how to find mole fraction

The Tautomerism of Heterocycles: Five-membered Rings with Two or More Heteroatoms

Vladimir I. Minkin , ... Olga V. Denisko , in Advances in Heterocyclic Chemistry, 2000

3 Imidazole, Benzimidazole, and Their Derivatives

In solution and in the solid state, imidazole and its N2-unsubstituted derivatives form large hydrogen-bonded associates 13 (Scheme 8) [76AHC(S1), p. 266; 84CHEC-I(5)345,84JPC5882; 96CHEC-II(3)77; 97JST(415)187].

Scheme 8.

Due to the perfect linear arrangement of the N-H⋯N hydrogen bridges in 13, extremely fast cooperative proton transfers occur in solution which lead to averaging of the positions 4 and 5 in the ring on the NMR time scale. In contrast with pyrazoles, only averaged ¹H, ¹³C, and ¹⁵N NMR spectra of effective C _2v symmetry were recorded so far for imidazole, benzimidazole, and many of their derivatives [75T1461, 82JOC5132, 96CHEC-II(3)77, 97MRC35]. Slow prototropic annular tautomerism of 2-perfluoroalkylimi-dazoles was attributed to intramolecular N-H⋯F bonds realized in these compounds (82JOC2867; 95OPP33). The proton exchange reaction is subject to general and specific acid and base catalysis [76AHC(S1), p. 266; 87AHC(41)187],

As in the case of pyrazole, relative energies of nonaromatic 2H 14c and 4 (or 5) 14d,e (R¹  =   R²  =   R³  =   H) isomers of imidazole are too high for them to contribute to the tautomeric equilibrium (Scheme 9). According to STO-3G calculations (86BSF429) these values are equal to 60.4 and 67.7   kJ mol⁻¹ respectively relative to 14a,b (R¹  =   R²  =   R³  =   H).

Scheme 9.

A major part of the information available on positions of tautomeric equilibria 14a ⇌ 14b of imidazoles stems from the basicity measurements of the N-unsubstituted imidazoles and N-methylated derivatives of tau- tomers 14a and 14b. The data obtained in earlier [76AHC(S1), p. 272] and more recent [96CHEC-II(3)77] studies are collected in Table III. These are generally in line with those obtained by the use of ¹H and ¹³C NMR [79JOC3864; 83T3797; 86JHC921; 92JCS(P1)2779; 98JCS(P2)1665], ¹⁵N NMR (79JOC3864; 83HCA1537) and UV-spectral (86ZC378) methods.

Table III. Equilibrium Constants (K_T   = 14a/14b) for Imidazole Annular Tautomerism as Determined from pK_a Measurements

Compound a	KT	Reference
4-CH3	1.1	96CHEC-II(3)77
4-Ph	10-37	76AHC(S1), p. 278
4-NO2	320-500	76AHC(S1), p. 278
4-Cl	90	96CHEC-II(3)77
4-Br	47	96CHEC-II(3)77
4-I	19	96CHEC-II(3)77
2-Br-4-NO2	210	76AHC(S1), p. 278
4-NO2-5-Cl	118	76AHC(S1), p. 278
2,4-(NO2)2	1.1-3.1	76AHC(S1), p. 278

a

Position of substituents is indicated according to the structure 14a.

The main conclusion on the influence of substituents in the imidazole ring on the state of the tautomeric equilibria 14a ⇌ 14b is that electron- withdrawing groups favor the 4-position, i.e., the tautomers 14a with R²  =   Hal, NO₂, and so on, are the energetically preferable species. Application of Charton's equation, K_T   =   [4-R Im]/[5-R Im]   =   3.2 σ _m, was discussed in detail [76AHC(S1); 96CHEC-II(3)77]. The equation was found to be in a qualitative agreement with the experimental data presented in Table III.

Theoretical studies of the relative stabilities of tautomers 14a and 14b were carried out mostly at the semiempirical level. AM1 and PM3 calculations [98JST(T)249] of the relative stabilities carried out for a series of 4(5)- substituted imidazoles 14 (R³  =   H, R²  =   H, CH₃, OH, F, NO₂, Ph) are mostly in accord with the conclusion based on the Charton's equation. From the comparison of the electronic spectra of 4(5)-phenylimidazole 14 (R²  =   Ph, R¹  =   R³  =   H) and 2,4(5)-diphenylimidazole 14 (R¹  =   R²  =   Ph, R³  =   H) in ethanol with those calculated by using π-electron PPP method for each of the tautomeric forms, it follows that calculations for type 14a tautomers match the experimentally observed spectra better (86ZC378). The AM1 calculations [92JCS(P1)2779] of enthalpies of formation of 4(5)- aminoimidazole 14 (R²  =   NH₂, R¹  =   R³  =   H) and 4(5)-nitroimidazole 14 (R²  =   NO₂, R¹  =   R³  =   H) point to tautomers 14a and 14b respectively as being energetically preferred in the gas phase. Both predictions are in disagreement with expectations based on Charton's equation and the data related to basicity measurements (Table III). These inconsistencies may be explained by the fact that tautomers 14b of 4(5)-aminoimidazole and 14a of 4(5)-nitroimidazole possess higher values of dipole moments compared to their more stable gas-phase tautomers. Accounting for the solvation energy should bridge the intrinsic energy gap.

Arguing that the MNDO method is more suitable than the AM1   method for predicting the heats of formation of five-membered nitrogenated aromatic rings, García and Vilarrasa (88H1803) calculated that 4-fluoroimidazole 14a (R²  =   F, R¹  =   R³  =   H) is 2.5   kJ mol⁻¹ more stable than its tautomer 14b, whereas 2,4-difluoroimidazole 14a (R¹  =   R²  =   F, R³  =   H) is 2.1   kJ mol⁻¹ less stable than 14b. The energy differences are very small and disappear when the correction terms related to the lone-pair repulsions between the vicinal pyridinelike nitrogens and fluorine substituents are taken into account.

A theoretical ab initio study of the gas-phase basicities of methyldiazoles (90JA1303) included a discussion of the 4(5)-methylimidazole tautomerism. The RHF/4-31G calculations led to the conclusion that the 4-methyl tautomeric form 14a (R²  =   Me, R¹  =   R³  =   H) is 5.2   kJ mol⁻¹ more stable than its 5-methyl counterpart 14b. It was emphasized that this result is to be considered as basic-set dependent. However, a recent theoretical study [94JST(T)45] showed that, starting from the RHF/6-31G* level, all the more accurate approximations indicate a higher intrinsic stability for the 4-methyl tautomer. At the MP2/6–31G* level, the total energy of the 4-methyl tautomer is 0.7   kJ mol⁻¹ lower than that of the 5-methyl tautomer. Inclusion of solvation effects can, thus, strongly affect the position of the tautomeric equilibrium 14a ⇌ 14b. Recently, a systematic theoretical study aimed at evaluating the molecular properties and tautomeric equilibrium of 4- and 5-methylimidazoles has been performed [98JST(T)197]. The K_T value for the gas-phase equilibrium was found to be rather sensitive to both basis set and correlation effects. To achieve good agreement with experimental data the use of extended basis sets (6-31+G*, 6-311+G*, and 6-31   ++G**) at MP2 and DFT levels is required.

Not much information has been added in recent years to the earlier studies of tautomeric equilibria of benzimidazoles based on basicity measurements [76AHC(S1), p. 292], For 5(6)- and 4(7)-substituted benzimidazoles and 2-methyl-5(6)-substituted benzimidazoles K_T values are very close to 1, which indicates near equivalence in the stability of N1(H) and N3(H) tautomers. The tautomeric equilibria of 2-substituted (H, NH₂, OMe, CN) 5-nitrobenzimidazoles and 4-nitrobenzimidazoles were analyzed with the use of semiempirical MINDO/3 and INDO methods. It was predicted that electron-releasing groups in position 2 shifted the equilibria to the 6-NO₂ and 4-NO₂ tautomers, respectively.

