2.4
状态函数、全微分与非全微分
状态函数是一种热力学性质，它仅取决于系统的当前状态，而与系统的历史或达到该状态的过程无关。这些函数用大写字母表示，例如 U、H 和 S，分别代表内能、焓和熵。
例如，内能的值取决于系统的状态变量，而不受过程路径的影响。这意味着无论系统是通过一个线性过程还是通过一系列复杂步骤达到当前状态，其内能都保持不…
考虑一个系统从具有特定内能的初始状态经历绝热膨胀到达具有不同内能的最终状态。在此过程中，系统对外所做的功记为 w。
如果过程在保持相同初态和末态的情况下变为非绝热过程，则内能的变化量保持不变。然而，不同路径下的热量和功有所不同，这表明内能与路径无关，而功和热量则依赖于路径。
仅取决于系统当前状态的性质，例如内能，被称为状态函数。
另一方面，功和热等依赖于路径的物理量被称为路径函数。
状态函数和路径函数分别用大写字母和小写字母表示。
功、热量和内能的微小变化分别用微分 δw、δq 和 dU 表示。
当进行积分时，δw 和 δq 给出过程中涉及的功和热量的绝对值，因此被归类为不完全微分；而 dU 给出 U 的变化量，属于完全微分。
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Q1: What is a state function in thermodynamics?
A state function is a thermodynamic property that depends solely on the system's current state, not on how it arrived there. Examples include internal energy (U), enthalpy (H), and entropy (S), represented by capital letters. Whether a system follows a linear or complex path, its state function values remain identical at the same initial and final conditions.
Q2: How do path functions differ from state functions?
Path functions depend on the specific route a system takes between states, while state functions do not. Work (w) and heat (q) are path functions, represented by lowercase letters, and their values vary depending on the process pathway. In contrast, state functions like internal energy remain constant regardless of the path taken.
Q3: What is the difference between exact and inexact differentials?
Exact differentials, like dU for internal energy, integrate to give path-independent values. Inexact differentials, like δw and δq for work and heat, integrate to give path-dependent values. When integrated, dU yields the change in internal energy (ΔU), while δw and δq yield absolute amounts of work and heat that depend on the process path.
Q4: Why does internal energy remain unchanged when a process changes from adiabatic to nonadiabatic?
Internal energy is a state function, depending only on initial and final states, not the process type. When a system transitions from adiabatic to nonadiabatic expansion while maintaining the same starting and ending states, the change in internal energy stays identical. However, the individual values of heat and work redistribute between the two paths.
Q5: What does the cyclic integral of a state function equal?
For a cyclic process where the system returns to its initial state, the cyclic integral of any state function equals zero. This reflects the path-independent nature of state functions. Conversely, cyclic integrals of path functions like heat and work are not necessarily zero, since these quantities depend on the specific process pathway.
Q6: Why are state functions represented by capital letters and path functions by lowercase letters?
This notation convention distinguishes between two fundamental thermodynamic property types. Capital letters (U, H, S) denote state functions that depend only on system state. Lowercase letters (w, q) denote path functions whose values depend on the process route. This symbolic distinction helps clarify whether a quantity is path-independent or path-dependent.
Q7: How do infinitesimal changes in work, heat, and internal energy differ mathematically?
Infinitesimal changes are represented as δw, δq, and dU respectively. The δ symbol denotes inexact differentials for work and heat, indicating path-dependence. The d symbol denotes the exact differential for internal energy, indicating path-independence. When integrated, δw and δq yield absolute amounts dependent on the process, while dU yields the change in internal energy independent of path.