Equation and State Function
Thermodynamics reveals how to describe and connect the state of a system through concepts like state functions and equations. How are properties such as pressure, temperature, and the ideal gas equation related? This content will guide you to understand the mathematical and physical rules that govern the equilibrium of the universe, challenging your intuition and broadening your perspective.
Learning Objectives:
By the end of this class, the student will be able to
- Describe the concept of a state function and its relationship with thermal equilibrium in thermodynamic systems.
- Distinguish between state functions and magnitudes that are not state functions, identifying concrete examples of both categories.
- Analyze how state equations, such as the ideal gas equation, connect state functions in systems at equilibrium.
TABLE OF CONTENTS:
The State Function and Systems in Thermal Equilibrium
The Equation and State Variables
Mathematical Notion of a State Function
The Ideal Gas Equation as a State Equation
One of the key concepts of classical thermodynamics is the idea of state equations and state functions. Through these, we can describe the state of systems in thermodynamic equilibrium.
The State Function and Systems in Thermal Equilibrium
In thermodynamics, a system is defined as the portion of the universe chosen for study, with the surroundings near the system. Recall that a system is in thermal equilibrium with its surroundings if its macroscopic observables (e.g., pressure, temperature) remain unchanged over time. For example, if we consider a gas inside a container and its temperature remains stable over time, then we say the gas is in thermodynamic equilibrium, and that set of macroscopic observables determines its state. Conversely, if a large amount of heat is suddenly applied to one side of the container, then at least for a while, the gas will be in a state distinct from thermal equilibrium, and its state will vary over time.
The Equation and State Variables
When a system is in thermal equilibrium, we can recognize two types of magnitudes: some depend on the way the system reached that equilibrium, and others are independent of such a process. The latter are what we call state functions (sometimes referred to as state variables). A state function is any physical quantity with a well-defined value for each equilibrium state of the system. Thus, in thermal equilibrium, these variables do not depend on time. Some examples include:
- pressure
- temperature
- volume
- internal energy
Examples of quantities that are not state functions include the position of particle number “umpteenth” in the system, the total work, and heat applied to the system. This can be understood intuitively by observing that your hands can reach the same temperature (and therefore the same state) through two different processes: one by applying a certain amount of work, rubbing them together, or by applying heat, placing your hands in warm water.
Mathematical Notion of a State Function
With these intuitive ideas, to achieve a more complete understanding, we only need a more mathematical development of what lies behind a state function. Consider a system as being described by a vector of parameters \vec{x}=(x_1, x_2, x_3, \cdots), and let f(\vec{x}) be a state function. Then, if the system’s parameters change from an initial value \vec{x}_i to a final value \vec{x}_f, the variation of the function f will be:
\Delta f = \displaystyle \int_{\vec{x}_i}^{\vec{x}_f}df = f(\vec{x}_f) - f(\vec{x}_i)
If things occur in this way, the variation of the state function only depends on the initial and final values of \vec{x}. This happens when the quantity df is an exact differential. All state functions have exact differentials; conversely, a quantity whose differential is not exact cannot be a state function.
The Ideal Gas Equation as a State Equation
In general, it is always possible to find, at least approximately, a state equation that connects state functions. An example is the ideal gas equation of state f(P,V,T)=0, which takes the form:
f(P,V,T) = PV - nRT = 0