As mentioned, the frequencies of proton jumps leading to establishment of the equilibrium 14a ⇌ 14b in solutions of imidazoles and benzimidazoles are too high for these forms to be observed on the NMR time scale with ¹³C [the chemical-shift differences for C4 and C5 in N-alkyl derivatives are about 10   ppm (88MRC134)] and even with ¹⁵N NMR wherein the nitrogen resonances of the pyrrolelike and pyridinelike nitrogens have chemical- shift differences up to 100   ppm (97MRC35). This indicates that the average lifetime of an individual tautomer before its prototropic rearrangement is about 10⁻⁴–10⁻⁵ s. This rearrangement is frozen in the solid state. Well-resolved solid-state ¹³C CPMAS NMR spectra were recorded for imidazole [81JA6011, 81JCS(CC)1207], 2-methylimidazole [87H(26)333], benzimidazole [83H1713; 96CHEC-II(3)77], and 2,2′-bis-1H-imidazole [96CHEC-II(3)77]: the C(4) and C(5) signals were assigned by comparison with the spectra of the corresponding N-methyl derivatives. Recently, an ¹H and ¹³C NMR spectroscopic study of 2,2′-bisbenzimidazolyl has been performed in DMSO solution in a range of temperatures between 293 and 383   K and an activation energy of 67 - 69   kJ mol⁻¹ was estimated for the proton-transfer process [98JPOC411], This value falls in the range of those characteristic for tautomerization of other cyclic oxalamidines (89JA5946; 94JA1230), which implies a possibility of a two-step intermolecular mechanism of the tautomeric rearrangement.

The ¹⁵N NMR spectrum of solid imidazole shows two separate signals for the NH (–210   ppm) and the pyridinelike(–138   ppm) nitrogen, indicating that proton exchange is retarded in the solid state (82JA1192). Taking into consideration the very large difference between the ¹⁵N chemical shifts of the two nitrogens in imidazole ring, ¹⁵N NMR spectroscopy may be regarded as the most accurate and reliable approach for the determination of the K_T values for the tautomeric equilibria 14a ⇌ 14b (82JA3945).

Mole fractions of tautomers 14a (X _a) and 14b (X _b) can be calculated using the equation

$P_{\text{obs}}=X_{\text{a}}{P}_{\text{a}}+X_{\text{b}}{P}_{\text{b}},$

where P _obs is an observed averaged chemical shift (or coupling constant) and P _a and P _b are chemical shifts (or coupling constants) of pyrrolelike and pyridinelike nitrogens in 14 under the condition of slow proton exchange (e.g., in solid) or in its NMe derivative. The procedure is not highly accurate because uncertainty always exists about how well the model values match those of the individual tautomers; however, it is generally considered (80JA2881; 82JA3945, 82JOC5132; 93JA6813) as more accurate than that based on basicity measurements. Using averaged values of the ¹⁵N chemical shifts of N1 and N3 nitrogens of histamine 15 and a series of its analogs with ability to form a cyclic hydrogen bond within a particular tautomer, Roberts et al. (82JA3945) calculated K_T values for the pH range that corresponds to existence of the compounds 15–21 in aqueous solution in the form subject to N1(H) ⇌ N3(H) tautomeric equilibria. ¹

As seen from the values of K_T , the effect of the cyclic hydrogen bonds in compounds 15,17, and 18 is not strong enough to shift the tautomeric equilibria to the N3(H) side. This fact was assigned to the unfavorable eclipsing of methylene groups in the side-chain and to water competing with the side- chain group for formation of a hydrogen bond with an N-H group. Elimination of the first factor in cis-urocanic acid 19 and 20 resulted in marked stabilization of the N(3)H tautomers, as confirmed by pK_a measurements [98JCS(P2)1665].

Special attention has been given to the study of tautomeric equilibria in solutions of histidine 22 because the key functional role of such equilibria in proteins is recognized. In aqueous solutions the tautomers of histidine rapidly interconvert and only a single averaged signal is observed for each ring nitrogen (Scheme 10).

Scheme 10.

In earlier studies, values for these ¹⁵N chemical shifts were estimated from model N-methylated compounds (78JA8041; 80JA2881; 82JA3945) or from ¹⁵N NMR spectroscopy of the tautomeric forms of histidine in the solid state where tautomeric exchange is slow (82JA1192). The use of 80% ethanol/water as solvent allowed Bachovchin et al. (93JA6813) to observe directly the individual tautomeric forms of histidine in ¹⁵N NMR spectra at –55°C. The solvent was found to show a little effect on the intrinsic ¹⁵N shifts of the individual tautomers and the tautomeric equilibria of the imidazole ring. In the pH interval where the imidazole ring is neutral, three of the four possible forms of histidine were observed in ¹⁵N NMR spectra at –55°C. Only 22b was not directly detected. Noteworthy is that only forms 22a and 23a were observed in the ¹⁵N NMR solid-state spectra of histidine (82JA1192). The mole fraction of the tautomeric form 22b calculated from ¹⁵N chemical shifts in solutions at pH ~8.2 where zwitterion 22 predominates is 0.18. This value agrees well with earlier estimates (80JA2881) based on the values of the ² J _NH coupling constants. Stabilization of 22a over 22b is apparently caused by formation of a hydrogen bond with participation of the N3 center. At pHs higher than 9.2, where deprotonation of the α-amino group does not donate a hydrogen bond to the N3 center, tautomer 23a still remains the energetically preferable form. The mole fraction was estimated to be 0.35 (93JA6813). Recently an efficient method for determining the pro-tonated and deprotonated forms and the tautomeric states of histidine residues, even in large-molecular-weight proteins (up to 30–40   kDa), has been developed based on measurements of coupling constants between ¹⁵N and ¹³C in the imidazole ring (98JA10998). In all histidine-containing proteins the 22a and 23a type tautomeric forms were found to be dominant.

Tautomeric equilibria between two degenerate pairs (A′, A″ ⇌ B′, B″) were observed in low-concentration DMSO-d₆ solutions of benzodiimid-azoles, bearing similar substituents in the both imidazole rings [98H(48)113]. In these equilibria, the unsymmetrical tautomers A′ and A″ predominate.

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Characterizing Hydrocarbon-Plus Fractions

Tarek Ahmed PhD, PE , in Equations of State and PVT Analysis (Second Edition), 2016

Behrens and Sandler's Lumping Scheme

Behrens and Sandler (1986) used the semicontinuous thermodynamic distribution theory to model the C_{7   +} fraction for equation-of-state calculations. The authors suggested that the heptanes-plus fraction can be fully described with as few as two pseudocomponents.

A semicontinuous fluid mixture is defined as one in which the mole fractions of some components, such as C ₁ through C₆, have discrete values, while the concentrations of others, the unidentifiable components such as C_{7   +}, are described as a continuous distribution function, F(I). This continuous distribution function F(I) describes the heavy fractions according to the index I, chosen to be a property of individual components, such as the carbon number, boiling point, or molecular weight.

For a hydrocarbon system with k discrete components, the following relationship applies:

${\displaystyle \sum _{i=1}^{{\mathrm{C}}_6}}{z}_i+z_{7+}=1.0$

The mole fraction of C_{7   +} in this equation is replaced with the selected distribution function, to give

(2.109) ${\displaystyle \sum _{i=1}^{{\mathrm{C}}_6}}{z}_i+{\displaystyle {\int }_A^{B}F\left(I\right)dI=1.0}$

where

A  =   lower limit of integration (beginning of the continuous distribution, eg, C₇)

B  =   upper limit of integration (upper cutoff of the continuous distribution, eg, C₄₅)

This molar distribution behavior is shown schematically in Fig. 2.21. The figure shows a semilog plot of the composition z_i versus the carbon number n of the individual components in a hydrocarbon system. The parameter A can be determined from the plot or defaulted to C₇; that is, A  =   7. The value of the second parameter, B, ranges from 50 to infinity; that is, 50   ≤ B  ≤   ∞. However, Behrens and Sandler pointed out that the exact choice of the cutoff is not critical.

■ Figure 2.21. Schematic illustration of the semicontinuous distribution model.

