Question

(a) The energy of a gas is increased by heating it. Using \(\mathrm{CO}_{2}\) as an example, illustrate the different ways in which additional energy can be distributed among the molecules of the gas. (b) You are told that the number of microstates for a system increases. What does this tell you about the entropy of the system?

Short answer

When a gas like CO2 is heated, its energy is distributed among its molecules in several ways, such as translational, rotational, and vibrational motions. As the number of microstates (specific arrangements of energy and particle positions) increases, so does the entropy of the system, according to Boltzmann's Entropy Formula: \( S = k_B \ln{W} \), where \(kB\) is the Boltzmann constant, and \(W\) is the number of microstates. Thus, an increase in the number of microstates corresponds to an increase in the entropy of the system.

Step-by-step solution

Part (a): Understanding Energy Distribution in Gas Molecules

When a gas is heated, its energy is increased. In the case of CO2, the energy can be distributed among the molecules in several ways. The additional energy can manifest in different energy levels of motion such as translational, rotational, and vibrational motions. Let's discuss them briefly: 1. Translational motion: Molecules gain kinetic energy and move in different directions in a linear manner. 2. Rotational motion: Molecules rotate about their axes causing an increase in their rotational energy. 3. Vibrational motion: Molecules vibrate causing an increase in their vibrational energy. These three types of motions contribute to the internal energy of the gas molecules. When CO2 is heated, its energy is distributed among these motions, leading to different energy states for the individual molecules.

Part (b): Microstates and Entropy

A microstate is a specific arrangement of energy and particle positions in a system. When you are told that the number of microstates has increased, it means there are more possible arrangements or configurations of particles in the system. Entropy is a measure of the disorder or randomness in a system. According to the second law of thermodynamics, the natural tendency of a system is to go from an ordered state to a disordered state (i.e., an increase in entropy). So, when the number of microstates increases, it typically means that the entropy of the system has also increased. The relation between entropy (S) and the number of microstates (W) is given by Boltzmann's Entropy Formula: \[ S = k_B \ln{W} \] Where kB is the Boltzmann constant, and W is the number of microstates. From this equation, we can see that an increase in the number of microstates will cause an increase in entropy.

Key concepts

Energy Distribution in Gases

When we think about energy distribution in gases, it helps to consider how gas molecules can store energy in different forms. More energy is introduced when we heat a gas, like carbon dioxide (CO₂). This added energy is not randomly contained but instead is distributed across several distinct types of molecular motion.

• Translational Motion: This is the most straightforward movement, where molecules zoom around in space. Their kinetic energy increases as they move in various straight-line paths.
• Rotational Motion: Here, the molecules spin around one or more axes. This increases the molecule's rotational energy, much like a spinning top builds momentum.
• Vibrational Motion: As molecules oscillate, they absorb energy through vibrations. This type of energy augmentation involves stretching and compressing the bonds between atoms within a molecule.

All these motions collectively contribute to the internal energy of the gas. As CO₂ molecules gain energy, they can exist in multiple states thanks to these forms of motion. Understanding these distributions helps us grasp why gases behave differently when subjected to heat and pressure.

Microstates

The term "microstate" refers to a unique arrangement of particles in a system. Each microstate represents a specific configuration, including how energy is spread among particles. Imagine these as detailed snapshots of molecular positions and energy levels.

With more microstates available, the system can be in numerous possible configurations. This could mean one particle might have high energy while others have low energy, or all particles might share energy equally. Each distinct configuration represents a different microstate.

• Increase in Microstates: When you hear that the number of microstates is growing, it tells us there's an increase in potential configurations of the system. This means there are more possible ways to distribute energy among particles.
• Implication of More Microstates: An increase in microstates often implies an increase in the system's complexity or disorder. This is closely tied to the concept of entropy, which we will discuss further.

Understanding microstates not only informs us about the possible states a system can occupy but also highlights the system's entropy dynamics.

Boltzmann's Entropy Formula

Boltzmann's Entropy Formula connects the number of microstates to a very important concept known as entropy. Entropy is often associated with the level of disorder or randomness within a system. When a system can access more states or configurations, its entropy tends to increase.

The formula is expressed as follows: \[ S = k_B \ln{W} \]Here,

• S is the entropy.
• k_B stands for Boltzmann's constant, a fundamental physical constant that relates energy to temperature.
• W represents the number of microstates the system can adopt.

As the number of microstates (W) rises, the natural logarithm of W also increases, leading to an increase in entropy (S). This formula succinctly encapsulates the idea that greater potential configurations of molecular energies and positions result in higher disorder, thus more entropy.

Boltzmann's equation beautifully bridges statistical molecular behavior with macroscopic thermodynamic properties, illustrating how deep microscopic changes affect the overall system. By understanding this relationship, we appreciate how intrinsic randomness at the molecular level influences the larger thermodynamic landscape.