Natural variables Helmholtz energy

For example, from the right (G) edge, the arrow from natural variable T ends up at — S (taken minus because it lies at the tail of the arrow), whereas that from variable P ends up at +V (taken plus because it lies at the head of the arrow), giving the differential form dG = (—S)dT + (+V)dP. Similarly, from the top (A) edge, we can read the differential form for the Helmholtz energy as dA = ( S) dT + (— P) dV, because both coefficients fall at arrow tails. 

In thermodynamics, we focus on the most important variables needed to describe a system. Although we are interested in the size of a system (or of a phase), we usually do not concern ourselves with the shape of the system. One way in which the shape of a system does influence its thermodynamic properties is through its surface area. The surface of a phase is a different environment than its bulk region. Molecules on the surface of a material do not experience attractive interactions to other molecules in all directions and, therefore, have higher energy than molecules in the bulk of the material. Energy is increased when the surface area of a condensed system (usually a liquid) is increased at constant volume and temperature. Because the Helmholtz free energy has T and V as its natural variables, we can immediately write... 

From E q. 11.1 Ob, it is seen that H is a characteristic function with natural variables S and P. Additional relations can be derived from the first and second laws when other experimental conditions are more easily controlled. For example, for a system of constant composition, the Helmholtz free energy, A (sometimes denoted as F), has natural variables T and V, and the Gibbs free energy, G, has natural variables T and P, as shown by E qs. 11.11b and 11.12b below ... 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

We can easily arrive at a new function that has different natural variables by performing a Legendre transform. For example, to arrive at a new state properly that posesses the independent variables T and V, we define the Helmholtz free energy A as ... 

From this expression, we see that the natural variables of the Helmholtz free energy A are the temperature, volume, and total number of moles of the system. 

Once one of the free energies of a system is known as a function of its natural variables, then all the other thermodynamic properties of the system can be derived. For these equations of states, the Helmholtz free energy is the relevant quantity. In the following, we demonstrate how to determine the Helmholtz free energy from an equation and then proceed to show how to derive other properties from it. 

The free energy- that has temperature, volume, and mole numbers as its natural variables is the Helmholtz free energy. Before we stated that once the Gibb s free energy of a system is known as a function of temperature, pressure, and mole numbers G(T,p, N, N2,..all the thermodynamics of the system are known. This is equivalent to the statement that once the Helmholtz free energy is known as a function of temperature, volume, and mole numbers of the system A(T, V, Ni,N2, -all the thermodynamics of the system are known. The fundamental equation of thermodynamics can be written in terms of the Helmholtz free energy as... 

Helmholtz energy is displayed as the natural energy function of its natural variables T and V. The geometry of the equilibrium surface is expressed by the coefficients of the differentials. 

Finally, we look for the natural energy function for the variables T and p. The Helmholtz energy of the natural variables T and V offers a starting point. We define the state function Gibbs free energy G by... 

We will begin the derivation with Helmholtz energy, as it is the natural energy function for the independent variables T and V of equations of state. By the fundamental differential equation for A, Equation (4.81)... 

Both the above are satisfied by a thermodynamic potential, when the thermodynamic process happens imder conditions where its natural variables remain constant. For example, for a closed system N = constant) undergoing a process under controlled temperature and volume (T, V are also constant), the Helmholtz potential F = F(N, V, T) is the corresponding free energy similarly, for the same system undergoing a process where NPT are constant, the Gibbs potential G = G(N, P, T) is now the free energy in this case and so on. 

From statistical mechanical theory, a simple model for a hypothetical hard-sphere liquid (spherical molecules of finite size without attractive intermolecular forces) gives the following expression for the Helmholtz energy with its natural variables T, V, and n as the independent... 

If the extremum of a function such as 5 ((/) predicts equilibrium, the variable U is called the natural variable of 5. T is not a natural variable of 5. Now we show that (T,V,N) are natural variables of a function F, the Helmholtz free energy. An extremum in f (T, V,N) predicts equilibria in systems that are constrained to constant temperature at their boundaries. 

The internal energy, the enthalpy, and the Helmholtz energy have their own sets of natural independent variables. From Eq. (4.5-3), Eq. (4.5-8), and the relation G = H-TS,... 

In fluid mechanics it might be natural to employ mass based thermodynamic properties whereas the classical thermodynamics convention is to use mole based variables. It follows that the extensive thermodynamic functions (e.g., internal energy, Gibbs free energy, Helmholtz energy, enthalpy, entropy, and specific volume) can be expressed in both ways, either in terms of mass or mole. The two forms of the Gibbs-Duhem equation are ... 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

The analogue to one-component thermodynamics applies to the nature of the variables. So Ay S, U and V are all extensive variables, i.e. they depend on the size of the system. The intensive variables are n and T -these are local properties independent of the mass of the material. The relationship between the osmotic pressure and the rate of change of Helmholtz free energy with volume is an important one. The volume of the system, while a useful quantity, is not the usual manner in which colloidal systems are handled. The concentration or volume fraction is usually used ... 

A natural starting point is the fundamental equation in a canonical ensemble of c components, for which the independent variables are the temperature , the volume V, and the number of molecules of each species, Nj,i = 1,..., c the potential is the Helmholtz free energy A... 

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