Question

Why does the entropy of a gas increase when it expands into a vacuum?

Short answer

The entropy of a gas increases when it expands into a vacuum because the number of possible microstates for the particles increases, which corresponds to a more disordered state.

Step-by-step solution

Understanding Entropy

Entropy is a measure of the disorder or randomness of a system. A state with higher entropy is more disordered, and thus there are more ways to arrange the particles without changing the system's overall energy.

The Gas Expansion

When a gas expands into a vacuum, the number of possible positions and velocities of its molecules increases. There is no external work done, and no heat is transferred, as the gas is expanding into a vacuum.

Increase in Entropy

Because there are more ways to arrange the particles in the larger volume, the entropy of the system increases. This increase corresponds to an increase in the number of microstates compatible with the macrostate of the system.

Key concepts

Entropy in Thermodynamics

Entropy is one of the fundamental concepts in thermodynamics tied to the idea of disorder or randomness in a system. It's often associated with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time. Imagine entropy as a measure of how many different ways a system can be arranged without changing its overall energy. The more ways there are to arrange the system (the more microstates), the higher the entropy. In practical terms, entropy can be thought of as the degree of unpredictability or the number of choices available at a microscopic level. It's like having a large crowd in a room: if everyone is seated in assigned seats (low entropy), there's not much disorder. But if everyone is walking around randomly (high entropy), the room is in a higher state of disorder. Higher entropy typically leads to less available energy to do work, as the energy is more spread out and less concentrated. This is why high entropy states, such as heat energy, are often less useful than other forms of energy, like mechanical or electrical energy, which are more ordered and can be directed to perform tasks efficiently.

Gas Expansion into Vacuum

When a gas expands into a vacuum, it undergoes a specific process that inherently increases its entropy. Imagine a syringe filled with gas, where the plunger moves backward without any external force; the gas then occupies more volume as it expands into the additional space. This expansion into a vacuum—where there is no opposing pressure—occurs without any exchange of heat or work being done by the gas.

Why Does Entropy Increase?

During this free expansion, the gas molecules now have more space to move around and can be found in a larger number of positions and states (more microstates), all while maintaining the same total energy. Because no energy leaves or enters the system, we can consider the gas' expansion into a vacuum as a reversible process—an ideal scenario often used in thermodynamics to understand entropy changes.

Microstates and Macrostates

The concepts of microstates and macrostates are integral to understanding entropy. A macrostate is the overall, observable state of a system, such as the pressure, volume, and temperature of a gas. Microstates, on the other hand, are the countless configurations that the system's molecules can assume while still corresponding to the same macrostate.

Counting the Possibilities

For example, if you take a snapshot of a gas at a certain temperature and pressure, there are trillions of microstates in which the molecules can be arranged to give those exact temperature and pressure readings. The increase in entropy when a gas expands into a vacuum correlates to an increase in the number of microstates. More volume means more positions and velocities available to each molecule. Therefore, as the gas expands and the number of possible microstates increases, so does the entropy because there are more ways for the system's energy to be distributed.