Choose the substance which has higher possible entropy (per mole) at a given temperature. (a) solid carbon dioxide (b) nitrogen gas at 1 atm (c) nitrogen gas at \(0.01\) atm (d) nitrogen gas at \(0.00001\) atm

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy in a system. This field is crucial for understanding how energy is converted in chemical reactions and physical changes. One of the central concepts in thermodynamics is entropy, which is used to measure the disorder or randomness in a system. The second law of thermodynamics states that in any spontaneous process, the total entropy of the system and its surroundings always increases, leading to the notion that systems naturally progress toward a state of maximum entropy. This explains why heat flows from hot to cold and why gases expand to fill their containers.

The three traditional states of matter are solid, liquid, and gas. Each state is distinguished by particle arrangement, energy, and movement. In solids, atoms or molecules are tightly packed in a structured pattern, with little movement besides vibration. Solids have a definite shape and volume. Liquids have a definite volume but take the shape of their container, as their particles have more freedom to move than in solids. Gases have neither a fixed volume nor shape and will expand to fill their container. Particles in a gas move rapidly and are widely spaced apart, which often results in higher entropy compared to solids and liquids at the same temperature.

Entropy in a system is greatly affected by pressure. In a gaseous system, reducing pressure allows gas molecules to spread out over a larger volume, thereby increasing the number of possible configurations or microstates the system can occupy. This increases the entropy of the system, reflecting a higher degree of randomness. Conversely, when pressure is increased, the gas molecules are compressed into a smaller volume, reducing the number of possible microstates and, consequently, decreasing entropy. Thus, pressure and entropy are inversely related in gases; as one decreases, the other increases, and vice versa.

A microstate is a specific configuration that a system can assume based on the arrangement and states of its individual particles. In simple terms, each microstate represents a unique way in which the system's particles can be distributed in terms of position and energy. In the realm of thermodynamics, entropy can be thought of as the logarithm of the number of microstates (W) available to a system at a given energy. The more microstates available, the higher the entropy, indicating a larger amount of disorder. In the context of our gaseous nitrogen example, a sample at a lower pressure has a greater volume to spread out, leading to many more microstates and therefore a higher entropy.