Selecting the index, I, of the distribution function F(I) to be the carbon number, n, Behrens and Sandler proposed the following exponential form of F(I):

(2.110) $F\left(n\right)=D\left(n\right){\mathrm{e}}^{\alpha n}dn$

with

$A\le n\le B$

in which the parameter α is given by the following function f(α):

(2.111) $f\left(\alpha \right)=\frac{1}{\alpha }–{\overline{c}}_n+A–\frac{\left[A–B\right]{\mathrm{e}}^{–B\alpha }}{{\mathrm{e}}^{–A\alpha }–{\mathrm{e}}^{–B\alpha }}=0$

where ${\overline{c}}_n$ is the average carbon number as defined by the relationship

(2.112) ${\overline{c}}_n=\frac{M_{{\mathrm{C}}_{7+}}+4}{14}$

Eq. (2.111) can be solved for α iteratively by using the method of successive substitutions or the Newton-Raphson method, with an initial value of α as

$\alpha =\left[1/{\overline{c}}_n\right]–A$

Substituting Eq. (2.112) into Eq. (2.111) yields

${\displaystyle \sum _i^{{\mathrm{C}}_6}}{z}_i+{\displaystyle {\int }_A^{B}D\left(n\right){\mathrm{e}}^{–\alpha n}dn=1.0}$

or

$z_{7+}={\displaystyle {\int }_A^{B}D\left(n\right){\mathrm{e}}^{–\alpha n}dn}$

By a transformation of variables and changing the range of integration from A and B to 0 and c, the equation becomes

(2.113) $z_{7+}={\displaystyle {\int }_0^{c}D\left(r\right){\mathrm{e}}^{–r}dr}$

where

(2.114) $\begin{array}{l}c=\left(B–A\right)\alpha \\ r=\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{integration}\end{array}$

The authors applied the "Gaussian quadrature numerical integration method" with a two-point integration to evaluate Eq. (2.113), resulting in

(2.115) $z_{7+}={\displaystyle \sum _{i=1}^2}D\left(r_i\right)w_i=D\left(r_1\right)w_1+D\left(r_2\right)w_2$

where r_i   =   roots for quadrature of integrals after variable transformation and w_i   =   weighting factor of Gaussian quadrature at point i. The values of r ₁, r ₂, w ₁, and w ₂ are given in Table 2.9.

Table 2.9. Behrens and Sandler Roots and Weights for Two-Point Integration

c	r 1	r 2	w 1	w 2	c	r 1	r 2	w 1	w 2
0.30	0.0615	0.2347	0.5324	0.4676	4.40	0.4869	2.5954	0.7826	0.2174
0.40	0.0795	0.3101	0.5353	0.4647	4.50	0.4914	2.6304	0.7858	0.2142
0.50	0.0977	0.3857	0.5431	0.4569	4.60	0.4957	2.6643	0.7890	0.2110
0.60	0.1155	0.4607	0.5518	0.4482	4.70	0.4998	2.6971	0.7920	0.2080
0.70	0.1326	0.5347	0.5601	0.4399	4.80	0.5038	2.7289	0.7949	0.2051
0.80	0.1492	0.6082	0.5685	0.4315	4.90	0.5076	2.7596	0.7977	0.2023
0.90	0.1652	0.6807	0.5767	0.4233	5.00	0.5112	2.7893	0.8003	0.1997
1.00	0.1808	0.7524	0.5849	0.4151	5.10	0.5148	2.8179	0.8029	0.1971
1.10	0.1959	0.8233	0.5932	0.4068	5.20	0.5181	2.8456	0.8054	0.1946
1.20	0.2104	0.8933	0.6011	0.3989	5.30	0.5214	2.8722	0.8077	0.1923
1.30	0.2245	0.9625	0.6091	0.3909	5.40	0.5245	2.8979	0.8100	0.1900
1.40	0.2381	1.0307	0.6169	0.3831	5.50	0.5274	2.9226	0.8121	0.1879
1.50	0.2512	1.0980	0.6245	0.3755	5.60	0.5303	2.9464	0.8142	0.1858
1.60	0.2639	1.1644	0.6321	0.3679	5.70	0.5330	2.9693	0.8162	0.1838
1.70	0.2763	1.2299	0.6395	0.3605	5.80	0.5356	2.9913	0.8181	0.1819
1.80	0.2881	1.2944	0.6468	0.3532	5.90	0.5381	3.0124	0.8199	0.1801
1.90	0.2996	1.3579	0.6539	0.3461	6.00	0.5405	3.0327	0.8216	0.1784
2.00	0.3107	1.4204	0.6610	0.3390	6.20	0.5450	3.0707	0.8248	0.1754
2.10	0.3215	1.4819	0.6678	0.3322	6.40	0.5491	3.1056	0.8278	0.1722
2.20	0.3318	1.5424	0.6745	0.3255	6.60	0.5528	3.1375	0.8305	0.1695
2.30	0.3418	1.6018	0.6810	0.3190	6.80	0.5562	3.1686	0.8329	0.1671
2.40	0.3515	1.6602	0.6874	0.3126	7.00	0.5593	3.1930	0.8351	0.1649
2.50	0.3608	1.7175	0.6937	0.3063	7.20	0.5621	3.2170	0.8371	0.1629
2.60	0.3699	1.7738	0.6997	0.3003	7.40	0.5646	3.2388	0.8389	0.1611
2.70	0.3786	1.8289	0.7056	0.2944	7.70	0.5680	3.2674	0.8413	0.1587
2.80	0.3870	1.8830	0.7114	0.2886	8.10	0.5717	3.2992	0.8439	0.1561
2.90	0.3951	1.9360	0.7170	0.2830	8.50	0.5748	3.3247	0.8460	0.1540
3.00	0.4029	1.9878	0.7224	0.2776	9.00	0.5777	3.3494	0.8480	0.1520
3.10	0.4104	2.0386	0.7277	0.2723	10.00	0.5816	3.3811	0.8507	0.1493
3.20	0.4177	2.0882	0.7328	0.2672	11.00	0.5836	3.3978	0.8521	0.1479
3.30	0.4247	2.1367	0.7378	0.2622	12.00	0.5847	3.4063	0.8529	0.1471
3.40	0.4315	2.1840	0.7426	0.2574	14.00	0.5856	3.4125	0.8534	0.1466
3.50	0.4380	2.2303	0.7472	0.2528	16.00	0.5857	3.4139	0.8535	0.1465
3.60	0.4443	2.2754	0.7517	0.2483	18.00	0.5858	3.4141	0.8536	0.1464
3.70	0.4504	2.3193	0.7561	0.2439	20.00	0.5858	3.4142	0.8536	0.1464
3.80	0.4562	2.3621	0.7603	0.2397	25.00	0.5858	3.4142	0.8536	0.1464
3.90	0.4618	2.4038	0.7644	0.2356	30.00	0.5858	3.4142	0.8536	0.1464
4.00	0.4672	2.4444	0.7683	0.2317	40.00	0.5858	3.4142	0.8536	0.1464
4.10	0.4724	2.4838	0.7721	0.2279	60.00	0.5858	3.4142	0.8536	0.1464
4.20	0.4775	2.5221	0.7757	0.2243	100.00	0.5858	3.4142	0.8536	0.1464
4.30	0.4823	2.5593	0.7792	0.2208	∞	0.5858	3.4142	0.8536	0.1464

The computational sequences of the proposed method are summarized in the following steps:

Step 1.

Find the endpoints A and B of the distribution. As the endpoints are assumed to start and end at the midpoint between the two carbon numbers, the effective endpoints become

(2.116) $A=\left(\mathrm{starting}\mathrm{carbon}\mathrm{number}\right)–0.5$

(2.117) $B=\left(\mathrm{ending}\mathrm{carbon}\mathrm{number}\right)+0.5$

Step 2.

Calculate the value of the parameter α by solving Eq. (2.111) iteratively.

Step 3.

Determine the upper limit of integration c by applying Eq. (2.114).

Step 4.

Find the integration points r ₁ and r ₂ and the weighting factors w ₁ and w ₂ from Table 2.9.

Step 5.

