Chain rule Volume

An alternative, simpler expression can be obtained as follows. Using the chain rule, the temperature derivative of the volume for constant value of the relaxation time is expressed as... 

From the definition (6.14) and the general chain rule under conditions of constant T and P, we can write the general differential of volume variation dV as... 

The chain rule for differentiating the volume integral of eqn. (10.69) and the following identities [50]... 

The situation becomes different if we want to remove some gas from the system, keeping the volume V constant, and not the total entropy S, but the molar entropy 5 = C3. Then we have to apply the chain rule for functions with multiple arguments... 

We remark that the energy of the gas, e.g., in the system (") changes with the loss of matter under the condition of constant molar entropy S" and constant molar volume V". Whenever a certain amount of volume escapes through the porous plug, it takes with it the entropy content in this particular volume and the volume diminishes accordingly, because in addition the plunger moves. Thus, S = n"S" and V" = n"V". Using the chain rule... 

Prom such an action functional, one seeks to determine the local Kohn-Sham potential through a series of chain rules for functional derivatives. The procedure is called the optimized effective potential (OEP) or the optimized potential method (OPM) for historical reasons [15,16]. The derivation of the time-dependent version of the OEP equations is very similar to the ground-state case. Due to space limitations we will not present the derivation in this chapter. The interested reader is advised to consult the original paper [13], one of the more recent publications [17,18], or the chapter by E. Engel contained in this volume. The final form of the OEP equation that determines the EXX potential is... 

The packing parameter expresses the ratio of the insoluble chain molecular volume to the volume actually occupied by the copolymer in the assembly, thus usually dictating the most likely self-assembled morphology. As a general rule. 

The sales strategy needs to decide what product to be sold in which sales market representing the sales location in the value chain network. New markets needs to be evaluated for their attractiveness and the own competitive position with respect to existing products or the capabilities in the development of new products for the respective demand. Sales business rules include decisions on the strategic share of contracted business volumes vs. flexible spot business volumes. These business rules often depend on sales channels and frame contracts with customers. The sales strategy can be matched with classical marketing mix decisions on products, prices, promotion and communication, as well as sales channel decisions. 

While some objects like production and distribution locations match the tactical level of value chain planning with distribution center or plant, aggregation has to be validated by aggregation rules considering the focus of value chain planning on volumes and values as illustrated in the following. 

The general conclusion of the mentioned works was that the appearance of the jump on the dependences of network volume on the composition of the solvent or on temperature is reached only at some definite content of ionic groups in the network chains. For neutral networks with flexible chains, the collapse is usually not observed. Exceptions to this rule were reported for poly(isopropylacrylamide) (PIPAA) [16], poly(vinylcaprolactam) and poly-(2-vinylpyrrolidone) [17] gels. The specific feature of these systems is that the transition takes place in structured solvents water or concentrated aqueous solutions of aluminium sulfate. 

The competition between these opposite effects (increase and decrease of pendant double bond reactivity) depends on the concentration of multifunctional monomers, the length and flexibility of the primary chains, and the quality of thermodynamic interactions between monomers and macromolecules. As a rule, cyclization is more effective at the beginning of polymerization, whereas the steric excluded-volume effects are more effective at the later stages. 

---