Question

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces. (a) The two arrangements of top faces shown here can be viewed as two possible microstates of the system. Explain. (b) To which state does each microstate correspond? (c) How many possible states are there for the system? (d) Which state or states have the highest entropy? Explain. (e) Which state or states have the lowest entropy? Explain. (f) Calculate the absolute entropy of the two-dice system.

Short answer

The two arrangements of top faces can be viewed as microstates, representing two different arrangements of the two-dice system. There are 36 possible microstates and 11 possible states (sums) in this system. The state with the highest entropy (most randomness) is the sum 7, while the states with the lowest entropy (least randomness) are the sums 2 and 12. The absolute entropy of the two-dice system is approximately \(1.29 \times 10^{-22} \, \text{J/K}\).

Step-by-step solution

(a) Explain the two arrangements as possible microstates of the system

A microstate is a specific configuration or arrangement of a system, while a macrostate is a collection of microstates with the same macroscopic properties. In this problem, the arrangement of the dice with specific numbers shown on their top faces are the microstates of the system. Since each dice face corresponds to a value from 1 to 6, there are a total of 36 possible microstates (6 possible values for each die, combined).

(b) Identify the state to which each microstate corresponds

Recall that the state of our system is given by the sum of the values on the top faces of the two dice. For each arrangement (microstate), add up the values of the two top faces to find the corresponding state (also known as macrostate).

(c) Find the number of possible states

To find the number of possible states, we count all the possible sums of the top faces of the dice. The minimum sum is 2 (when both dice show a 1), and the maximum sum is 12 (when both dice show a 6). Thus, there are 11 possible states in total (corresponding to the sums 2, 3, 4, ..., 12).

(d) Identify the state or states with the highest entropy

Entropy is a measure of disorder or randomness in a system. The state with the highest entropy is the one that has the most microstates associated with it since more microstates mean more possible random configurations. In the case of the two-dice system, the state with the highest entropy is the one with the sum 7, as it has the most possible microstate combinations (1+6, 2+5, 3+4, 4+3, 5+2, and 6+1). There is only one state (sum 7) with the highest entropy.

(e) Identify the state or states with the lowest entropy

The state with the lowest entropy has the least number of microstates associated with it (i.e., the least random arrangements possible). For the two-dice system, these are the states with the lowest sums (2 and 12). Each of these states has only one possible microstate combination (1+1 for the sum 2 and 6+6 for the sum 12). Thus, there are two states (sum 2 and sum 12) with the lowest entropy.

(f) Calculate the absolute entropy of the two-dice system

To calculate the absolute entropy of the system, we can use Boltzmann's entropy formula: \(S = k_B \ln W\), where \(S\) is the entropy, k_B is Boltzmann's constant (1.38 x 10^-23 J/K), and \(W\) is the number of microstates in the system. In this case, the total number of microstates (combinations of top face values) is 36. So, the absolute entropy is: \(S = (1.38 \times 10^{-23} \, \text{J/K}) \ln(36) \approx 1.29 \times 10^{-22} \, \text{J/K}\). The absolute entropy of the two-dice system is approximately \(1.29 \times 10^{-22} \, \text{J/K}\).

Key concepts

Microstates

In statistical mechanics, a microstate is a specific detailed configuration of a system. Imagine two dice, each with six faces showing numbers from 1 to 6. By throwing these dice, you create different arrangements of top faces, known as microstates. Each unique combination, such as one die showing 1 and the other showing 2, represents a distinct microstate. When we consider two dice, there are a total of 6 possible outcomes per die, leading to a total of 36 microstates (because 6 multiplied by 6 equals 36).
Microstates provide a microscopic picture of a system's state. The benefit of understanding microstates is that they offer insights into how a system behaves at a fundamental level. By exploring different microstates, one can predict the macroscopic behavior of the system.

Macrostates

While a microstate is the detailed, microscopic configuration, a macrostate is defined by macroscopic properties like temperature, pressure, and in this case, the sum of the dice faces. Macrostates represent collections of one or more microstates that result in the same observable outcome.
For our two-dice system, macrostates are represented by the possible sums of the numbers on the top faces, ranging from 2 to 12. Each macrostate encompasses several microstates. For example, the macrostate of sum 7 can arise from the microstates of (1+6), (2+5), (3+4), (4+3), (5+2), or (6+1). In this way, macrostates offer a simplified view of a system's possible states by grouping together similar outcomes from different microstates.

Entropy

Entropy, in simple terms, is a measure of disorder or randomness within a system. The concept is a cornerstone in both thermodynamics and statistical mechanics, reflecting the number of ways a system can be arranged while still resulting in the same overall state.
In our two-dice system, entropy is linked to the number of microstates associated with a particular sum (macrostate). For instance, the state with a sum of 7 has the highest entropy because it can be formed in more ways (6 different microstates) than any other state. The states with sums of 2 and 12 have the lowest entropy, each having only one microstate (1+1 and 6+6, respectively). Thus, entropy provides insight into the possible configurations of a system and the likelihood of a system being in a particular state.

Boltzmann's Entropy Formula

Ludwig Boltzmann, a pioneering physicist, formulated a crucial equation to understand entropy: \[ S = k_B \ln W \]Here, \( S \) represents the entropy of a system, \( k_B \) is Boltzmann's constant, approximately \( 1.38 \times 10^{-23} \, \text{J/K} \), and \( W \) stands for the number of microstates accessible to the system.
In the context of our two-dice example, Boltzmann's formula helps calculate the absolute entropy of the system. Since there are 36 potential microstates, the entropy can be calculated as: \[ S = (1.38 \times 10^{-23} \, \text{J/K}) \ln(36) \approx 1.29 \times 10^{-22} \, \text{J/K} \]This value provides a quantitative measure of the system's disorder and serves as a foundation for understanding how entropy changes influence the behavior and energetics of physical systems.