Find the pseudocomponent carbon numbers, n_i , and mole fractions, z_i , from the following expressions:

For the first pseudocomponent,

(2.118) $\begin{array}{l}{n}_1=\frac{r_1}{\alpha }+A\hfill \\ z_1=w_1{z}_{7+}\hfill \end{array}$

For the second pseudocomponent,

(2.119) $\begin{array}{l}{n}_2=\frac{r_2}{\alpha }+A\hfill \\ z_2=w_2{z}_{7+}\hfill \end{array}$

Step 6.

Assign the physical and critical properties of the two pseudocomponents from Table 2.2.

Example 2.18

A heptanes-plus fraction in a crude oil system has a mole fraction of 0.4608 with a molecular weight of 226. Using the Behrens and Sandler lumping scheme, characterize the C_{7   +} by two pseudocomponents and calculate their mole fractions.

Solution

Step 1.

Assuming the starting and ending carbon numbers to be C₇ and C₅₀, calculate A and B from Eqs. (2.116) and (2.117):

$\begin{array}{l}A=\left(\mathrm{starting}\mathrm{carbon}\mathrm{number}\right)–0.5\hfill \\ A=7–0.5=6.5\hfill \\ B=\left(\mathrm{ending}\mathrm{carbon}\mathrm{number}\right)+0.5\hfill \\ B=50+0.5=50.5\hfill \end{array}$

Step 2.

Calculate ${\overline{c}}_n$ from Eq. (2.112):

$\begin{array}{l}{\overline{c}}_n=\frac{M_{{\mathrm{C}}_{7+}}+4}{14}\hfill \\ {\overline{c}}_n=\frac{226+4}{14}=16.43\hfill \end{array}$

Step 3.

Solve Eq. (2.111) iteratively for α, to give

$\begin{array}{l}\frac{1}{\alpha }–{\overline{c}}_n+A–\frac{\left[A–B\right]{\mathrm{e}}^{–B\alpha }}{{\mathrm{e}}^{–A\alpha }–e^{–\mathit{Ba}}}=0\hfill \\ \frac{1}{\alpha }–16.43+6.5–\frac{\left[6.5–50.5\right]{\mathrm{e}}^{–50.5\alpha }}{{\mathrm{e}}^{–6.5\alpha }–{\mathrm{e}}^{–50.5\alpha }}=0\hfill \end{array}$

Solving this expression iteratively for α gives α  =   0.0938967.

Step 4.

Calculate the range of integration c from Eq. (2.114):

$\begin{array}{l}c=\left(B–A\right)\alpha \hfill \\ c=\left(50.5–6.5\right)0.0938967=4.13\hfill \end{array}$

Step 5.

Find integration points r_i and weights w_i from Table 2.9:

$\begin{array}{l}{r}_1=0.4741\hfill \\ r_2=2.4965\hfill \\ w_1=0.7733\hfill \\ w_2=0.2267\hfill \end{array}$

Step 6.

Find the pseudocomponent carbon numbers n_i and mole fractions z_i by applying Eqs. (2.118) and (2.119).

For the first pseudocomponent,

$\begin{array}{l}{n}_1=\frac{r_1}{\alpha }+A\hfill \\ n_1=\frac{0.4741}{0.0938967}+6.5=11.55\hfill \\ z_1=w_1{z}_{7+}\hfill \\ z_1=\left(0.7733\right)\left(0.4608\right)=0.3563\hfill \end{array}$

For the second pseudocomponent,

$\begin{array}{l}{n}_2=\frac{r_2}{\alpha }+A\hfill \\ n_2=\frac{2.4965}{0.0938967}+6.5=33.08\hfill \\ z_2=w_2{z}_{7+}\hfill \\ z_2=\left(0.2267\right)\left(0.4608\right)=0.1045\hfill \end{array}$

The C_{7   +} fraction is represented then by the two pseudocomponents below.

Pseudocomponent	Carbon Number	Mole Fraction
1	C11.55	0.3563
2	C33.08	0.1045

Step 7.

Assign the physical properties of the two pseudocomponents according to their number of carbon atoms using the Katz and Firoozabadi generalized physical properties as given in Table 2.2 or by calculations from Eq. (2.6). The assigned physical properties for the two fractions are shown below.

Pseudocomponent	n	T b (°R)	γ	M	T c (°R)	p c (psia)	ω
1	11.55	848	0.799	154	1185	314	0.437
2	33.08	1341	0.915	426	1629	134	0.921

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Fundamentals of electrochemistry

Majid Ghassemi , ... Robert Steinberger-Wilckens , in Fundamentals of Heat and Fluid Flow in High Temperature Fuel Cells, 2020

4.1.1 Mass fractions and mole fractions

Mass and mole fraction are the two important components of any specific gas mixtures. They are used to describe gas concentrations as well as to determine the vapor pressures of mixtures of similar liquids.

Mass fraction (y _i ), also known as weight fraction, is the ratio of the ith species the mass, m _i , to the total mass of the mixture, m _tot :

(4.1) $y_i=\frac{m_i}{m_{tot}}$

The total mass of the mixture, the sum of the individual masses of the different species that existed in the mixture, is equal to one:

(4.2) $\sum _i{y}_i=1$

Similarly, the mole fraction, x _i , is the ratio of mole of an ith species in the mixture, n _i , to the total mole of the mixture, n _tot , as formulated by

(4.3) $x_i=\frac{n_i}{n_{tot}}$

where the number of moles of ith species in a mixture, n _i , is the ratio of the ith species mass to its molar mass, M _i :

(4.4) $n_i=\frac{m_i}{M_i}$

Substituting Eq. (4.4) into Eq. (4.3) and considering the fact that the total number of moles of the mixture is the sum of the number of moles of each components that existed in the mixture, the mole fraction equation becomes

(4.5) $x_i=\frac{\frac{m_i}{M_i}}{\sum _i\frac{m_i}{M_i}}$

Again, the total number of moles of the mixture, the sum of the number of moles of all species that existed in the mixture, is equal to one:

(4.6) $\sum _i{x}_i=1$

Also, by substituting Eq. (4.4) into Eq. (4.1) the mass fraction equation becomes as follows:

(4.7) $y_i=\frac{n_i{M}_i}{n_{tot}{M}_{mix}}$

When dealing with SOFC it is more convenient to use molar mass for chemical reaction. However, in practice and usually the amount of mass fractions of the components in the mixture are known. The total molar mass of the mixture, M _mix , is calculated by

(4.8) $M_{mix}=\sum _i{x}_i{M}_i$

Combining the mole fraction relation, Eq. (4.3), and Eq. (4.8) yields

(4.9) $y_i=\frac{x_i{M}_i}{\sum x_i{M}_i}$

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Absorption of Gases

J.F. RICHARDSON , ... J.R. BACKHURST , in Chemical Engineering (Fifth Edition), Volume 2, 2002

12.7.6. Height based on overall coefficients

If the driving force based on the gas concentration is written as (Y — Y_e ) and the overall gas transfer coefficient as Kg , then the height of the tower for dilute concentrations becomes:

(12.51) $Z=\frac{G_m}{K_{G}aP}{\int }_{Y_1}^{Y_2}\frac{\text{d}Y}{Y_e-Y}$

or in terms of the liquor concentration as:

(12.52) $Z=\frac{L_m}{K_{L}a{C}_T}{\int }_{X_1}^{X_2}\frac{\text{d}X}{X-X_e}$

Equations for dilute concentrations

As the mole fraction is approximately equal to the molar ratio at dilute concentrations then considering the gas film:

(12.53) $Z=\frac{G_m}{K_{G}aP}{\int }_{Y_1}^{Y_2}\frac{\text{d}Y}{Y_e-Y}=\frac{G_m}{K_{G}aP}{\int }_{Y_1}^{Y_2}\frac{\text{d}y}{y_e-y}$

and considering the liquid film:

(12.54) $Z=\frac{L_m}{K_{L}a{C}_T}{\int }_{X_1}^{X_2}\frac{\text{d}X}{X-X_e}=\frac{L_m}{K_{L}a{C}_T}{\int }_{x_1}^{x_2}\frac{\text{d}x}{x-x_e}$

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Compressor fundamentals

Maurice Stewart , in Surface Production Operations, 2019

7.2.3.1 Gas mixtures

Knowing the mole fractions in a mixture leads to calculation of several important properties of the mixture:

•

Molecular weight, M _m

•

Molal specific heat, MC _p(m)

•

Critical pressure, P _c(m)

•

Critical temperature, T _c(m)

Fig. 7.11 shows a sample calculation of gas mixture properties.

