Total entropy

This equation represents the formation of an aqueous solution of the salt ammonium nitrate, NH4N03, a widely used fertilizer. The product ions have much more freedom of movement when dissolved in water than they do when bound together within the pure solid. With this greater freedom of movement they are able to absorb more dispersed energy. The products, therefore, have larger entropy totals than the reactants. According to these ideas, this reaction should be favorable. 

Translational entropy Rotational entropy Vibrational entropy Electronic entropy Total... 

Change in total entropy = Total entropy in Total entropy out + Total entropy produced Entropy balance in the rate form is given by... 

Thermodynamic data for condensed states are derived from either measured or estimated information. The thermochemical table is obtained normally by the appropriate integration of heat capacity data. At the appropriate temperatures, transition enthalpies are added to the enthalpy total, while the quotient of the transition enthalpy divided by the thermodynamic temperature is added to the entropy total. 

This is known as the Clausius inequality and has important applications in irreversible processes. For example, dS > (dQ/T) for an irreversible chemical reaction or material exchange in a closed heterogeneous system, because of the extra disorder created in the system. In summary, when we consider a closed system and its surroundings together, if the process is reversible and if any entropy decrease takes place in either the system or in its surroundings, this decrease in entropy should be compensated by an entropy increase in the other part, and the total entropy change is thus zero. However, if the process is irreversible and thus spontaneous, we should apply Clausius inequality and can state that there is a net increase in total entropy. Total entropy change approaches zero when the process approaches reversibility. 

The entropy, total energy, and free energy are thus given by Ss = kN hi f+kNTd]nf/dT,... 

Applied to a two-phase system, this says that the change in pressure with temperature is equal to the change in entropy at constant temperature as the total volume of the system (a + P) is increased, which can only take place if some a is converted to P ... 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

This expression gives the average entropy change per chain to get the average for the sample, we multiply by the number v of subchains in the sample. The total entropy change is... 

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

Although the right-hand sides of Eqs. (8.27) and (8.28) are the same, the former applies to the mixture (subscript mix), while the latter applies to the mixing process (subscript m). The fact that these are identical emphasizes that in Eq. (8.27) we have calculated only that part of the total entropy of the mixture which arises from the mixing process itself. This is called the configurational entropy and is our only concern in mixing problems. The possibility that this mixing may involve other entropy effects—such as an entropy of solvation-is postponed until Sec. 8.12. 

The ring-opening polymerization of is controUed by entropy, because thermodynamically all bonds in the monomer and polymer are approximately the same (21). The molar cycHzation equihbrium constants of dimethyl siloxane rings have been predicted by the Jacobson-Stockmayer theory (85). The ring—chain equihbrium for siloxane polymers has been studied in detail and is the subject of several reviews (82,83,86—89). The equihbrium constant of the formation of each cycHc is approximately equal to the equihbrium concentration of this cycHc, [(SiR O) Thus the total... 

Because the engine operates in cycles, it experiences no change in its own properties therefore the total entropy change of the engine and its associated heat reservoirs is given by equation 8 ... 

The systems of interest in chemical technology are usually comprised of fluids not appreciably influenced by surface, gravitational, electrical, or magnetic effects. For such homogeneous fluids, molar or specific volume, V, is observed to be a function of temperature, T, pressure, P, and composition. This observation leads to the basic postulate that macroscopic properties of homogeneous PPIT systems at internal equiUbrium can be expressed as functions of temperature, pressure, and composition only. Thus the internal energy and the entropy are functions of temperature, pressure, and composition. These molar or unit mass properties, represented by the symbols U, and S, are independent of system size and are intensive. Total system properties, J and S do depend on system size and are extensive. Thus, if the system contains n moles of fluid, = nAf, where Af is a molar property. Temperature... 

The total heat released is the sum of the entropy contribution plus the irreversible contribution. This heat is released inside the battery at the reaction site. Heat release is not a problem for low rate appHcations however, high rate batteries must make provisions for heat dissipation. Failure to accommodate heat can lead to thermal mnaway and other catastrophic situations. 

The second law reqmres that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the Emit, where the process is reversible, the total entropy of the system plus its surroundings be constant. 

The total rate of entropy increase (in both system and surroundings) as a result of a process is... 

Thus, an analysis of the lost work, made by calculation of the fraction that each individual lost-work term represents of the total lost work, is the same as an analysis of the rate of entropy generation, made by expressing each individual entropy-generation term as a fraclion of the sum of all entropy-generation terms. 

To understand the flow in turbomachines, an understanding of the basic relationships of pressure, temperature, and type of flow must be acquired. Ideal flow in turbomachines exists when there is no transfer of heat between the gas and its surroundings, and the entropy of the gas remains unchanged. This type of flow is characterized as a rever.sible adiabatic flow. To describe this flow, the total and static conditions of pressure, temperature, and the concept of an ideal gas must be understood. 

The overall effieieney of a radial-inflow turbine is a funetion of effieieneies from various eomponents sueh as the nozzle and rotor. A typieal turbine expansion enthalpy/entropy diagram is shown in Figure 8-7. The total enthalpy remains eonstant through the nozzle, sinee neither work nor heat is transferred to or from the fluid. Within the rotor, the total enthalpy ehanges. Downstream of the rotor the total enthalpy remains eonstant. 

The work of the eompressor under ideal eonditions oeeurs at eonstant entropy. The aetual work oeeurs with an inerease in entropy thus the adiabatie effieieney ean be written in terms of the total ehanges in enthalpy ... 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

Real fan total pressure difference is smaller for the same volume flow than that of an isentropic, theoretical fan. This is a result of the fan losses. These arise from the entropy generation in adiabatic systems. We investigate the losses separately, i.e., entropy generation in the impeller and casing. 

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