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Chapter 14: Problem 79

For an \(N\) -molecule gas, show that the number of microstates with \(n\) molecules on the left side of the box is \(N ! /[n !(N-n) !]\). Hint: Consider the pattern established by the cases of two, three, four, and five molecules.

Short Answer

Expert verified
The number of microstates is \( \frac{N!}{n!(N-n)!} \).

Step by step solution

01

Understanding Microstates

In thermodynamics, a microstate refers to a specific configuration of a system's components. For a gas of molecules, the microstates represent the different ways the molecules can be distributed between two sides of a box. We are tasked with finding how many microstates correspond to having \(n\) molecules on the left side when there are \(N\) total molecules.
02

Analyzing Simple Cases

We begin by analyzing small numbers of molecules to identify a pattern. For 2 molecules, we can have 2 on the left, 1 on the left, or 0 on the left, giving microstates 1, 2, and 1 respectively. For 3 molecules, the configuration possibilities become 3 on the left, 2 on the left, 1 on the left, or 0 on the left, yielding microstates 1, 3, 3, and 1 respectively.
03

Using Combinatorics for Microstates

The distribution of molecules can be approached using combinatorics, specifically combinations. If you have \(n\) spots to fill out of \(N\) molecules, the number of ways to choose which \(n\) molecules are on one side of the box is given by combinations denoted \( \binom{N}{n} \), which is calculated as \( \frac{N!}{n!(N-n)!} \).
04

Generalizing the Pattern

From using combinations, we generalize the pattern observable from the cases we computed. Using combinatorial logic, placing \(n\) molecules on one side out of \(N\) is computed as \( \frac{N!}{n!(N-n)!} \), which matches our observed pattern in smaller numbers.
05

Conclusion

For an \(N\)-molecule gas, the number of microstates with \(n\) molecules on the left side of the box is indeed \( \frac{N!}{n!(N-n)!} \), aligning with both the pattern observed in simpler examples and the formula derived from combinatorial principles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Microstates
Imagine a box divided into two halves. Each half can contain a different number of molecules. In the world of thermodynamics, how the molecules are distributed between these two halves is what we call a "microstate." If we have a fixed number of molecules, say \( N \), each possible distribution is a unique microstate.
  • The term microstate refers to each distinct way of arranging the molecules between the two sides of the box.
  • For instance, if you have 3 molecules, and they can either all be on the left, all on the right, or some combination in-between.
Microstates are crucial because they help in understanding probabilities in thermodynamic systems. As you analyze how molecules are distributed, you're also considering how likely each arrangement is to occur. When thermodynamics terms are thrown around, think of this as looking at all the options available in how molecules can be shuffled.
Combinatorics
Combinatorics is essentially the mathematics of counting. It involves determining how many different ways we can arrange or select items from a larger set. In our case, with molecules in a box, we're interested in understanding how we can choose a subset of \( n \) molecules to be on one side of a box out of \( N \) total molecules.
  • By using combinations, we find out how many configurations exist for the molecules, given that they can switch sides.
  • This often involves using the formula for combinations: \( \binom{N}{n} = \frac{N!}{n!(N-n)!} \).
This formula calculates different ways to select molecules to place on one side of the container. The factorial \( N! \) signifies all possibilities, while the division by \( n! \) and \( (N-n)! \) corrects for over-counting. Having a solid grasp of combinatorial methods is invaluable when working through thermodynamic problems.
Thermodynamic Systems
When describing a thermodynamic system, you're talking about a specific material or group of materials within a set boundary. The behavior of such systems is influenced by temperature, pressure, and volume 鈥� all of which dictate energy exchanges within the system.
  • In thermodynamic systems, microstates represent different configurations of the system's components, like molecule arrangements.
  • This system's macroscopic properties are often averages over many different microstates.
Understanding thermodynamic systems requires a grasp of both microstates and the broader combinatorial structures. By looking at all possible configurations in a system (the possible microstates), we get insights into how the system behaves overall. This concept is central to fields like statistical mechanics, where microscopic behavior helps explain macroscopic properties.

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Most popular questions from this chapter

The McNeil Generating Station in Burlington, Vermont is one of the world's largest wood-fired electric power plants. The plant produces steam at \(950^{\circ} \mathrm{F}\) to drive its turbines, and condensed steam returns to the boiler at \(90^{\circ} \mathrm{F}\). (Note the temperatures in Fahrenheit, used in U.S. engineering situations.) Find McNeil's maximum thermodynamic efficiency and compare with its actual efficiency of \(25 \% .\) Note: Some of the difference comes from having to evaporate moisture out of the wood-chip fuel.

Find the net entropy change when a \(50-\mathrm{g}\) ice cube initially at \(0^{\circ} \mathrm{C}\) melts in (a) a \(10^{\circ} \mathrm{C}\) room and (b) a \(35^{\circ} \mathrm{C}\) room.

In an example of free expansion, a gas doubles its volume by expanding into a vacuum without pushing on anything. Discuss whether each of the following quantities increase, decrease, or remain the same: pressure, temperature, and entropy.

A standard deck of cards contains 52 different cards. A poker hand consists of five cards, chosen randomly. How many different poker hands are there?

A nuclear power plant has a maximum steam temperature \(\left(T_{\mathrm{H}}\right)\) of \(310^{\circ} \mathrm{C}\). It produces \(650 \mathrm{MW}\) of electric power in the winter, when its \(T_{\mathrm{C}}\) is effectively \(0^{\circ} \mathrm{C}\). (a) Find its maximum winter efficiency. (b) If its summertime \(T_{\mathrm{C}}\) is \(38^{\circ} \mathrm{C},\) what's its summertime electric power output, assuming nothing else changes?
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