Fig. 7.11. Sample calculation of gas mixture properties.

The mole fraction "X" can be determined from Eqs. (7.6A), (7.6B), and (7.6C).

(7.6A) $X_1=N_1/N_m$

(7.6B) $X_2=N_2/N_m$

(7.6C) $X_3=N_3/N_m$

where:

N _m   =   total moles in a mixture

N ₁, and so on   =   number of moles of each individual component

A "mole" is actually a number of molecules, approximately 6   ×   10²³. A "mole fraction" is the ratio of molecules of one component in a mixture. For example, if the mole fraction of methane in natural gas is 0.90, then this means that 90% of the molecules are methane. Since the volume fractions are equivalent to mole fractions, the mixture is also 90%, by, methane.

The mixture fractions could also be calculated on a mass or weight basis. The mole (volume) basis is used in compressor calculations because it is a simpler, less confusing method.

The molal specific heat is used to determine the "k" value, known as the ratio of specific heats, as follows. The "k" value is often called the adiabatic exponent and is a value used in the calculation of horsepower, adiabatic head, and adiabatic discharge temperature. (Refer to Isentropic (Adiabatic) compression). The "k" value is determined by Eq. (7.7).

(7.7) $k=\frac{C_p}{C_v}=\frac{{\mathit{MC}}_{p\left(m\right)}}{{\mathit{MC}}_{p\left(m\right)}-\frac{\mathit{Ro}}{778}}=\frac{{\mathit{MC}}_{p\left(m\right)}}{{\mathit{MC}}_{p\left(m\right)}-1.986}$

where:

MC _p(m)  =   molal specific heat (heat capacity) of mixture at constant pressure

778   =   conversion factor, ft-lb/BTU

C _p   =   specific heat at constant pressure

C _v   =   specific heat at constant volume

R _o  =   See Eq. (7.1) for R _o definition

MC_p(m) should be taken at the desired temperature (usually the average suction and discharge temperature). This aspect is covered in Isentropic (Adiabatic) Compression. Note that the "k" value of the mixture must be determined by first determining the molal heat capacity of the mixture (see Eq. 7.6). A common mistake is to multiply the:k: value of the individual gas components by their respective mole fraction to determine the "k" value of the mixture.

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Process Design of Solvent Extraction-Separation of Rare Earths

Dezhi Qi , in Hydrometallurgy of Rare Earths, 2018

4.2.5.5 Stage-Wise Mol-Fraction (Distribution) of Each Component in Aqueous and Organic Phases in the Steady State for Each Circuit

The stagewise mole fraction (distribution) of each component in aqueous and organic phases in steady state for each circuit should be provided by the designer for guiding the production operations. When a solvent extraction system of rare earth separation is in steady state or equilibrium state, the distribution of each component in each cell basically will not change; if the results of the routine assay for both organic phase and aqueous phase show that the distribution curves of each component have changed due to the variations of the flow rate of organic phase, scrubbing aqueous phase, or feed flow rate, the parameters of the process should be adjusted to change the curves back to the original position to ensure the quality of the products. Otherwise, once the unqualified product comes out from the outlets of the system, it will take a very long time to adjust the system to recover itself back to the original state; during this period, a lot of unqualified product may be produced.

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Solar Desalination Systems

Soteris A. Kalogirou , in Solar Energy Engineering (Second Edition), 2014

Exercises

8.1

Estimate the mole and mass fractions for the salt and water of seawater, which has a salinity of 42,000   ppm.

8.2

Estimate the mole and mass fractions for the salt and water of brackish water, which has a salinity of 1500   ppm.

8.3

Find the enthalpy and entropy of seawater at 35   °C.

8.4

A solar still has a water and glass temperature equal to 52.5   °C and 41.3   °C, respectively. The constants C and n are determined experimentally and are found to be C  =   0.054 and n  =   0.38. If the convective heat transfer coefficient from water surface to glass is 2.96   W/m²  K, estimate the hourly distillate output per square meter from the solar still.

8.5

An MSF plant has 32 stages. Estimate the M _f/M _d ratio if the brine temperature in the first effect is 68   °C and the temperature of the brine in the last effect is 34   °C. The mean latent heat is 2300   kJ/kg and the mean specific heat is 4.20   kJ/kg   K.

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Vapor-Liquid Phase Equilibria

Tarek Ahmed , in Reservoir Engineering Handbook (Fifth Edition), 2019

Ahmed's Method

Ahmed et al. (1985) devised a simplified method for splitting the C_{7   +} fraction into pseudo-components. The method originated from studying the molar behavior of 34 condensate and crude oil systems through detailed laboratory compositional analysis of the heavy fractions. The only required data for the proposed method are the molecular weight and the total mole fraction of the heptanes-plus fraction.

The splitting scheme is based on calculating the mole fraction z_n at a progressively higher number of carbon atoms. The extraction process continues until the sum of the mole fraction of the pseudo-components equals the total mole fraction of the heptanes-plus (z_{7   +}).

(15-175) ${\mathrm{z}}_{\mathrm{n}}={\mathrm{z}}_{\mathrm{n}+}\left[\frac{{\mathrm{M}}_{\left(\mathrm{n}+1\right)+}-{\mathrm{M}}_{\mathrm{n}+}}{{\mathrm{M}}_{\left(\mathrm{n}+1\right)+}-{\mathrm{M}}_{\mathrm{n}}}\right]$

where

z_n = mole fraction of the pseudo-component with a number of carbon atoms of n (z₇, z₈, z₉, etc.)

M_n = molecular weight of the hydrocarbon group with n carbon atoms as given in Table 1-1 in Chapter 1

M_n  + = molecular weight of the n   + fraction as calculated by the following expression:

(15-176) ${\mathrm{M}}_{\left(\mathrm{n}+1\right)+}={\mathrm{M}}_{7+}+\mathrm{S}\left(\mathrm{n}-6\right)$

where n is the number of carbon atoms and S is the coefficient of Equation 15-178 with these values:

Number of Carbon Atoms	Condensate Systems	Crude Oil Systems
n ≤ 8	15.5	16.5
n > 8	17.0	20.1

The stepwise calculation sequences of the proposed correlation are summarized in the following steps:

Step 1.

According to the type of hydrocarbon system under investigation (condensate or crude oil), select appropriate values for the coefficients.

Step 2.

Knowing the molecular weight of C_{7   +} fraction (M_{7   +}), calculate the molecular weight of the octanes-plus fraction (M_{8   +}) by applying Equation 15-176.

Step 3.

Calculate the mole fraction of the heptane fraction (z₇) using Equation 15-175.

Step 4.

Apply steps 2 and 3 repeatedly for each component in the system (C₈, C₉, etc.) until the sum of the calculated mole fractions is equal to the mole fraction of C_{7   +} of the system.

The splitting scheme is best explained through the following example.

Example 15-20

Rework Example 15-19 using Ahmed's splitting method.

Solution

Step 1.

Calculate the molecular weight of C₈  + by applying Equation 15-176:

${\mathrm{M}}_{8+}=141.25+15.5\left(7-6\right)=156.75$

Step 2.

Solve for the mole fraction of heptane (z ₇) by applying Equation 15-175:

${\mathrm{z}}_7={\mathrm{z}}_{7+}\left[\frac{{\mathrm{M}}_{8+}-{\mathrm{M}}_{7+}}{{\mathrm{M}}_{8+}-{\mathrm{M}}_7}\right]=0.0154\left[\frac{156.75-141.25}{156.75-96}\right]=0.00393$

Step 3.

