Question

(a) What is the difference between a stafe 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 cenclude about the sign of $\mathrm{\Delta }{S}_{\text{wart? }}^{\text{? }}$ ?

Short answer

(a) A state of a system describes its macroscopic properties, while a microstate represents a specific arrangement of particles or energy levels corresponding to a given state. (b) If the system's entropy decreases during a transition from state A to state B, the number of microstates in state B is less than in state A. (c) If the number of microstates decreases in a spontaneous process, $\mathrm{\Delta }{S}_{\text{sys}}<0$. However, to determine the sign of $\mathrm{\Delta }{S}_{\text{tot}}$, we must consider the entropy change in the surroundings.

Step-by-step solution

a. State and Microstate Definitions

: A state of a system refers to the macroscopic physical properties of the system, such as temperature, pressure, and volume. A microstate, on the other hand, is a specific arrangement of particles or energy levels within the system that corresponds to a given macroscopic state. There can be multiple microstates associated with a single macroscopic state of a system.

b. Entropy Change and Number of Microstates

: As a system transitions from state A to state B, its entropy decreases, which means there is a reduction in the level of disorder or randomness in the system. In terms of microstates, this implies that the number of microstates in state B is less than the number of microstates in state A. In other words, when entropy decreases, the system moves toward a more ordered and less probable arrangement of particles or energy levels.

c. Entropy Change in a Spontaneous Process with Decreased Microstates

: In a particular spontaneous process, if the number of microstates available to the system decreases, it suggests that the system is moving toward a more ordered state. This corresponds to a negative change in entropy, i.e., $\mathrm{\Delta }{S}_{\text{sys}}<0$. However, to fully determine the sign of $\mathrm{\Delta }{S}_{\text{tot}}$, we would need to know the change in entropy of the surrounding environment as well. If the decrease in the system's entropy is compensated by an increase in the entropy of the surroundings, the overall process can still be spontaneous.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, energy, and temperature. It is concerned with the macroscopic behavior of systems and encompasses principles that allow us to predict how systems respond to changes in their surroundings.

One of the fundamental concepts in thermodynamics is the distinction between states and microstates. A state can be identified by macroscopic variables such as pressure, volume, and temperature. Each state is made up of countless microstates, which are the specific ways particles can be arranged to achieve those macroscopic conditions. For instance, if we consider a gas in a container, the temperature and pressure define its state. However, many different arrangements of gas particles (microstates) can result in the same temperature and pressure.

In educational contexts, understanding the difference between a state and a microstate enhances the comprehension of thermodynamic processes and facilitates the grasp of more complex ideas such as entropy, which is intrinsic to the subject of thermodynamics.

Entropy Decrease

Entropy is a measure of the disorder or randomness in a system, and it is a key concept in the second law of thermodynamics. It dictates that, in an isolated system, the total entropy can never decrease over time. However, in non-isolated systems, such as those that exchange energy with the surroundings, entropy can indeed decrease.

Understanding that entropy is closely related to the number of microstates gives insight into why certain processes might lead to a decrease in entropy. If the entropy of a system decreases as it goes from state A to state B, it indicates a transition towards a more ordered state with fewer microstates. This is counter-intuitive in natural processes because it implies a move away from disorder. In real-world situations, a decrease in the system's entropy often involves an increase in the entropy of the surroundings, maintaining the overall trend towards increased disorder when viewed on a universal scale.

Breaking down the relationship between entropy and microstates helps students connect the theoretical aspects of thermodynamics with the tangible phenomena around them, providing a more solid understanding of how our universe tends toward chaos, albeit with pockets of order emerging occasionally.

Spontaneous Processes

Spontaneous processes are natural occurrences that happen without external intervention, driven by the tendency to move towards greater entropy or disorder. An example of such a process is the melting of ice at room temperature. In the realm of thermodynamics, spontaneity is often associated with an increase in entropy; however, there can be exceptions.

When the number of microstates decreases during a spontaneous process, it poses an interesting question: How can the process be spontaneous if it leads to reduced disorder? The answer lies in the wider system, including the surroundings. A spontaneous process that results in decreased entropy within the system must be increasing entropy somewhere else, usually in its surroundings, for the overall entropy across the universe to still increase.

This concept aids students in identifying the driving forces behind spontaneous processes and in reconciling seemingly contradictory observations, such as freezing water in a cold environment. Ultimately, the focus should remain on the universe’s entropy, which always increases, despite localized decreases within specific systems.