Question

Prove that Boltzmann's microscopic definition of entropy, \(S=k_{\mathrm{B}} \ln w,\) implies that entropy is an additive variable: Given two systems, \(A\) and \(B\), in specified thermodynamic states, with entropies \(S_{A}\) and \(S_{B},\) respectively, show that the corresponding entropy of the combined system is \(S_{A}+S_{B}\).

Short answer

Question: Prove that entropy is an additive variable using Boltzmann's microscopic definition of entropy. Answer: Entropy is an additive variable as the combined entropy of systems A and B equals the sum of the individual entropies of the two systems: S = S_A + S_B. This is proven through analyzing the accessible microstates of systems A and B, finding the number of microstates of the combined system, calculating the entropies of individual systems and the combined system, and using logarithmic properties to show the combined entropy equals the sum of the individual entropies.

Step-by-step solution

Determine the microstates of A and B

Out of all the accessible microstates of systems \(A\) and \(B\), system \(A\) has \(w_A\) microstates and system \(B\) has \(w_B\) microstates.

Find the number of microstates of the combined system

The number of microstates of the combined system, \(w\), is equal to the product of the individual microstates of the two systems, i.e., \(w = w_A \cdot w_B\).

Find entropies of individual systems and the combined system

According to Boltzmann's formula, the entropy of each system is given by \(S_A=k_B\ln w_A\), \(S_B=k_B\ln w_B\), and the entropy of the combined system is given by \(S=k_B\ln w\).

Plug in the expression for the combined system's microstates

We found in step 2 that the combined system's microstates are given by \(w=w_A\cdot w_B\). We can now plug this equation into the expression for the combined system's entropy: \(S = k_B\ln{(w_A\cdot w_B)}\).

Use logarithmic properties to simplify the expression for the combined entropy

Using the logarithmic properties, the expression for the combined entropy can be simplified as follows: \(S = k_B\ln{(w_A\cdot w_B)} = k_B(\ln{w_A} + \ln{w_B})\).

Prove the additivity of entropy

From step 5, we have \(S = k_B(\ln{w_A}+\ln{w_B})\). We already have \(S_A=k_B\ln w_A\) and \(S_B=k_B\ln w_B\). Thus the combined entropy is \(S=k_B\ln{w_A} + k_B\ln{w_B} = S_A + S_B\). We have successfully proven that entropy is an additive variable as the combined entropy of systems \(A\) and \(B\) equals the sum of the individual entropies of the two systems: \(S = S_A + S_B\).

Key concepts

Entropy Additivity

Understanding entropy as an additive property helps clarify how the disorder or randomness of a composite system is calculated. When we consider two separate systems, each with its own level of disorder, the total entropy should be the sum of the individual entropies if they are independent of each other.

Let's simplify the concept of entropy additivity with an analogy. Imagine two boxes filled with colored balls, one red and one blue, representing systems A and B respectively. If we have a certain number of ways to arrange the red balls (microstates for system A) and a certain number of arrangements for the blue balls (microstates for system B), when we combine both boxes without mixing the colors, the total number of arrangements is the product of the arrangements of each color. Translate this to thermodynamics: we multiply the microstates of system A by those of system B to determine the composite system's microstates.

In thermodynamics, entropy additivity is demonstrated through Boltzmann's entropy formula, shown by combining the entropies of two independent systems to get the entropy of the total system. The concept is central to understanding how macroscopic properties emerge from microscopic behavior. This principle is vital, for instance, in calculating the entropy of ideal gases, where the total entropy is the sum of the entropies of the individual gas components.

Microstates in Thermodynamics

In thermodynamics, a microstate is a specific detailed configuration of a system's particles, including their positions and velocities. One of the fundamental postulates of statistical thermodynamics is that at a given energy, each microstate of an isolated system is equally probable. This idea introduces a probabilistic aspect to the microscopic world, linking it to the macroscopic thermodynamic property of entropy.

The importance of microstates lies in their sheer number, which is extraordinarily high even for a small amount of any substance. The tally of these configurations, denoted as 'w' in the Boltzmann entropy formula, can be astronomical. However, even though we cannot count each microstate individually for a macroscopic system, we can still calculate the entropy changes because we're often dealing with relative differences in entropy, where the exact numbers of microstates cancel out.

Why is the number of microstates integral?

Each microstate corresponds to a unique way the system's particles can be arranged while satisfying the given energy conditions. Because entropy is closely linked to the number of microstates (more microstates mean higher entropy), understanding how microstates work and count them is crucial for analyzing the thermodynamic behavior of substances at a molecular level.

Logarithmic Properties

Logarithms transform the product of numbers into the sum of logarithms. This mathematical property is not only convenient for simplifying calculations but also serves as a fundamental aspect in the relationship between entropy and microstates. In the context of Boltzmann's entropy formula, the logarithmic property that the log of a product is equal to the sum of the logs allows the concept of entropy additivity to emerge.

To put it concretely, if we take two numbers representing the microstates of systems A and B, \( w_A \) and \( w_B \), the property states that \( \ln(w_A \times w_B) = \ln(w_A) + \ln(w_B) \). This property is pivotal in proving the additivity of entropy because it simplifies the expression of the combined microstates' entropy into a sum of individual entropies.

Understanding how logarithmic properties underpin the mathematical framework of thermodynamics equips students with the tools they need to unravel complex relationships between macroscopic measurements and the underlying microscopic phenomena. With logarithms, we can effectively 'decode' the behavior of entropy and gain a deeper appreciation for the elegant interplay between mathematics and the physical world.