B.4.2 Entropy and system evolution (HL only)

Entropy: A Measure of Disorder

Entropy is a fundamental concept in thermodynamics that describes the degree of disorder or randomness in a system.

Calculating Entropy Change

The change in entropy, $\mathrm{\Delta }S$, is defined as:

$$\mathrm{\Delta }S=\frac{\mathrm{\Delta }Q}{T}$$

where:

• $\Delta Q$ is the heat added to the system.
• $T$ is the absolute temperature (in Kelvin) at which the heat is added.

The Second Law of Thermodynamics

The second law of thermodynamics introduces the concept of entropy to explain why certain processes occur spontaneously while others do not.

Entropy Increases in Isolated Systems

1. In an isolated system (where no energy or matter is exchanged with the surroundings), the total entropy always increases or remains constant.
2. This is often stated as:
   • Entropy increases in realistic, irreversible processes.
   • Entropy remains constant in idealized, reversible processes.

Reversible and Irreversible Processes

In these processes, the entropy change of the system is exactly balanced by the entropy change of the surroundings, resulting in no net change in the total entropy.

These processes always result in an increase in the total entropy of the universe.

Entropy and Microstates

Entropy can also be understood from a statistical perspective, which connects macroscopic thermodynamic properties to microscopic configurations.

Microstates and Macrostates

1. A microstate is a specific arrangement of particles in a system, while a macrostate is defined by macroscopic properties such as temperature, pressure, and volume.
2. The number of microstates, $\Omega$, corresponding to a given macrostate determines the entropy of the system.

Boltzmann’s Entropy Formula

The entropy, $S$, of a system is given by Boltzmann’s formula:

$$S=k_B\ln \mathrm{\Omega }$$

where:

• $k_B$ is the Boltzmann constant ($1.38\times {10}^{-23}{\text{J K}}^{-1}$).
• $\Omega$ is the number of microstates.

Entropy and Probability

1. The statistical definition of entropy highlights its connection to probability.
2. A macrostate with a larger number of microstates is more probable and has higher entropy.
3. This explains why systems tend to evolve toward states of higher entropy—these states are simply more likely to occur.

Applications and Implications

1. Entropy has profound implications beyond thermodynamics.
2. It helps explain why certain processes are irreversible and why energy transformations are never 100% efficient.
3. Entropy also plays a critical role in fields such as information theory, where it measures the uncertainty or randomness of information.

Reflection and Self-Assessment

Entropy provides a powerful framework for understanding the directionality of natural processes and the limitations imposed by the second law of thermodynamics.