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State Function, Exact and Inexact Differentials

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Consider a system undergoing adiabatic expansion from an initial state with a certain internal energy to a final state with a different internal energy. Here, the work done by the system is denoted as w.

If the process changes to nonadiabatic while retaining the same initial and final states, the change in internal energy remains the same. However, the heat and work differ between the paths, reflecting that internal energy is path-independent, whereas work and heat are path-dependent.

Properties such as internal energy that depend only on the system's current state are called state functions.

On the other hand, path-dependent physical quantities, like work and heat, are called path functions.

The state and path functions are symbolized by capital and lowercase letters, respectively.

Infinitesimal changes in the work, heat, and internal energy are represented by differentials δw, δq, and dU.

When integrated, δw and δq give the absolute amount of work and heat involved in the process, categorizing them as inexact differentials, while dU gives the change in U, marking it as an exact differential.

Overview

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A state function is a thermodynamic property that depends solely on the current state of a system, irrespective of its history or how it arrived at that state. These functions are represented by capital letters, such as U, H, and S, which stand for internal energy, enthalpy, and entropy, respectively.

For instance, the value of internal energy depends on the system's state variables and remains unaffected by the process path. This means that whether the system underwent a linear process or a complex series of steps to reach its current state, the internal energy will remain the same.

Contrasting with state functions are path functions, which are thermodynamic properties that depend on the path taken by the system to reach its current state. Notable examples of path functions are work (w) and heat (q), symbolized by lowercase letters.

The concept of exact and inexact differentials emerges when dealing with infinitesimal changes in a system. The infinitesimal changes in work, heat, and internal energy are represented as δw, δq, and dU, respectively. When these infinitesimals are integrated over a complete process from initial to final conditions, there's a divergence in notation.

When δw and δq are integrated, the results represent the absolute amount of work and heat for the process. Conversely, the integration of dU does not yield the absolute U, but the change in U, denoted as ΔU. This reflects the path-independent nature of state functions.

The differentials δw and δq are known as inexact differentials, implying that their integrated values, w and q, are path-dependent. On the other hand, dU is an exact differential, signifying that its integrated value ΔU is path-independent.

For a cyclic process, where the system returns to its initial state after undergoing a series of changes, the cyclic integral of a state function is zero. This means that for such processes, the cyclic integral of dU is zero, reflecting the path-independent nature of state functions. However, the cyclic integrals of δq and δw yield q and w, respectively, indicating that heat and work are not necessarily zero for a cyclic process due to their path-dependent nature.

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