The Fundamental Thermodynamic Equations

It is an inconvenient quirk of history (for us) that thermodynamics evolved instead with the arrangement of variables known as the fundamental thermodynamic equation for energy  

We will see that the fundamental definitions of pressure, chemical potential, and temperature are based on the form U = U S,V,N). Unfortunately, while the main definitions of thermodynamics originate from the energy equation, the microscopic driving forces are better understood in terms of the entropy equation S = S U,V,W. So we need to switch back and forth betw-een S = S(U,V,N) and U = U(5, U,N). For example, in Chapter 2 we illustrated how S U) accounts for the tendency for heat to flow, but temperature is defined in terms of f/(S). This small inconvenience adds a few steps to our logic. 

Either equation. S = S(U, V,N) or U = U S,V,N), will completely specify the state of a simple system. Thermodynamics does not tell you the specific mathematical dependence of S on ([/, V,N) or U on (S, V,N). Equations of state, such as the ideal gas law, specify interrelations among these variables. Equations of state must come either from experiments or from microscopic models. They are imported into thermodyoiamics, which gives additional useful relationships. 

The fundamental equation for small changes in entropy S(U, V,N) can be expressed as 

When you use the fundamental energy equation instead, you can describe small changes in energy U (S, V, N) by 

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As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as ... 

Cvm and R constants) be regarded as the fundamental thermodynamic equation of an ideal gas. With the aid of the two laws of thermodynamics, show that Equations (5.1) and (5.2) are contained implicitly in Equation (6.134). 

These are the fundamental thermodynamic equations from which we can develop our energy balances in batch, stirred, and tubular reactors. 

First of all, we should emphasize that the compensatory enthalpy-entropy relationship cannot be derived directly from the fundamental thermodynamic equations but... 

This volume begins as Chapter 11 in the two-volume set. This Chapter summarizes the fundamental relationships that form the basis of the discipline of chemical thermodynamics. This chapter can serve as a review of the fundamental thermodynamic equations that are necessary for the more sophisticated applications described in the remainder of this book. This level of review may be all that is necessary for the practising scientist who has been away from the field for some time. For those who need more, references are given to the sections in Principles and Applications where the equations are derived. This is the only place that this volume refers back to the earlier one. 

The thermodynamic approach applied here considers the adsorbent plus the adsorbed gas, or vapor, as a solid solution (system aA). Applying this description, it is feasible to get the fundamental thermodynamic equation for the aA system [2,15,16,25]... 

The fundamental thermodynamic equation for a reversible electrochemical transformation is... 

Chemical interaction between solids involves the consumption of some compounds and the formation of others. These processes occur in agreement with physicochemical laws and can be characterized using the fundamental thermodynamic equations. The main of them is the equation... 

By combining with the fundamental thermodynamic equation for open systems (Equation... 

Just as with a bulk phase, the fundamental thermodynamic equations representing the combined first and second laws may be written in four equivalent ways in terms of the internal energy, enthalpy, Helmholtz free energy, or Gibbs free energy. For an adsorbed phase... 

All of the thermodynamic properties of a homogeneous phase may be obtained from the Helmholtz energy or from the Gibbs energy. When (T, V, ) are the independent variables, the Helmholtz energy A (T, V, ) is the appropriate choice and the fundamental thermodynamic equation for the phase is... 

How do we determine the fugacity experimentally We can start with the fundamental thermodynamic equation given in equation 4.49 ... 

The calculation of the chemical potentials of the components can be done by the intercept method which is graphically shown in Fig. 4.3. It relies on the fundamental thermodynamic equations ... 

Consider a two-dimensional membrane system. In the thermodynamic description of such systems, the area (A) plays the role that volume (V) does in three-dimensional systems, and the surface pressure (tt) replaces the pressure P. Analogously to three-dimensional systems, the work exerted in increasing the area is (TtdA). The fundamental thermodynamic equation for the energy is thus... 

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