Calculate the molecular weight of C_{9   +} from Equation 15-178:

${\mathrm{M}}_{9+}=141.25+15.8\left(8-6\right)=172.25$

Step 4.

Determine the mole fraction of C₈ from Equation 15.177:

$\begin{array}{c}{z}_8=z_{8+}\left[\left({\mathrm{M}}_{9+}-{\mathrm{M}}_{8+}\right)/\left({\mathrm{M}}_{9+}-{\mathrm{M}}_8\right)\right]\\ z_8=\left(0.0154-0.00393\right)\left[\left(172.5-156.75\right)/\left(172.5-107\right)\right]\\ =0.00276\end{array}$

Step 5.

This extracting method is repeated as outlined in the above steps to give:

Component	n	Mn  +   Equation 15-176	Mn(Table 1-1)	znEquation 15-175
C7	7	141.25	96	0.000393
C8	8	156.25	107	0.00276
C9	9	175.25	121	0.00200
C10	10	192.25	134	0.00144
C11	11	209.25	147	0.00106
C12	12	226.25	161	0.0008
C13	13	243.25	175	0.00061
C14	14	260.25	190	0.00048
C15	15	277.25	206	0.00038
C16   +	16   +	294.25	222	0.00159⁎

⁎

Calculated from Equation 15-169 .

Step 6.

The boiling point, critical properties, and the acentric factor of C_{16   +} are then determined by using the appropriate methods, to

$\begin{array}{l}\mathrm{M}=222\\ \mathrm{\gamma }=0.856\\ {\mathrm{T}}_{\mathrm{b}}={1174.6}^{°}\mathrm{R}\\ {\mathrm{p}}_{\mathrm{c}}=175.9\text{psia}\\ {\mathrm{T}}_{\mathrm{c}}={1449.3}^{°}\mathrm{R}\\ \mathrm{\omega }=0.742\end{array}$

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METHODS FOR THE MEASUREMENT OF SOLVENT ACTIVITY OF POLYMER SOLUTIONS

CHRISTIAN WOHLFARTH , in Handbook of Solvents (Second Edition), Volume 1, 2014

4.4.2 NECESSARY THERMODYNAMIC EQUATIONS

Here, the thermodynamic relations are summarized which are necessary to understand the following text. No derivations will be made. Details can be found in good textbooks, e.g., Prausnitz et al. ⁴⁹

The activity of a component i at a given temperature, pressure, and composition can be defined as the ratio of the fugacity of the solvent at these conditions to the solvent fugacity in the standard state; that is, a state at the same temperature as that of the mixture and at specified conditions of pressure and composition:

[4.4.1a] ${\text{a}}_{\text{i}}\left(\text{T},\text{P},\text{x}\right)\equiv {\text{f}}_{\text{i}}\left(\text{T},\text{P},\text{x}\right){/\text{f}}_{\text{i}}\left({\text{T},\text{P}}^{\text{0}}{,\text{x}}^{\text{0}}\right)$

where:

ai	activity of component i
T	absolute temperature
P	pressure
x	mole fraction
fi	fugacity of component i

In terms of chemical potential, the activity of component i can also be defined by:

[4.4.1b] ${\text{a}}_{\text{i}}\left(\text{T},\text{P},\text{x}\right)\equiv exp\left\{\frac{{\mu }_{\text{i}}\left(\text{T},\text{P},\text{x}\right)-{\mu }_{\text{i}}\left({\text{T},\text{P}}^{\text{0}}{,\text{x}}^{\text{0}}\right)}{\text{RT}}\right\}$

where:

μi	chemical potential of component i
R	gas constant

P⁰ and x⁰ denote the standard state pressure and composition. For binary polymer solutions the standard state is usually the pure liquid solvent at its saturation vapor pressure at T. The standard state fugacity and the standard state chemical potential of any component i are abbreviated in the following text by their symbols ${\text{f}}_{\text{i}}^0$ and ${\mu }_{\text{i}}^0$ , respectively.

Phase equilibrium conditions between two multi-component phases I and II require thermal equilibrium,

[4.4.2a] ${\text{T}}^{\text{I}}={\text{T}}^{\text{II}}$

mechanical equilibrium,

[4.4.2b] ${\text{P}}^{\text{I}}={\text{P}}^{\text{II}}$

and the chemical potential of each component i must be equal in both phases I and II.

[4.4.3] ${\mu }_{\text{i}}^{\text{I}}={\mu }_{\text{i}}^{\text{II}}$

For Equation [4.4.3] to be satisfied, the fugacities of each component i must be equal in both phases.

[4.4.4] ${\text{f}}_{\text{i}}^{\text{I}}={\text{f}}_{\text{i}}^{\text{II}}$

Applying fugacity coefficients, the isochemical potential expression leads to:

[4.4.5] ${\varphi }_{\text{i}}^{\text{I}}{\text{x}}_{\text{i}}^{\text{I}}={\varphi }_{\text{i}}^{\text{II}}{\text{x}}_{\text{i}}^{\text{II}}$

where:

ϕi	fugacity coefficient of component i.

Fugacity coefficients can be calculated from an equation of state by:

[4.4.6] $\text{ln}{\varphi }_{\text{i}}=\frac{1}{\text{RT}}\underset{\text{v}}{\overset{\infty }{\int }}\left[{\left(\frac{\partial \text{P}}{\partial {\text{n}}_{\text{i}}}\right)}_{{\text{T},\text{V},\text{n}}_{\text{j}}}-\frac{\text{RT}}{\text{V}}\right]\text{dV}-\text{ln}\frac{\text{PV}}{\text{RT}}$

where a pressure explicit equation of state is required to use Equation [4.4.6]. Not all equations of state for polymers and polymer solutions are also valid for the gaseous state (see section in Subchapter 4.4.4), however, and a mixed gamma-phi approach is used by applying Equation [4.4.7]. Applying activity coefficients in the liquid phase, the isochemical potential expression leads, in the case of the vapor-liquid equilibrium (superscript V for the vapor phase and superscript L for the liquid phase), to the following relation:

[4.4.7] ${\varphi }_{\text{i}}^{\text{v}}{\text{y}}_{\text{i}}\text{P}={\gamma }_{\text{i}}{\text{x}}_{\text{i}}^{\text{L}}{\text{f}}_{\text{i}}^0$

where:

yi	mole fraction of component i in the vapor phase with partial pressure Pi = yiP
γi	activity coefficient of component i in the liquid phase with activity ${\text{a}}_{\text{i}}={\text{x}}_{\text{i}}^{\text{L}}{\gamma }_{\text{i}}$

or in the case of liquid-liquid equilibrium to

[4.4.8] ${\gamma }_{\text{i}}^{\text{I}}{\text{x}}_{\text{i}}^{\text{I}}{\text{f}}_{\text{i}}^{01}={\gamma }_{\text{i}}^{\text{II}}{\text{x}}_{\text{i}}^{\text{II}}{\text{f}}_{\text{i}}^{0\text{II}}$

If the standard state in both phases is the same, the standard fugacities cancel out in Equation [4.4.8]. Equation [4.4.8] also holds for solid-liquid equilibria after choosing appropriate standard conditions for the solid state, but they are of minor interest here.

All expressions given above are exact and can be applied to small molecules as well as to macromolecules. The one difficulty is having accurate experiments to measure the necessary thermodynamic data and the other is finding correct and accurate equations of state and/or activity coefficient models to calculate them.

Since mole fractions are usually not the concentration variables chosen for polymer solutions, one has to specify them in each case. The following three quantities are most frequently used:

[4.4.9a] $\begin{array}{ll}\text{mass}\text{fractions}& {\text{w}}_{\text{i}}={\text{m}}_{\text{i}}/\sum {\text{m}}_{\text{k}}\end{array}$

[4.4.9b] $\begin{array}{ll}\text{volume}\text{fractions}& {\phi }_{\text{i}}={\text{n}}_{\text{i}}{\text{V}}_{\text{i}}/\sum {\text{n}}_{\text{k}}{\text{V}}_{\text{k}}\end{array}$

[4.4.9c] $\begin{array}{ll}\text{segments}\left(\text{hard}-\text{core}\text{volume}\right)\text{fractions}& {\psi }_{\text{i}}={\text{n}}_{\text{i}}{\text{V}}_{\text{i}}^{*}/\sum {\text{n}}_{\text{k}}{\text{V}}_{\text{k}}^{*}\end{array}$

where:

mi	mass of component i
ni	amount of substance (moles) of component i
Vi	molar volume of component i
Vi *	molar hard-core (characteristic) volume of component i.

