Understanding Entropy Changes in Different Processes

Entropy is a fundamental thermodynamic property used to quantify the degree of randomness or molecular disorder in a system. Its behaviour during various thermodynamic processes is crucial for understanding the second law of thermodynamics, predicting spontaneity, and solving problems in JEE Main and Advanced examinations.

Definition and Meaning of Entropy

Entropy, denoted by $S$, is defined as a state function that measures the degree of disorder or randomness in a thermodynamic system. It describes how energy in a system becomes more uniformly spread out at a given temperature.

In statistical terms, entropy measures the number of microscopic arrangements (microstates) corresponding to a macroscopic state. For chemical and physical processes, entropy is central to predicting the direction of change. Further conceptual details are discussed in Thermodynamics Overview.

Entropy Change and Its Mathematical Expression

The change in entropy $(\Delta S)$ for a reversible process is defined by the equation:

$\Delta S = \dfrac{q_\text{rev}}{T}$

Here, $q_\text{rev}$ is the heat transfer during a reversible process, and $T$ is the absolute temperature in Kelvin. The SI unit of entropy is joules per kelvin (J/K). For further understanding, refer to Second Law of Thermodynamics.

Entropy Changes in Different Thermodynamic Processes

Entropy change varies with the type of thermodynamic process. The detailed expressions for entropy change in principal processes are given below.

Entropy Change in Isobaric Process

An isobaric process occurs at constant pressure. For an ideal gas heated from temperature $T_1$ to $T_2$ at constant pressure:

$\Delta S_\text{isobaric} = m\,C_p \ln \left(\dfrac{T_2}{T_1}\right)$

Here, $m$ is the mass, and $C_p$ is the specific heat at constant pressure. Entropy increases as temperature increases under constant pressure conditions.

Entropy Change in Isochoric Process

An isochoric process takes place at constant volume. For an ideal gas with temperature increasing from $T_1$ to $T_2$ at constant volume:

$\Delta S_\text{isochoric} = m\,C_v \ln \left(\dfrac{T_2}{T_1}\right)$

Here, $C_v$ is the specific heat at constant volume. Entropy also increases with temperature at constant volume.

Entropy Change in Isothermal Process

An isothermal process occurs at constant temperature. For one mole of an ideal gas undergoing expansion or compression from volume $V_1$ to $V_2$ at temperature $T$:

$\Delta S_\text{isothermal} = nR \ln \left(\dfrac{V_2}{V_1}\right)$

Alternatively, using the ideal gas equation, this can be written as $\Delta S = nR \ln \left(\dfrac{P_1}{P_2}\right)$. Entropy increases during isothermal expansion.

Entropy Change in Adiabatic Process

In an adiabatic process, no heat is exchanged with the surroundings ($q=0$). For a reversible adiabatic process:

$\Delta S_\text{adiabatic,\,rev} = 0$

There is no change in the entropy of the system since the process occurs without heat transfer.

Entropy in Irreversible Processes

For an irreversible process, the total entropy of the universe (system plus surroundings) always increases. The mathematical expression is:

$\Delta S_\text{irreversible} \geq \dfrac{q}{T}$

In irreversible adiabatic processes, although $q=0$ for the system, entropy increases due to dissipative effects such as friction or non-equilibrium changes.

For comprehensive details about reversibility, visit Entropy in Reversible Processes.

Entropy in Quasi-Static Processes

A quasi-static process occurs infinitely slowly, allowing the system to remain nearly in equilibrium throughout. Entropy change follows:

$dS = \dfrac{dQ}{T} + I$

Here, $I$ represents irreversible entropy generation, which is zero for purely reversible (quasi-static) steps but positive if any irreversibility is involved.

The relationship between various processes and entropy change facilitates understanding the fundamental laws governing energy transformations. Connections to energy transfer mechanisms are discussed in First Law of Thermodynamics.

Summary Table: Entropy Change in Major Processes

Physical Examples of Entropy Change

Entropy changes occur in diverse physical situations, such as heating solids and liquids, phase transitions, and mixing substances. Everyday examples include the melting of ice, diffusion of gases, and thermal conduction.

For instance, the entropy increases when a solid melts or when a solute dissolves in a solvent due to an increase in disorder. More on energy transfer can be found in Understanding Heat Pumps.

Key Properties of Entropy

• State function, independent of path
• Extensive property, depends on system size
• SI unit is joule per kelvin (J/K)
• Entropy increases with temperature
• In isolated system, entropy does not decrease
• Symbol is $S$, while $S^\circ$ represents standard entropy

The properties and calculations of entropy are often used to predict spontaneity and equilibrium in thermodynamic systems and can be useful for understanding entropy of discrete random variables and distributions.

Conclusion on Entropy of Different Processes

Entropy serves as a crucial parameter to assess the direction and feasibility of thermodynamic transformations. In all natural processes, total entropy increases or remains constant, but never decreases for an isolated system. The study of entropy across different processes aids significantly in mastering concepts essential for thermodynamics topics in JEE examinations.

Further insights into work and entropy transformations can be explored in Compression Work Explained.