Problem 30 - Chemistry: The Central Science in SI Units, Global Edition

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 \(\Delta S\) surr?

Step-by-Step Solution

Answer

(a) A state of a system refers to its macroscopic properties (e.g., temperature, pressure, volume) while a microstate refers to a specific arrangement of particles at a given energy level compatible with a particular macroscopic state. (b) If entropy decreases as the system goes from state A to state B, the number of microstates corresponding to state B is less than that for state A. (c) If the number of microstates decreases in a spontaneous process, the sign of \(\Delta S_{surr}\) is positive.

Step 1: a) State and microstate definition

A state of a system refers to the macroscopic properties of the system such as temperature, pressure, and volume. The macroscopic properties describe the overall behavior of the system. On the other hand, a microstate refers to a specific arrangement of particles in the system at a given energy level, which is compatible with a particular macroscopic state. Microstates represent all the possible arrangements of particles that can lead to the same macroscopic properties.

Step 2: b) Entropy and the number of microstates

Entropy (S) is a measure of the number of microstates (W) that correspond to a particular macroscopic state. The relationship between entropy and the number of microstates is given by the Boltzmann's entropy formula: \(S = k_B \ln W\), where \(k_B\) is Boltzmann's constant. As the system goes from state A to state B, if its entropy decreases, it means that the number of microstates corresponding to state B is less than the number of microstates for state A. This can be seen from the formula as if W decreases, then S will also decrease.

Step 3: c) Sign of entropy change in surroundings

For a spontaneous process, the total entropy change \(\Delta S_{tot}\) is positive. The total entropy change can be expressed as the sum of the entropy change of the system (\(\Delta S_{sys}\)) and the entropy change of the surroundings (\(\Delta S_{surr}\)): $$\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr}$$ In the given spontaneous process, if the number of microstates available to the system decreases, it means the system's entropy is decreasing, i.e., \(\Delta S_{sys} < 0\). Therefore, for the total entropy change to be positive, the entropy change of the surroundings must be positive, i.e., $$\Delta S_{surr} > 0$$ Hence, we can conclude that the sign of \(\Delta S_{surr}\) is positive when the number of microstates available to the system decreases in a spontaneous process.

Key Concepts

Macroscopic Properties

Macroscopic properties refer to the large-scale characteristics of a system that can be measured directly or observed easily. These properties include:

• Temperature: A measure of the average kinetic energy of the particles within a system. It determines the direction of thermal energy transfer.
• Pressure: The force exerted by particles colliding within the walls of its container, measured per unit area.
• Volume: The space occupied by a system. It is crucial in defining the system's state in thermodynamics.

These macroscopic properties give us an overview of the system’s state without detailing the individual motion or arrangement of particles. They are essential in establishing how a system behaves under various conditions, allowing us to predict changes and the outcomes of different interactions.

Microstates

Microstates are the different possible configurations that particles within a system can adopt, which result in the same macroscopic properties.

• Each microstate represents a unique arrangement of every particle and its energy state in the system.
• A single macroscopic state can correspond to numerous microstates.

The concept of microstates is critical to understanding statistical mechanics and thermodynamics. By analyzing all possible microstates, we can deduce important information about the system's entropy. Small changes in the number or energy of particles can shift the balance between these microstates, influencing the system's observable properties.

Entropy Change

Entropy is a measure of disorder or randomness, closely tied to the number of microstates of a system.

• Entropy change \(\Delta S\) indicates the level of randomness in a process.
• According to Boltzmann's formula \(S = k_B \ln W\), an increase in the number of microstates \(W\) results in higher entropy.

When a system's entropy decreases, as it transitions from state A to B, fewer microstates are available. This simplification often results from energy constraints or external interactions that limit particle freedom. Understanding entropy change is crucial for determining spontaneity in processes. For a process to occur spontaneously, the total entropy, which includes both the system and its surroundings, should increase. If the system loses entropy, the surroundings must gain it to satisfy this condition: \(\Delta S_{surr} > 0\).