Question

] Consider a system that conststs 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-dice system has 11 possible states, with the highest entropy state being 7 (six possible microstates) and the lowest entropy states being 2 and 12 (one possible microstate each). An exact calculation for the system's absolute entropy cannot be made without specific temperatures or probability distributions provided; however, understanding the relationship between microstates, states, and entropy is essential in this exercise.

Step-by-step solution

Identify and explain the microstates

A microstate refers to a specific way in which the system can be arranged, given a particular macroscopic state. In the case of two dice, a microstate is a specific arrangement of numbers on the top faces. For example, if we have two dice with top face values 2 (die 1) and 5 (die 2), this particular arrangement is considered a microstate of the system.

Determine the state corresponding to each microstate

The state of the system is defined by the sum of the values shown on the top faces. For each microstate, we must add the values of the top faces of each die together. For example, if die 1 has a face value of 2 and die 2 has a face value of 5, then the state of the system corresponding to this microstate is 7 (2 + 5).

Count the possible states for the system

To count the number of possible states for the system, we must consider that a die has six possible outcomes for each roll. When two dice are rolled, there is a total of 6*6 = 36 possible outcomes, as each die's face value is combined with every other face value. However, the minimum sum of top faces' values for two dice is 2, and the maximum is 12. Therefore, there are 11 possible states for the system (from 2 to 12, inclusive).

Determine the state(s) with the highest entropy

Entropy is a measure of the number of microstates that correspond to a given state, and higher entropy indicates more possible microstates. The highest entropy state(s) correspond to having the largest number of microstates. For the two-dice system, the highest entropy state is 7, with six microstates contributing to it (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). No other state has more microstates associated with it.

Determine the state(s) with the lowest entropy

The lowest entropy state(s) correspond to having the least number of microstates. For this two-dice system, the lowest entropy states occur when the sum of the top faces is either 2 (1+1) or 12 (6+6), because each of these has only one microstate contributing to it.

Calculate the absolute entropy of the two-dice system

The absolute entropy of a system can be computed using the Boltzmann's entropy formula: \(S = k_B \ln W\), where \(S\) is the entropy, \(k_B\) is the Boltzmann constant, and \(W\) is the number of microstates for a given state. In this problem, we are not given any specific temperatures or probability distributions, so an exact calculation for the system's absolute entropy is not feasible. However, understanding the principles and relationship between microstates, states, and entropy should be sufficient for this exercise.

Key concepts

Microstates and Macrostates

In the realm of chemical systems, the concepts of microstates and macrostates are crucial to understanding entropy. A microstate represents a specific, detailed arrangement at the molecular level within a system. For example, when you toss two dice, each possible result (like a 2 on the first die and a 5 on the second die) is considered a unique microstate.

On the other hand, a macrostate is the overall state of the system characterized by macroscopic properties such as pressure, temperature, and volume—in the case of the dice, the macrostate would be the sum of the top faces. The connection between microstates and macrostates lies at the heart of statistical thermodynamics, as the number of microstates associated with a particular macrostate determines its entropy—an important measure of disorder or randomness in a system.

If we consider a two-dice system, each roll has a definite number of microstates corresponding to the sum of the dice. For instance, a sum of 7 has more microstates than a sum of 2 or 12, since there are more combinations of dice rolls that result in a 7. Thus, the entropy, which we can think of as the level of 'spreading out' or 'dispersal' of energy, would be higher for a sum of 7 than for a sum of 2 or 12.

Boltzmann's Entropy Formula

When discussing statistical thermodynamics, one cannot overlook the seminal contribution of Ludwig Boltzmann. Boltzmann's entropy formula is a pillar in this field, and it provides a quantitative measure of the disorder within a system. The formula is expressed as \(S = k_B \ln W\), where \(S\) represents the entropy, \(k_B\) is the Boltzmann constant (approximately \(1.38 \times 10^{-23} J/K\)), and \(W\) stands for the number of microstates associated with a given macrostate.

Returning to our two-dice example, if we look at the state with a sum of 7, this state's entropy can be calculated using Boltzmann's formula, considering the number of microstates (different dice combinations) leading to this sum. Since entropy increases with the number of microstates, the state where the dice sum to 7, with the most ways to occur, would have the highest entropy. While we might not calculate the exact numeric value of entropy in this specific exercise, understanding Boltzmann's formula helps explain why certain states have higher entropy based on their number of microstates.

Statistical Thermodynamics

Statistical thermodynamics is a branch of physical science that melds the principles of thermodynamics with statistical methods. It addresses how the properties of atoms and molecules translate into the macroscopic properties observed in bulk material. In other words, it bridges the gap between the micro-level behavior of particles and the macro-level physical properties.

As applied to our dice example, statistical thermodynamics would guide us in understanding how the probability of each roll (the microstate) determines the overall behavior of the system (the macrostate). It's an elegant framework that explains why certain outcomes (like a total of 7 from two dice) are more probable, and hence possess higher entropy, because there are more ways (microstates) for that outcome to manifest. In a chemical system, this branch of study provides insights into the way energy disperses at a molecular level and how this impacts observable phenomena like temperature and pressure.

Probability in Chemical Systems

Probability plays an integral role in chemical systems, particularly when discussing entropy and the behavior of particles at the quantum level. In fact, entropy can be thought of as a measure of uncertainty or the number of possible configurations (microstates) that a system can have.

In any chemical system, the likelihood of particular microstates factors heavily into the macroscopic behavior of the system. This can be exemplified by the possible arrangements of molecules in a gas, the alignment of spins in magnetic materials, or the various conformations of complex molecules like proteins. The higher the probability for a particular macrostate, the greater the number of microstates and, consequently, the higher the entropy associated with that state.

Considering the dice analogy, we assign probabilities to each possible roll outcome, with certain sums having higher probabilities due to the greater number of microstates that can yield that total. Likewise, in chemical systems, states with higher entropy are more statistically likely because the probability of the system being in one of many microstates is higher, highlighting the deep-seated connection between probability and thermodynamic properties.