Subscripts fractions

In a similar fashion, a fraction of the velocity of the molecules with first-order coupling is transmitted to other molecules entangled with the latter. This is called second-order coupling (subscript 2). Still higher orders of effect radiate from the original molecule in the manner suggested by Fig. 2.13. Because of the... 

Force per unit area along the axis of the deformation is called the uniaxial tension or stress. We shall use the symbol a as a shorthand replacement for F/A and attach the subscript t to signify tension. The elongation, expressed as a fraction of the original length AL/Lq is called the strain. We shall use 7j as the symbol for the resulting strain (subscript t for tension). Both o... 

In this context the repeat units in a polymer may be divided into two classes those at the ends of the chain (subscript e) and the others which we view as being in the middle (subscript m) of the chain. The mole fraction of each category in a sample is Xg and x j, respectively. Since all segments are of one type or the other. 

Since the fractions of crystalline (subscript c) and amorphous (subscript a) polymer account for the entire sample, it follows that we may measure whichever of the two is easiest to determine and obtain the other by difference. Generally, it is some property P, of the crystalline phase that we are able to... 

Next we assume that the state designated by the subscript 2 in Eq. (4.61) corresponds to Tg, we designate the fraction free volume at Tg by fg. Likewise,... 

Dividing both sides of Eq. (6.58) by [M-], the total radical concentration, gives the number fraction of n-mer radicals in the total radical population. This ratio is the same as the number of n-mers in the sample containing a total of N (no subscript) polymer molecules ... 

In these expressions Xc is the critical (subscript c) value of x which marks the threshold at which immiscibility sets in, and 1 - 0j or 0j is the volume fraction of the solvent in the solution at this point. Rearranging Eq. (8.56), we obtain... 

Osmotic pressure is one of four closely related properties of solutions that are collectively known as colligative properties. In all four, a difference in the behavior of the solution and the pure solvent is related to the thermodynamic activity of the solvent in the solution. In ideal solutions the activity equals the mole fraction, and the mole fractions of the solvent (subscript 1) and the solute (subscript 2) add up to unity in two-component systems. Therefore the colligative properties can easily be related to the mole fraction of the solute in an ideal solution. The following review of the other three colligative properties indicates the similarity which underlies the analysis of all the colligative properties ... 

Such a coil is said to be nondraining, since the interior of its domain is unaffected by the flow. We anticipate using Eq. (1.58) to describe the molecular weight dependence of In view of this, we replace rg by (rg ) and attach a subscript 0 to the latter as a reminder that, under 0 conditions, solvent and excluded-volume effects cancel to give a true value. With these ideas in mind, the volume fraction of the nondraining coil is written... 

The subscript v is attached to both of these intensities as a reminder that the foregoing analysis is based on the assumption of vertical polarization for the incident light beam. The ratio of these intensities gives the fraction of light scattered per molecule by vertically polarized light ... 

Note that this also involves the assumption of isotropic molecules, which have the same polarizability in all directions. Unpolarized light consists of equal amounts of vertical and horizontal polarization, so the fraction of light scattered in the unpolarized (subscript u) case is given by... 

The volume fraction, sometimes called holdup, of each phase in two-phase flow is generally not equal to its volumetric flow rate fraction, because of velocity differences, or slip, between the phases. For each phase, denoted by subscript i, the relations among superficial velocity V, in situ velocity Vj, volume fraclion Rj, total volumetric flow rate Qj, and pipe area A are... 

O As subscript, referring to inlet stream Y Cumulative fraction by weight undersize ... 

LK = subscript for light key Nn, = minimum theoretical stages at total reflux Xhk = mol fraction of heavy key component Xlk = mol fraction of the light key component otLK/HK = relative volatility of component vs the heavy key component... 

The capital N s obviously represent fractional standard counting errors, and the subscripts S and U denote standard and unknown. 

As long as the subscripted variable (the variable held constant in the partial derivative) remains the same, partial derivatives can be treated in many ways like fractions. For example... 

In the case of vinyl/divinyl copolymerization, the subscript 1 is used to designate mono-vinyl monomer, 2 is used for divinyl monomer. f Q indicates the initial mole fraction of divinyl monomer in the monomer mixture, instantaneous mole fraction of monomer i bound in the polymer chain. 

In the model equations, A represents the cross sectional area of reactor, a is the mole fraction of combustor fuel gas, C is the molar concentration of component gas, Cp the heat capacity of insulation and F is the molar flow rate of feed. The AH denotes the heat of reaction, L is the reactor length, P is the reactor pressure, R is the gas constant, T represents the temperature of gas, U is the overall heat transfer coefficient, v represents velocity of gas, W is the reactor width, and z denotes the reactor distance from the inlet. The Greek letters, e is the void fraction of catalyst bed, p the molar density of gas, and rj is the stoichiometric coefficient of reaction. The subscript, c, cat, r, b and a represent the combustor, catalyst, reformer, the insulation, and ambient, respectively. The obtained PDE model is solved using Finite Difference Method (FDM). 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

The subscripts m, L, S, and G will represent the local two-phase mixture, liquid phase, solid phase and gas phase, respectively. The definitions below are given in terms of solid-liquid (S-L) mixtures, where the solid is the more dense distributed phase and the liquid the less dense continuous phase. The same definitions can be applied to gas-liquid (G-L) flows if the subscript S is replaced by L (the more dense phase) and the L by G (the less dense phase). The symbol

volume fraction of the more dense phase, and s is the volume fraction of the less dense phase (obviously (p = 1 — e). An important distinction is made between ([Pg.444]

A second approach also considers three populations free (unquenched) donors No, free acceptors NA, and a population engaged in FRET pairs Ns that transfer energy with characteristic efficiency E (between 0 and 1). However, in this case, the Ns population emits both donor fluorescence (quenched by a fraction (1 - E)) and sensitized emission (proportional to ENS). To keep in line with the treatment and terminology in other chapters in this volume, this latter approach will be followed here. Note, however, that in other chapters the population of FRET pairs is indicated by the subscript DA whereas we stick to the notation Ns to indicate that this quantity is based on photons emitted from sensitized emission (S image) and to keep the close synonymy with the former approach. Thus, our Io-s equals Ida and Is +1 a equals IAo- Both ways yield essentially identical results. 

The fluorescent components are denoted by I (intensity) followed by a capitalized subscript (D, A or s, for respectively Donors, Acceptors, or Donor/ Acceptor FRET pairs) to indicate the particular population of molecules responsible for emission of/and a lower-case superscript (d or, s) that indicates the detection channel (or filter cube). For example, / denotes the intensity of the donors as detected in the donor channel and reads as Intensity of donors in the donor channel, etc. Similarly, properties of molecules (number of molecules, N quantum yield, Q) are specified with capitalized subscript and properties of channels (laser intensity, gain, g) are specified with lowercase superscript. Factors that depend on both molecular species and on detection channel (excitation efficiency, s fraction of the emission spectrum detected in a channel, F) are indexed with both. Note that for all factorized symbols it is assumed that we work in the linear (excitation-fluorescence) regime with negligible donor or acceptor saturation or triplet states. In case such conditions are not met, the FRET estimation will not be correct. See Chap. 12 (FRET calculator) for more details. 

In this equation < > and -q, are the volume fraction and viscosity of the pure polymer at the temperature under consideration. The subscript 2 refers to the solvent or plasticizer. 

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