Energy natural independent variables

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 ... 

Because pressure and temperature are the natural independent variables for G, the Gibbs free energy will play the leading role in the discussion of equilibrium, to which we shall turn after considering the variation of the heat of reaction under other than standard conditions. 

Free energies also have an additional, fundamental importance. Once the matlicmatical form of the free energy of a system is known in terms of its natural independent variables (c.g., the Gibbs free energ as a fimction of T, p, and N), then all the thermodynamic properties of the system can be determined. In the remainder of the course, we will be learning how to both develop approximate models for the free energy and how to use these models to estimate the thermodynamic behavior of various systems. 

The Legendre transform (10) thus allows one to define a thermodynamic potential with natural independent variables which are more accessible (T instead of s in the present case). The quantity having the unit of energy per unit volume per unit time, is called total dtsstpation. It represents the transformation of non-thermal energy into heat via frictional processes, which then becomes less mailable. 

However, the partial derivatives in this equation are not equal to any simple thermodynamic variables as are the partial derivatives in Eq. (4.5-2). We therefore say that the natural independent variables for the Gibbs energy are r, k i, na,..., c ... 

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,... 

Legendre transformation does not affect the essential nature of a function and all of the different potentials defined above still describe the internal energy, not only in terms of different independent variables, but also on the basis of different zero levels. In terms of Euler s equation (2) the internal energy consists of three components... 

The thermodynamic functions have been defined in terms of the energy and the entropy. These, in turn, have been defined in terms of differential quantities. The absolute values of these functions for systems in given states are not known.1 However, differences in the values of the thermodynamic functions between two states of a system can be determined. We therefore may choose a certain state of a system as a standard state and consider the differences of the thermodynamic functions between any state of a system and the chosen standard state of the system. The choice of the standard state is arbitrary, and any state, physically realizable or not, may be chosen. The nature of the thermodynamic problem, experience, and convention dictate the choice. For gases the choice of standard state, defined in Chapter 7, is simple because equations of state are available and because, for mixtures, gases are generally miscible with each other. The question is more difficult for liquids and solids because, in addition to the lack of a common equation of state, limited ranges of solubility exist in many systems. The independent variables to which values must be assigned to fix the values of all of the... 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

The transformed Gibbs energy provides the criterion for spontaneous change and equilibrium in systems of enzyme-catalyzed reactions when the independent variables for the system are T, P, pH, and Wc - Notice that making this Legendre transform has introduced ) as a natural variable, but it has not changed the number of natural variables because there is now one less component that is conserved, the hydrogen atom component. 

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 ... 

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)... 

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... 

The important set of independent variables needed to represent Cp in terms of Jacobians is T, p and all N,. However, the total differential of extensive internal energy in terms of its natural variables via equation (29-4) and the definition of Cy ... 

While the meaning of distance from the amine receptor to the micelle surface is degraded by its temporally dynamic and spatially convoluted nature, we can estimate the relative microlocation of a given member of a sensor series on a scale between the limits of bulk water and the less polar micelle interior in terms of the hydrophobicity constant (P) (24) of the variable component of the family 2a-f i,e. the tail group. In other words, we can use hydrophobic free energy as the independent variable instead of a space coordinate for the mapping of membrane bounded proton densities. Such... 

For the moment we shall confine our attention to closed systems with one component in one phase. The total differential of the internal energy in such a system is given by Eq. 5.2.2 dt/ = T dS — pdV. The independent variables in this equation, S and V, are called the natural variables of U. 

So far we have set up the energy and enthalpy in differential form, where E = E S,V,n ) and H = H S,P,ni) are specified in terms of the indicated independent variables. These particular functions are said to involve natural coordinates. However, as already stated, the differentials of E and H are set up in terms of another function of state, the entropy S, which makes it unclear how we are to proceed further. As a remedy, we express the entropy in its terms of its dependence on temperature, volume or pressure, and composition either as 5 = 5(r,K i) or as S = S T,P,ni). We then introduce the differential form (1.3.19) for dS, whereby the first law may be recast in the form... 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

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 ... 

In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term Sio, called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. 

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