Combined first and second laws

Many of the usefiil applications of thermodynamics result from combining the first and second laws in terms of appropriate fimctions under the assumption of reversible conditions. The second law may be combined with the first law by substituting Equation (1.23) for the heat absorbed by the system into Equation (1.9), giving 

The combined first and second laws may also be written in terms of other state functions, which are defined for convenience in solving certain types of problems. These include the enthalpy (Equation (1.15)) such that 

Two useful relations result from Equations (1.29) and (1.31). Firstly, under isothermal conditions Equation (1.29) becomes 

The relation in Equation (1.32) indicates that the work done by the system will always be less than or equal to the negative of the change in Helmholtz free energy. Thus the maximum isothermal work which can be obtained from the system will correspond to the equality, i.e. reversible conditions. Similarly under isothermal, isobaric conditions Equation (1.31) becomes 

Multicomponent and open systems require additional terms in Equations (1.26), (1.27), (1.29) and (1.31) to include the contributions to the various functions made by adding or removing matter from the system. These terms are the chemical potentials of each component in the system, defined by 

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The treatments that are concerned in more detail with the nature of the adsorbed layer make use of the general thermodynamic framework of the derivation of the Gibbs equation (Section III-5B) but differ in the handling of the electrochemical potential and the surface excess of the ionic species [114-117]. The derivation given here is after that of Grahame and Whitney [117]. Equation III-76 gives the combined first- and second-law statements for the surface excess quantities... 

The combined first and second laws state that, at constant T and V, a system seeks to minimize A until dA for any subsequent change is zero (Equation 4.14), and likewise, at constant T and P, the Gibbs free energy decreases until dG for any subsequent change equals zero (Equation 4.20). One then recognizes that the condition for equilibrium is exactly the same at constant T and V as it is at constant T, P. 

Equation (6.111) is sometimes called the combined first and second laws of thermodynamics, and this equation suggests that S and V are natural independent variables for U. Conversely, we can say that U and V are natural variables for S. One can also... 

We can begin with Equation (7.1) as a statement of the combined first and second laws, which were rearranged to... 

The combined first and second laws of thermodynamics state how an increment of mechanical work (fdX = dW) done on the system can produce an increase in the internal energy dE or a decrease in the entropy ... 

We cannot write down a priori a generic balance relation for the entropy of a fluid. We can, however, derive a result that can be placed in the same form as Eq. (3) and therefore recognized as a balance relation. By working with the combined first and second laws of thermodynamics, one... 

Equation (1.75) is a mathematical statement of the second law of thermodynamics for reversible processes. The introduction of the integrating factor for Sq causes the thermal energy to be split into an extensive factor. S and an intensive factor T. Introducing Eq. (1.75) into Eq. (1.56) yields the combined first and second laws... 

A similar program is used for reacting systems. In 7.4 we extend the combined first and second laws to closed systems xmdergoing chemical reactions, then in 7.5 we show how the combined laws apply to reactions in open systems. In 7.6 we formulate the thermodynamic criterion for identifying reaction equilibria. By presenting... 

Equation (7.1.11) is a general form of the combined first and second laws applied to closed systems we call it the combined laws. Since Nw, Nv, and Ns are extensive properties of the system while and Pg are properties of the sinroundings, (7.1.11) applies both to homogeneous systems and to heterogeneous systems. If the system is heterogeneous, but composed of homogeneous parts, then (7.1.11) can be written as a sum over the homogeneous parts, as in (7.1.2). 

Furthermore, throughout any reversible change all system temperatures are the same, Tj = = T, the equality in the combined first and second laws (7.1.11) applies,... 

The interface itself has negligible mass compared to the masses of the phases, and during processes, states of the interface may be undefined or undefinable. We will treat the interface as an open system and interpret each phase as a "port" for the other phase that is, the open-system energy and entropy balances from 2.4 will apply. In what follows, we first derive the combined first and second laws ( 7.2.1). Then we find limits on the directions ( 7.2.2) and magnitudes ( 7.2.3) of mass and energy transfers between phases a and p. 

Equation (7.2.13) is a form of the combined first and second laws describing processes in which material and energy cross an interface between bulk phases that are each at their own fixed T and P. When only energy can be transferred between the phases, then (7.2.13) reduces to (7.2.10). We now deduce limitations on the directions and magnitudes of transfers by considering special cases of (7.2.10) and (7.2.13) the special cases arise by applying constraints to the interface. 

In this section we obtain the combined first and second laws for reacting systems. The development parallels that presented in 7.1 for nonreacting systems. However, the development here is more elaborate than the earlier one because our analysis must account for the fact that, during reactions, chemical species are not conserved. This problem is addressed in 7.4.1 and examples are offered in 7.4.2 and 7.4.3 then in 7.4.4 we derive the combined laws for reactions in closed systems. 

With the notation and stuff equations from the previous section, we can now extend the combined first and second laws from unreacting systems ( 7.1 and 7.2) to reacting systems. To facilitate the presentation, it is useful to introduce a new set of property differences that apply to reacting systems. For any extensive property F in a reacting system, we define a change in F for each reaction j by the intensive quantity... 

Equation (7.5.10) is the combined first and second laws for open systems undergoing chemical reactions with T and P constant in each phase. It imposes limitations on the combined effects of reactions, material transfers, and energy transfers across an interface between bulk phases a and p. 

In this chapter we formulated the combined first and second laws for closed and open systems, both with and without chemical reactions. We found that each form of the combined laws imposes limitations on the directions and magnitudes of processes particular forms apply to particular kinds of processes and systems. In addition, the combined laws provide the conditions that must be satisfied when all processes are complete and equilibrium has been established. This means that the material in this chapter can serve as the starting point for any thermodynamic analysis. 

Again, it is emphasized that the first law of thermodynamics cannot be used to determine minimum or maximum energy transfer to or from a system. Instead, we must use the second law or the availability balance (combined first and second laws). For the propane refrigeration cycle, the availability balance of Eq. 19.321 simplifies to... 

Then (Sq/T) becomes a state function called entropy and T the absolute temperature. As a state function, entropy is path-independent. Eqn (1.25) is a mathematical statement of the second law of thermodynamics. The introduction of the integrating factor for 8q causes the thermal energy to be split into an extensive factor S and an intensive factor T. Clausius defined the entropy with the integrating factor of the inverse of absolute temperature in T 8q) = dS. Similarly, integrating factor 1/P in IP 6W) = dV leads to exact differential dV, which is formulated by Clapeyron in 1834. Introducing Eqn (1.25) into the first law of thermodynamics dU =8q + yields the combined first and second laws of thermodynamics... 

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... 

The combined first and second laws all have the same mathematical form. If reversible conditions are invoked, the general form is that of a perfect differential,... 

Consider the combined first and second laws in terms of the Gibbs free energy, Equation (2.21). How many Maxwell reciprocal relations can be obtained from this equation Write each of them and comment on their physical significance. 

The integrated form of the combined first and second laws for the entire system shown in Figure 7.1 is... 

The combined first and second laws of thermodynamics may be written as... 

THE COMBINED FIRST AND SECOND LAW STATEMENT REVERSIBLE WORK... 

The combined first and second law statement allows us to calculate the reversible work for any process, actual or hypothetical. It also leads to the definitions of several useful convenience functions. 

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