltzmann entropy formula implies that entropy is a measure of the number of accessible microstates for a given macrostate
• The formula suggests that entropy increases with the number of particles in a system, as a larger number of particles leads to a larger number of possible microstates
• The relationship between entropy and the number of microstates helps explain the second law of thermodynamics
  • Systems naturally tend towards states of higher entropy (more microstates) because these states are more probable
• The Boltzmann entropy formula provides a link between the microscopic world of particles and the macroscopic thermodynamic properties of a system

Calculating Entropy

Entropy of Systems with Distinguishable Particles

• To calculate the entropy of a simple system using the Boltzmann entropy formula, one needs to determine the number of microstates ($W$) corresponding to the macrostate of interest
• For a system of $N$ distinguishable particles, each of which can be in one of two states (coin toss), the number of microstates is $W = 2^N$
  • In this case, the entropy is $S = k \ln (2^N) = Nk \ln 2$
• Example: Consider a system of 4 distinguishable particles (A, B, C, D), each of which can be in one of two states (0 or 1). The number of microstates is $W = 2^4 = 16$, and the entropy is $S = 4k \ln 2$

Entropy of Systems with Indistinguishable Particles

• For a system of $N$ indistinguishable particles, each of which can be in one of two states (a lattice of $N$ sites, each of which can be empty or occupied), the number of microstates is $W = \binom{N}{n}$, where $n$ is the number of particles in one of the two states
  • In this case, the entropy is $S = k \ln \binom{N}{n}$
• Example: Consider a lattice of 6 sites, with 3 indistinguishable particles occupying 3 of the sites. The number of microstates is $W = \binom{6}{3} = 20$, and the entropy is $S = k \ln 20$
• When calculating entropy using the Boltzmann entropy formula, it is essential to correctly identify the relevant microstates and to use the appropriate formula for distinguishable or indistinguishable particles