Question

A two-sided room contains six particles, \(a, b, c, d\). \(e .\) and \(f\), with two on the left and four on the right. (a) Describe the macrostate. (b) Identify the possible microstates. (Note: With only six particles, this isn't a thermodynamic system, but the general idea stitl applies. and the number of combinations is tractable.)

Short answer

The macrostate is described as two particles on the left and four particles on the right. The microstates, which are the possible configurations of the particles that result in this macrostate, are \(15\) in number: \(\{(a,b),(a,c),(a,d),(a,e),(a,f),(b,c),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f),(d,e),(d,f),(e,f)\}\).

Step-by-step solution

Identifying the Macrostate

A macrostate can be described by macroscopic properties. In this case, the macrostate would be described by the number of particles on each side of the room. The description of the macrostate would thus be: ‘Two particles are on the left side of the room, four particles are on the right side of the room.’

Considering Possible Microstates

Microstates are the distinct configurations that the system can take on while still in the described macrostate. Here, the microstate would be any arrangement of the six particles across the two sides of the room that preserves the totals on each side.

Listing All Possible Microstates

Given six distinct particles \(a, b, c, d, e, f\), one can enumerate all the possible ways that two particles could be chosen to be on the left side. The microstates are as follows: \(\{(a,b),(a,c),(a,d),(a,e),(a,f),(b,c),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f),(d,e),(d,f),(e,f)\}\).

Counting the Microstates

The number of possible microstates corresponds to the number of the combinations listed in Step 3. By counting the combinations, we find there are 15 possible microstates.

Key concepts

Macrostate

Understanding the macrostate in thermodynamics is akin to viewing a large-scale snapshot of a system's overall state, much like glancing at a forest and describing its general appearance rather than the individual trees. In our given exercise, where a two-sided room contains six distinct particles, the macrostate is described by the number of particles on each side of the room.

This simplification allows us to describe the system with fewer details, focusing only on aggregate, macroscopic properties like volume, pressure, temperature, and, in this case, particle distribution - two particles on the left side and four on the right. It represents a high-level perspective without concern for specifics of particle identity or position.

Microstate

Diving into the detailed configurations of our system leads us to microstates. These are the specific, microscopic arrangements of particles that are consistent with the given macrostate. For the exercise, each microstate would show a unique placement of the six individual particles, labeled as \(a, b, c, d, e, f\) in a way that respects the overall particle distribution of the macrostate.

It is vital to recognize that although different microstates might look dissimilar at a microscopic level (with particles in different positions), they are considered equivalent if they result in the same macroscopic properties. Hence, all configurations with two particles on the left and four on the right belong to the same macrostate, despite their differences in positioning. This concept is fundamental in understanding how large numbers of particles behave in thermodynamics, underpinning the statistical nature of this field.

Particle Distribution

In thermodynamics, particle distribution is a term that refers to the arrangement of particles within a given space or system. Our textbook exercise illustrates this concept through a simple scenario: particles distributed between two sides of a room.

Effective teaching of this concept requires illustrating the vast number of ways particles can be arranged while retaining certain macroscopic properties, showcasing the interplay between the macrostate and microstate perspectives. For the six distinct particles in our exercise, the macroscopic descriptor is 'two on the left, four on the right,' yet numerous microstates reflect this scenario with different particle arrangements. Such distribution patterns are integral to many physical laws and phenomena, including entropy and probability.

Combinatorics in Physics

Combinatorics, the branch of mathematics dealing with counting, arranging, and combination, plays a pivotal role in physics, especially in thermodynamics. The textbook exercise provides a clear demonstration of applying combinatorial principles to determine the number of microstates. Here, instead of merely combining numbers, we are counting the distinct ways particles can be arranged, or combined, to achieve the defined macrostate.

Each unique arrangement -- from \(a, b\) on the left to \(e, f\) on the left -- is a result of combinatorics. In this context, combinatorics helps us understand the statistical foundations of thermodynamics. By enumerating all possible microstates through combinatorial calculations, physicists can infer the likelihood of a system's state and make predictions about its macroscopic behavior.