System state function

The laws of thermodynamics are the cornerstones of any description of a system at equilibrium. The First Law (the energy conservation law) introduces H, the enthalpy, as a state function of a system. State functions depend only on the state of the system and not on how it managed to arrive at that state. 

Atomic and molecular simulation methods can generally be categorized as either equilibrated or dynamic. Static simulations attempt to determine the structural and thermodynamic properties such as crystal structure, sorption isotherms, and sorbate binding. Structural simulations are often carried out using energy minimization schemes that are similar to molecular mechanics. Elquihbrium prop>erties, on the other hand, are based on thermodynamics and thus rely on statistical mechanics and simulating the system state function. Monte Carlo methods are then used to simulate these systems stochastically. 

A state functions is a property of a system that depends only on its present state, determined by variable such as pressure, temperature and is independent of any previous history of the system. History of the system means the previous conditions under which it existed. Thermodynamic functions like internal energy U, enthalpy H, entropy S etc., are all state functions. The change in their values depends only upon the initial and final states of the system. State functions give exact or perfect differentials. 

Chapter 2. Thermodyneunic concepts contains an overview of the most essential concepts and definitions included in the thermodynamic description of substances. In the explanation of thermodynamics a number of precise terms concerning systems, state functions and process types are employed. The meaning of these concepts and their application to practical substance systems are explained. 

The relationship between this M avefunction (sometimes called state function) and the location of particles in the system fonus the basis for a second postulate ... 

There exists a state function S, called the entropy of a system, related to the heat Dq absorbedfrom the surroundings during an infinitesimal change by the relations... 

As one raises the temperature of the system along a particular path, one may define a heat capacity C = D p th/dT. (The tenn heat capacity is almost as unfortunate a name as the obsolescent heat content for// alas, no alternative exists.) However several such paths define state functions, e.g. equation (A2.1.28) and equation (A2.1.29). Thus we can define the heat capacity at constant volume Cy and the heat capacity at constant pressure as... 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

State Functions State functions depend only on the state of the system, not on past history or how one got there. If r is a function of two variables, x and y, then z x,y) is a state function, since z is known once X and y are specified. The differential of z is... 

Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, andtne functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume which can be measured, and leads immediately to the conclusion that there exists an equation of. state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primaiy tool in apphcations of thermodyuamics. 

It is convenient to consider the operability of a system as a function of its components /=/(C, c .c i). If a components operability is identified as 1 for operating and 0 for failed, the status of the components at any time may be represented by a system state vector =(1,1,1, 0, 0) meaning that components 1, 2, and 3 are operating and components 4 and 5 have failed. Requirements for system operability may be represented by a matrix 101 that has Is where components are required and Os where they are non-essential the result is ( ) = lOlilt, where the rules... 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

Representation theory for nonunitary groups.—Before proceeding we should consider what is meant by a unitary and an anti-unitary operator.5 -6 If the hamiltonian of a system commutes with the operators u and a of the group 0, and T and O are state functions of the system, u is unitary if... 

The second thing to note about the thermodynamic variables is that, since they are properties of the system, they are state functions. A state function Z is one in which AZ = Zi — Z that is, a change in Z going from state (l) to state (2), is independent of the path. If we add together all of the changes AZ, in going from state (1) to state (2), the sum must be the same no matter how many steps are involved and what path we take. Mathematically, the condition of being a state function is expressed by the relationship... 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

Entropy S like internal energy, volume, pressure, and temperature is a fundamental property of a system. As such, it is a function of the state of the system and a state function so that... 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

In earlier days, A was called the work function because it equals the work performed on or by a system in a reversible process conducted at constant temperature. In the next chapter we will quantitatively define work, describe the reversible process and prove this equality. The name free energy for A results from this equality. That is, A A is the energy free or available to do work. Work is not a state function and depends upon the path and hence, is often not easy to calculate. Under the conditions of reversibility and constant temperature, however, calculation of A A provides a useful procedure for calculating u ... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Equation (2.66) indicates that the entropy for a multipart system is the sum of the entropies of its constituent parts, a result that is almost intuitively obvious. While it has been derived from a calculation involving only reversible processes, entropy is a state function, so that the property of additivity must be completely general, and it must apply to irreversible processes as well. 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

Quantities like V, U, S, H< A, and G are properties of the system. That is, once the state of a system is defined, their values are fixed. Such quantities are called state functions. If we let Z represent any of these functions, then it does not matter how we arrive at a given state of the system, Z has the same value. If we designate Z to be the value of Z at some state l, and Z to be the value of Z at another state 2, the difference AZ = Z2 - Z in going from state l to state 2 is the same, no matter what process we take to get from one state to the other. Thus, if we go from state l through a series of intermediate steps, for which the changes in Z are given by AZ, AZ . AZ,-. and eventually end up in state 2,... 

Because enthalpy is a state function, the enthalpy change of a system depends only on its initial and final states. Therefore, we can carry out a reaction in one step or visualize it as the sum of several steps the reaction enthalpy is the same in each case. 

Similarly, heat is not a state function. The energy transferred as heat during a change in the state of a system depends on how the change is brought about. For example, suppose we want to raise the temperature of 100 g of water from 25°C to 30°C. One way to raise the temperature would be to supply energy as heat by... 

The first law of thermodynamics states that the internal energy of an isolated system is constant. A state function depends only on the current state of a system. The change in a state function between two states is independent of the path between them. Internal energy is a state function work and heat are not. 

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