With the necessary care, all thermodynamic expressions given above can be formulated with mass or volume or segment fractions as concentration variables instead of mole fractions. This is the common practice within polymer solution thermodynamics. Applying characteristic/hard-core volumes is the usual approach within most thermodynamic models for polymer solutions. Mass fraction based activity coefficients are widely used in Equations [4.4.7 and 4.4.8] which are related to activity by:

[4.4.10] ${\Omega }_{\text{i}}={\text{a}}_{\text{i}}/{\text{w}}_{\text{i}}$

where:

Ωi	mass fraction based activity coefficient of component i
ai	activity of component i
wi	mass fraction of component i

Classical polymer solution thermodynamics often did not consider solvent activities or solvent activity coefficients but usually a dimensionless quantity, the so-called Flory-Huggins interaction parameter χ. ^{44, 45} The χ is not only a function of temperature (and pressure), as was evident from its foundation, but it is also a function of composition and polymer molecular mass. ^{5, 7, 8} As pointed out in many papers, it is more precise to call it χ–function (what is in principle a residual solvent chemical potential function). Because of its widespread use and its possible sources of mistakes and misinterpretations, the necessary relations must be included here. Starting from Equation [4.4.1b], the difference between the chemical potentials of the solvent in the mixture and in the standard state belongs to the first derivative of the Gibbs free energy of mixing with respect to the amount of substance of the solvent:

[4.4.11] $\Delta {\mu }_1={\mu }_1-{\mu }_1^0={\left(\frac{\partial \text{n}{\Delta }_{\text{mix}}\text{G}}{\partial {\text{n}}_{\text{1}}}\right)}_{{\text{T},\text{P},\text{n}}_{\text{j}\ne \text{1}}}$

where:

ni	amount of substance (moles) of component i
n	total amount of substance (moles) of the mixture: n = Σni
ΔmixG	molar Gibbs free energy of mixing.

For a truly binary polymer solutions, the classical Flory-Huggins theory leads to: ^{46, 47}

[4.4.12a] ${\Delta }_{\text{mix}}{\text{G}/\text{RT}=\text{x}}_{\text{1}}\text{ln}{\phi }_{\text{1}}+{\text{x}}_{\text{2}}\text{ln}{\phi }_{\text{2}}+{\text{gx}}_{\text{1}}{\phi }_{\text{2}}$

or

[4.4.12b] ${\Delta }_{\text{mix}}\text{G}/\text{RTV}=\left({\text{x}}_{\text{1}}/{\text{V}}_{\text{1}}\right)\text{ln}{\phi }_{\text{1}}+\left({\text{x}}_{\text{2}}/{\text{V}}_{\text{2}}\right)\text{ln}{\phi }_{\text{2}}+\text{BRT}{\phi }_{\text{1}}{\phi }_{\text{2}}$

where:

xi	mole fraction of component i
φi	volume fraction of component i
g	integral polymer-solvent interaction function that refers to the interaction of a solvent molecule with a polymer segment, the size of which is defined by the molar volume of the solvent V1
B	interaction energy-density parameter that does not depend on the definition of a segment but is related to g and the molar volume of a segment Vseg by B = RTg/Vseg
V	molar volume of the mixture, i.e., the binary polymer solution
Vi	molar volume of component i

The first two terms of Equation [4.4.12] are named combinatorial part of Δ_mixG, the third one is then a residual Gibbs free energy of mixing. Applying Equation [4.4.11] to [4.4.12], one obtains:

[4.4.13a] $\Delta {\mu }_1/\text{RT}=\text{ln}\left(1-{\phi }_2\right)+\left(1+\frac{1}{\text{r}}\right){\phi }_2+\chi {\phi }_2^2$

or

[4.4.13b] $\chi =\left[\Delta {\mu }_1/\text{RT}-\text{ln}\left(1-{\phi }_2\right)-\left(1-\frac{1}{\text{r}}\right){\phi }_2\right]/{\phi }_2^2$

or

[4.4.13c] $\chi =\left[{\text{lna}}_{\text{1}}-ln\left(1-{\phi }_2\right)-\left(1-\frac{1}{\text{r}}\right){\phi }_2\right]/{\phi }_2^2$

where:

r	ratio of molar volumes V2/V1, equal to the number of segments if Vseg = V1
χ	Flory-Huggins interaction function of the solvent

The segment number r is, in general, different from the degree of polymerization or from the number of repeating units of a polymer chain but proportional to it. One should note that Equations [4.4.12 and 4.4.13] can be used on any segmentation basis, i.e., also with $\text{r}={\text{V}}_2^{*}/{\text{V}}_1^{*}$ on a hard-core volume segmented basis and segment fractions instead of volume fractions, or with r = M₂/M₁ on the basis of mass fractions. It is very important to keep in mind that the numerical values of the interaction functions g or χ depend on the chosen basis and are different for each different segmentation!

From the rules of phenomenological thermodynamics, one obtains the interrelations between both parameters at constant pressure and temperature:

[4.4.14a] $\chi =\text{g}+{\phi }_1\frac{\partial \text{g}}{\partial {\phi }_1}=\text{g}-\left(1-{\phi }_2\right)\frac{\partial \text{g}}{\partial {\phi }_2}$

[4.4.14b] $\text{g}=\frac{1}{{\phi }_1}\underset{0}{\overset{{\phi }_1}{\int }}\chi \text{d}{\phi }_{\text{1}}$

A discussion of the g-function was made by Masegosa et al. ⁵⁰ Unfortunately, g– and χ–functions were not always treated in a thermodynamically clear manner in the literature. Sometimes they were considered to be equal, and this is only true in the rare case of composition independence. Sometimes, and this is more dangerous, neglect or misuse of the underlying segmentation basis is formed. Thus, numerical data from literature has to be handled with care (using the correct data from the reviews ^{5–8, 11} is therefore recommended).

A useful form for their composition dependencies was deduced from lattice theory by Koningsveld and Kleintjens: ⁵¹

[4.4.15] $\text{g}=\alpha +\frac{\beta }{1-\gamma {\phi }_2}\text{and}\chi =\alpha +\frac{\beta \left(1-\gamma \right)}{{\left(1-\gamma {\phi }_2\right)}^2}$

where:

α	acts as constant within a certain temperature range
β	describes a temperature function like β = β0 + β1/T
γ	is a constant within a certain temperature range.

Quite often, simple power series are applied only:

[4.4.16] $\chi =\sum _{\text{i}=0}^{\text{n}}{\chi }_{\text{i}}{\phi }_2^{\text{i}}\text{and}\text{g}=\sum _{\text{i}=0}^{\text{n}}\left(\frac{{\chi }_1}{\text{i}+1}\right)\left(\frac{1-{\phi }_2^{\text{i}+1}}{1-{\phi }_2}\right)$

where:

χi	empirical fitting parameters to isothermal-isobaric data

Both interaction functions are also functions of temperature and pressure. An empirical form for these dependencies can be formulated according to the rules of phenomenological thermodynamics:

[4.4.17] $\text{g}={\beta }_{\text{00}}+{\beta }_{01}/\text{T}+\left({\beta }_{10}+{\beta }_{11}/\text{T}\right)\text{P}\text{or}\chi =\text{a}+\text{b}/\text{T}+\left(\text{c}+\text{d}/\text{T}\right)\text{P}$

where:

a, b, c, d	empirical fitting parameters for the χ–function
β00, β01, β10, β11	empirical fitting parameters for the g–function
T	absolute temperature
P	pressure

All these fitting parameters may be concentration dependent and may be included in Equations [4.4.15 or 4.4.16]. Details are omitted here. More theoretical approaches will be discussed in Subchapter 4.4.4.

