Question

An isolated molecular system exists in two states of equal energy. State A has high multiplicity, whereas State B has low multiplicity. Without any external agents, a system in State B will spontaneously convert to State A. True/False

Short answer

True, the system will move towards higher entropy.

Step-by-step solution

Identify System Properties

In this problem, we have two states, A and B, within an isolated molecular system. Both states have equal energy, but they differ in their multiplicity values, with State A having high multiplicity and State B having low multiplicity.

Understand Principles of Thermodynamics

In an isolated system, processes occur that increase the entropy of the system. The multiplicity of a state is directly related to its entropy, as multiplicity represents the number of ways a state can be achieved.

Apply Concepts of Entropy and Multiplicity

Since State A has higher multiplicity compared to State B, it also has higher entropy. Without external influences, systems naturally progress towards states of higher entropy.

Determine the Transition Outcome

Given that State A has a higher multiplicity and thus higher entropy compared to State B, a system that is initially in State B will spontaneously convert to State A because isolated systems tend to move towards a state of higher entropy.

Key concepts

Molecular States

Molecular states refer to the various configurations that a small or large group of molecules can assume. These configurations are defined by the distribution of individual molecules within different energy levels or positions. Each molecular state represents a particular arrangement of molecules, contributing to the system's overall properties such as density, energy, and temperature.
In the example of the isolated molecular system, States A and B are two possible configurations that have the same energy but differ in how the molecules are arranged. Since both have equal energy, one might initially assume they are equally probable. However, other factors like entropy and multiplicity can affect the likelihood of the system being in one state versus the other.
Throughout thermodynamics, understanding these states is crucial because they help determine what transitions could happen naturally in isolated systems as they move towards equilibrium.

Multiplicity

Multiplicity is a key concept in statistical thermodynamics. It refers to the number of ways a particular macroscopic state can be realized by different microscopic arrangements of molecules.
Imagine a simple dice roll. A sum of 7, for example, can be achieved in more ways (combinations of dice rolls) than a sum of 2 or 12. In this analogy, the higher number of ways represents higher multiplicity.
For molecular states, having higher multiplicity means more possible arrangements that result in the same observable state. Greater multiplicity corresponds to higher entropy, as measured by Boltzmann's formula for entropy: \[ S = k \ln(\Omega) \]where \( S \) is the entropy, \( k \) is Boltzmann's constant, and \( \Omega \) is the multiplicity.
Thus, in an isolated system, the state with higher multiplicity (like State A in the exercise) is favored because systems tend to move towards maximum entropy, which corresponds to maximum disorder and more possible arrangements.

Isolated Systems

An isolated system is one that does not exchange matter or energy with its surroundings. This means that any change happening within such a system is due to its internal processes, without any influence from outside. When analyzing thermodynamic changes in isolated systems, we apply the second law of thermodynamics, which states that the entropy of an isolated system never decreases over time. This implies that isolated systems, when left to their own devices, naturally progress towards states with higher entropy.
In the exercise, the initial state (State B) has lower entropy due to its lower multiplicity. Over time, the system will shift towards State A, which has a higher multiplicity and thus higher entropy, because an increase in entropy signifies a more probable configuration in the long term.