Natural variables internal energy equation

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... 

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... 

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. 

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 ... 

Since the specific internal energy is a state function and is supposed to be entirely determined by the state variables, we conclude from the differential form in (8) that e depends naturally on and s (i.e. e = e( ,s)) and that the following state equations hold ... 

Because Newton s equations of motion are conservative, the natural ensemble is NVE (micro-canonical), Aat is one in which the internal energy rather than the temperature is held constant. This is inconvenient if one wishes to compare with experiment where it is the temperature that is generally controlled. In order to perform molecular dynamics in the canonical ensemble, a thermostat must be applied to the system. This is accomplished by constructing a pseudo-Lagrangian. Many forms for temperature-conserving Lagrangians have been proposed, most of which can be written in a form that adds a frictional (velocity-dependent) term to the equations of motion (Allen and Tildesley 1989). Physically, the thermostat can be thought of as a heat bath to which the system is coupled. In the NPT ensemble, in which the pressure is held constant, the cell size and shape fluctuates. The choice of dynamical variables is critical. If the lattice parameters are chosen as in the method of Parrinello and Rahman (1981), the time evolution may depend on the chosen size or shape of the supercell. This difficulty is... 

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. 

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). 

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 ... 

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... 

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