Energy natural variable equations

It is worth stating again how useful the natural variable equations are. If we know how any one of the energies varies in terms of its natural variables, we can use the various definitions and equations from the laws of thermodynamics to construct expressions for any other energy. The mathematics of thermodynamics is becoming powerful indeed. 

Starting with the natural variable equation for dU, derive an expression for the isothermal volume dependence of the internal energy, (dUldV)j, in terms of measurable properties (T, V, or p) and a and/or k. Hint You will have to invoke the cyclic rule of partial derivatives (see Chapter 1). 

Third, if we want to consider the natural variable equation for dG for a liquid system whose surface area is changing, we must include the change in the Gibbs energy due to surface area change ... 

In this chapter we introduce a more useful equation for the surface tension. This we do in two steps. First, we seek an equation for the change in the Gibbs free energy. The Gibbs free energy G is usually more important than F because its natural variables, T and P, are constant in most applications. Second, we have just learned that, for curved surfaces, the surface tension is not uniquely defined and depends on where precisely we choose to position the interface. Therefore we concentrate on planar surfaces from now on. 

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 successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

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... 

The equation displays the Gibbs energy as the natural energy function of its natural variables T and p. The coefficients of the differentials in this equation reflect the geometry of the equilibrium surface. 

Step 2. Use the total differential of specific enthalpy in terms of its natural variables, via Legendre transformation of the internal energy from classical thermodynamics, to re-express the pressure gradient in the momentum balance in terms of enthalpy, entropy, and mass fractions. Then, write the equation of change for kinetic energy in terms of specific enthalpy and entropy. 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

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 ... 

There is a potential pitfall in the use of the Legendre transformation of Eulerian homogeneous equations. We start from the representation of the energy [/ of a single phase in natural variables as a total differential. We use purposely a sloppy notation, as sometimes common,... 

Often, the energy in terms of temperature, volume, and mol number U T,V, n) is addressed as the caloric state equation, whereas the volume in terms of temperature, pressure, and mol number V T, p,n) is addressed as the thermal state equation. On the other hand, the energy exclusively expressed in terms of the corresponding natural variables U S, V, n) belongs to the type of a fundamental equation or fundamental form. 

The equation above shows that the natural variables of G are two intensive properties of T, P, and extensive property At,. On the other hand when the Gibbs energy is defined by... 

The proportionality factor D, is called the diffusivity (or diffusion coefficient) it is a characteristic of the corpuscle and of the medium. In the Formal Graph theory, this law applies to a space pole in the corpuscular energy variety. Naturally, this equation can be rewritten with global variables instead of localized ones as... 

An example of a completely classical approach can be found in Feynman et al. (2006) and Jackson (1998). Their approach can be summarized as follows. It relies on the electric polarization of the medium, which absorbs part of the energy owing to the separation of charges induced by the electric field acting on the charged material. To take into account this effect, called dielectric loss, the free space (vacuum) permittivity Eq is replaced by a material permittivity e equal to the free space permittivity multiplied by the relative permittivity e,. The Maxwell equations are written with this material permittivity e but without spatial constraint or conduction current. In doing so, the wave pulsation co and the wave-vector k are unchanged, equal to the natural variables of the oscillator cOq. and kq. The wave velocity u is therefore equal to the natural velocity Uq which directly depends on the material permittivity e, thus (continued)... 

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. 

The Gibbs free energy is one of the most important fundamental functions. Constant temperature and pressure arc the easiest constraints to impose in the laboratory, because the atmosphere provides them. T, p, and N are the natural variables for the Gibbs free energy G = G(T, p,N), which has a minimum at equilibrium. To find the fundamental equation, start with the enthalpy, H = H(S, p,N). Now we want to replace the dS term with a dT term in the equation dH = TdS -t Vdp + Zjii Define a function G ... 

Finding a fundamental equation. While the Gibbs free energy G is the fundamental function of the natural ariables (T, p, N), growing biological cells often regulate not the numbers of molecules N, but the chemical potentials Pi- That is, they control concentrations. What is the fundamental function Z of natural variables T,p,p)7... 

These equations are important because when the behaviors of these energies on their natural variables are known, all thermodynamic properties of the system can be determined. 

In words, this equation says that the change in the Gibbs energy is directly proportional to the change in area of a liquid. If we consider an isothermal, isobaric process (that is, dp = 0 and dT = 0 these conditions are necessary when you consider the natural variable expression in equation 22.7), the process is spontaneous if AG is negative. Because surface tension must be a positive number, this implies that AA for a spontaneous process must be negative a spontaneous process must occur with a corresponding decrease in surface area. 

The Gibbs-Duhem equation is derived from the Euler and Gibbs equations (e.g., Prausnitz et al. [108], app D Slattery [132], p. 443). The mass based form of the Gibbs-Duhem equation is outlined next. The total differential of the Gibbs free energy G in terms of the natural variables (i.e., T, p, ms) is ... 

Note that the fundamental equations (f) express a change of internal energy dll when the entropy dS and the volume dV are changed. It is said that S, V) are the natural variables of the internal energy U = U S,V), because dU has a particularly simple relation to dS and dV. Correspondingly, S,p) axe the natural variables for enthalpy H = and (T,p) are the natural variables for the... 

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. 

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