Question

(a) What is the difference between a state and a microstate of a system? (b) As a system goes from state A to state \(B,\) its entropy decreases. What can you say about the number of microstates corresponding to each state? (c) In a particular spontaneous process, the number of microstates available to the system decreases. What can you conclude about the sign of \(\Delta S_{\text { surr }}\) ?

Short answer

(a) A state is a set of macroscopic properties defining a system's condition (e.g., temperature, pressure, volume), while a microstate is a specific microscopic configuration of particles associated with a particular state, having the same macroscopic properties but varying positions and velocities of particles. (b) If entropy decreases when going from state A to state B, the number of microstates for state B is less than for state A, as entropy is related to microstates through \(S = k_{B} \ln{W}\). (c) If microstates decrease in a spontaneous process, \(\Delta S_{\text{sys}}\) is negative, and for the process to be spontaneous, \(\Delta S_{\text{surr}}\) must be positive, indicating an increase in the entropy of the surroundings.

Step-by-step solution

Part(a): Define State and Microstate

A state of a system is a specific set of macroscopic properties that uniquely define its condition, such as temperature, pressure, and volume. On the other hand, a microstate is a specific microscopic configuration of particles associated with a particular state. Each microstate has the same macroscopic properties (i.e., it corresponds to the same state), but the individual positions and velocities of the particles may vary.

Part(b): Entropy and Number of Microstates

When the system transitions from state A to state B with a decrease in entropy, it implies that the number of possible microstates for the system has decreased. This is because entropy is a measure of the disorder or randomness in a system, and it is directly related to the number of microstates through the formula \(S = k_{B} \ln{W}\), where \(S\) is the entropy, \(k_{B}\) is the Boltzmann constant, and \(W\) is the number of microstates. If the entropy decreases, then the number of microstates corresponding to state B would be less than the number of microstates corresponding to state A.

Part(c): Sign of \(\Delta S_{\text{surr}}\) in a Spontaneous Process

If the number of microstates available to the system decreases during a spontaneous process, it implies that the entropy of the system, \(S_{\text{sys}}\), decreases. For a process to be spontaneous, the total entropy change, which includes both the system and its surroundings, must be positive: \(\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0\). Since \(\Delta S_{\text{sys}}\) is negative (due to decreasing microstates), for the process to be spontaneous, we must have \(\Delta S_{\text{surr}}\) positive (i.e., the entropy of the surroundings must increase) to satisfy the inequality and maintain a positive total entropy change.

Key concepts

Understanding Entropy

Entropy is a fundamental concept in thermodynamics that relates to the amount of disorder or randomness in a system. Put simply, it's a measure of how many different ways you can rearrange the components of the system without changing its macroscopic properties, like temperature and pressure.

• Imagine a messy room: a high degree of messiness means high entropy because there are many ways to organize things.
• On the other hand, a perfectly organized room has low entropy due to fewer possible arrangements.

The mathematical representation of entropy is given by the formula \(S = k_B \ln W\), where \(S\) is the entropy, \(k_B\) is Boltzmann's constant, and \(W\) is the number of possible microstates. This equation shows that if the number of microstates increases, the entropy increases as well.
Thus, more order is associated with fewer microstates and less entropy. Understanding this relationship helps explain why a decrease in entropy often signals a decrease in possible microstates.

Exploring Microstates

A microstate is like a unique snapshot of a system at the microscopic level. While a state gives us the big picture by considering overall properties, microstates dive deeper into the particulars of how each particle in the system can be arranged.

• Think of a molecular assembly: each possible arrangement of molecules represents a different microstate.
• The crucial point is that many microstates can correspond to one macroscopic state, as they share the same observable properties (like total energy and volume).

To clarify, consider a gas in a box. The macroscopic state might define just temperature and pressure, whereas each precise arrangement and velocity of gas molecules denote a separate microstate.
Thus, microstates highlight the complexity and multitude of ways a system could potentially exist at a microscopic level, contributing to our understanding of entropy.

What is a Spontaneous Process?

In thermodynamics, a spontaneous process is one that occurs naturally without needing external energy input. Such processes are driven by an increase in the overall entropy of the universe. This includes both the system's entropy and the surrounding environment's entropy.

• For a spontaneous process, the total entropy change must be positive, denoted as \(\Delta S_{\text{total}} > 0\).
• This is given by the relationship \(\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}\).

If the system's entropy decreases, as can happen when microstates decrease, the surroundings must see an increase in entropy to ensure the total entropy keeps rising.
Thus, in such scenarios, even when \(\Delta S_{\text{sys}}\) is negative, \(\Delta S_{\text{surr}}\) should be positive enough to keep the process spontaneous, upholding the Second Law of Thermodynamics. Understanding this helps in predicting the direction of natural processes and their feasibility.