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 of a system refers to a set of macroscopic properties, while a microstate refers to a specific arrangement of particles exhibiting those macroscopic properties. (b) As entropy decreases, the number of microstates available to the system also decreases, meaning state A has more microstates than state B. (c) If the number of microstates decreases in a spontaneous process, then the entropy change of the system is negative, and the change in entropy of the surroundings must be positive to maintain spontaneity, making \(\Delta S_{\text{total}} \geq 0\).

Step-by-step solution

(a) State and Microstate:

A state of a system refers to a specific set of macroscopic properties such as temperature, pressure, or volume. In contrast, a microstate refers to a specific arrangement of particles in a system, describing the position and momentum of each particle, such that the system exhibits the given macroscopic properties.

(b) Entropy and Number of Microstates:

As the system goes from state A to state B, with its entropy decreasing, it means the number of microstates available to the system is decreasing. In other words, state A has a higher number of possible microstates compared to state B since a higher entropy implies more possible microscopic arrangements.

(c) Change in Surroundings' Entropy:

In a particular spontaneous process, if the number of microstates available to the system decreases, it implies that the entropy change of the system, \(\Delta S_{\text{sys}}\), is negative. However, since the process is spontaneous, it is only possible if the change in entropy of the surroundings, \(\Delta S_{\text{surr}}\), compensates for this decrease. Thus, \(\Delta S_{\text{surr}}\) must be positive, making the total change in entropy, \(\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}}\), greater than or equal to zero.

Key concepts

Microstates

In the realm of thermodynamics, "microstates" describe the myriad ways in which particles in a system can be arranged,resulting in the same macroscopic properties such as temperature or pressure. Think of microstates as the tiny building blocks that make up the larger structure of the state of a system.Each individual microstate corresponds to a unique way particles can be distributed in terms of position and momentum.For example, if you have a box with gas particles, a microstate could indicate where each particle is and how fast they're moving.

• Microstates capture individual configurations of particles.
• More microstates equate to higher entropy, as there's more disorder.
• Microstates are crucial for understanding the statistical nature of entropy.

The concept of microstates allows scientists to quantify the idea of randomness or disorder in a system by using the formula for entropy:\[ S = k \ln W \]where \( S \) is the entropy, \( k \) is the Boltzmann constant, and \( W \) is the number of microstates available to the system.

Macrostates

Macrostates refer to the broader, observable properties of a system, such as temperature, pressure, and volume. These are the characteristics we can directly measure. While each macrostate can be constituted by numerous microstates, a system typically only exhibits its macroscopic properties without exposing these underlying complexities. Imagine a macrostate as the overall condition of our box of gas particles, defined by the temperature and pressure you feel or measure. While you know the gas is hot or dense, the specific arrangements of particles forming this state remain unseen.

• Macrostates are defined by observable, measurable properties.
• They are simplified versions of the complex arrangements inherent in microstates.
• The shift from one macrostate to another can affect the system's entropy.

When understanding macrostates, it is essential to note that different configurations of microstates can result in the same macrostate. Thus, the number of microstates determines the macrostate's entropy level.

Spontaneous Process

A spontaneous process is one that occurs naturally without any external force helping it along.This type of process often results in an increase in the universe's total entropy.However, there can be cases where the entropy of the system itself decreases.When the number of microstates in a system decreases, such as in a system moving to a more ordered state, the system's entropy (\( \Delta S_{\text{sys}} \)) is negative.But for the process to remain spontaneous, the surroundings' entropy (\( \Delta S_{\text{surr}} \)) must increase.

• A spontaneous process can decrease a system's entropy but still proceed due to increased surroundings' entropy.
• This ensures the total entropy change (\( \Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \)) stays non-negative.
• Spontaneity aligns with the second law of thermodynamics, dictating that the total entropy of a system and its surroundings never decreases.

Thus, even if within a spontaneous process the entropy of the system drops, the surrounding environment makes up for it, driving the natural progression of these processes.