The χ–function can be divided into an enthalpic and an entropic parts:

[4.4.18] $\chi ={\chi }_{\text{H}}+{\chi }_{\text{S}}\text{with}{\chi }_{\text{H}}=-\text{T}{\left(\frac{\partial \chi }{\partial \text{T}}\right)}_{\text{p},\phi }\text{and}{\chi }_{\text{S}}={\left(\frac{\partial \chi \text{T}}{\partial \text{T}}\right)}_{\text{p},\phi }$

where:

χH	enthalpic part
χS	entropic part

An extension of all these equations given above to multi-component mixtures is possible. Reviews of continuous thermodynamics which take into account the polydisperse character of polymers by distribution functions can be found elsewhere. ^52–54

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STEADY-STATE LUMPED SYSTEMS

W. Fred Ramirez , in Computational Methods in Process Simulation (Second Edition), 1997

2.4.5 Mixer-Exchanger-Mixer Design

The production of methylamines from methanol and ammonia is economically affected by the ratio of the demands of the three products (monomethylamine, dimethylamine, and trimethylamine). The recycle of trimethylamine will reduce the production of dimethylamine and trimethylamine relative to the production of monomethylamine. Similarly, the dilution of the reaction mixture with water will result in a relative increase in the production of monomethylamine. A mixer-heat exchanger-mixer portion of a methylamine plant is a proposed capital investment which would allow the relative production of the methylamines to be varied to meet changes in demand.

A detailed schematic for the mixer-exchanger-mixer system is shown in Figure 2.29. Trimethylamine recycle enters in stream 4, is cooled in the heat exchanger, and is mixed with water from stream 1 in mixer 1. The trimethylamine–water mixture is used as the cold side fluid in the heat exchanger and is then mixed with the ammonia–methanol stream from the gas absorber in mixer 3. The mixture leaving mixer 3 is the reaction mixture, which feeds into the preheater of the existing plant. A preliminary estimate of the cost of installing the mixer–heat exchanger–mixer system is desired. A generic mixer–exchanger–mixer problem has been studied by Ramirez and Vestal (1972).

Figure 2.29. Mixer–Exchanger–Mixer System.

The installation is modeled by the 49 equations presented in Table 2.2 all of which fit an acceptable form from Table 2.1. There are four redundant material balance equations in this equation set (equations 12, 13, 14, and 16) and three trivial equations (equations 1, 2, and 4). Each of the 55 variables is assigned a number for convenience (Table 2.3). Removing the redundant and trivial equations (with their variables) reduces the problem to 42 equations and 52 variables. The installation of the mixer–exchanger–mixer system into the existing plant tends to specify certain of the process variables. The temperatures of the entering streams (T ₁, T ₄ and T ₆) are known. Likewise it is desired to produce a flow rate and composition of the exit stream which meets the proper specifications. Hence, values are set for variables F ₇, x ₁7, x _2,7, x _3,7, and x _4,7 . Only three of the four exit mole fractions are independent and may be specified as known variables. Therefore, of the ten degrees of freedom, seven are essentially specified as known variables via the problem. With seven known variables, three additional variables are required as design variables. The residence times for the two mixers ( t ₁ and t ₃) and the cold end temperature difference of the heat exchanger (w ₁) were preliminarily selected as possible design variables. These variables are not necessarily the best design variables for solution simplicity but are ones for which the designer can best assign reasonable values. Engineering experience, flow sheet analysis, and knowledge of the process mathematical model are used to construct the preliminary set of design variables, and are chosen whenever possible in the structural analysis. The preliminary set of design variables could contain any number of variables and does not have to exactly constrain the system.

Table 2.2. Equations for Mixer–Exchanger–Mixer System

Material Balance Equations

1. x 1,1 − 1 = 0 2. x 2,2 − 1 = 0 3. [1] x 1,3 + x 2,3 − 1 = 0 4. x 2,4 − 1 = 0 5. [2] x 1,5 + x 2,5 − 1 = 0 6. [3] x 3,6 + x 4,6 = 1 7. [4] x 1,7 + x 2,7 + x 3,7 + x 4,7 − 1 = 0

Flow Balance Equations

8. [5] F 1 + F 2 − F 3 = 0 9. [6] F 2 − F 4 = 0 10. [7] F 3 − F 5 = 0 11. [8] F 5 + F 6 − F 7 = 0

Component Balance Eqns.

12. x 1,1 F 1 − y 1 = 0 13. x 1,3 F 3 − y 1 = 0 14. x 1,5 F 5 − y 1 = 0 15. [9] x 1,7 F 7 − y 1 = 0 16. x 2,2 F 2 − y 2 = 0 17. [10] x 2,3 F 3 − y 2 = 0 18. [11] x 2,4 F 4 − y 2 = 0 19. [12] x 2,5 F 5 − y 2 = 0 20. [13] x 2,7 F 7 − y 2 = 0 21. [14] x 3,6 F 6 − y 3 = 0 22. [15] x 4,7 F 7 − y 3 = 0 23. [16] x 4,6 F 6 − y 4 = 0 24. [17] x 4,7 F 7 − y 4 = 0

Energy Balance Equations

25. [18] F 1 T 1 − z 1 = 0 26. [19] F 2 T 2 − z 2 = 0 27. [20] F 3 T 3 − z 3 = 0 28. [21] F 4 T 4 − z 4 = 0 29. [22] F 5 T 5 − z 5 = 0 30. [23] F 6 T 6 − z 6 = 0 31. [24] F 7 T 7 − z 7 = 0 32. [25] C 1 z 1 + C 2 z 2 − C 3 z 3 = 0 33. [26] C 4 z 4 − Q 2 − C 2 z 2 = 0 34. [27] C 3 z 3 + Q 2 − C 5 z b = 0 35. [28] C 5 z 5 + C 6 z 6 − C 7 z 7 = 0

Equipment Specification Equations

36. [29] V 1 − (F 3 t 1) / ρ3 = 0 37. [30] Q 2 − U 2 A 2 ΔT lm 2 = 0 38. [31] w 1 + T 3 − T 2 = 0 39. [32] w 2 + T 5 − T 4 = 0 40. [33] w 2 w 3 − w 1 = 0 41. [34] w 4 − ln(w 3) = 0 42. [35] w 4 ΔT lm 2 + w 2 − w 1 = 0 43. [36] V 3 − (F 7 t 3) / ρτ = 0

Capital Cost Estimation Equations

44. [37] log(S 1)-0.515 log(v 1)-3.354 = 0 45. [38] v 1 − V 1 = 0 46. [39] log(S 2)-0.699 log(A 2)-2.414 = 0 47. [40] log(S 3)-0.515 log(v 3)-3.354 = 0 48. [41] v 3 − V 3 = 0 49. [42] S − S 1 − S 2 − S 3 = 0

Table 2.3. Variable Assignments

Variable	Designated Number	Variable	Designated Number	Variable	Designated Number
x 1,1	1	y 1	21[14]	A 2	41[31]
x 2,2	2	y 2	22[15]	V 1	42[32]
x 1,3	3 [1]	y 3	23[16]	V 3	43[33]
x 2,3	4 [2]	y 4	24[171	t 1	44[43]
x 2,4	5	T 1	25	t 3	45[44]
x 1,5	6 [3]	T 2	26[18]	w 1	46[45]
x 2,5	7 [4]	T 3	27[19]	w 2	47[34]
x 3,6	8 [5]	T 4	28	w 3	48[35]
x 4,6	9 [6]	T 5	29[20]	w 4	49[36]
x 1,7	10	T 6	30	v 1	50[37]
x 2,7	11	T 7	31[21]	v 3	51138]
x 3,7	12	z 1	32[22]	S 1	52[39
x 4,7	13 [7]	z 2	33[23]	S 2	53[40]
F 1	14 [8]	z 3	34[24]	S 3	54[41]
F 2	15 [9]	z 4	35[25]	S	55[42]
F 3	16[10]	z 5	36[26]
F 4	17[11]	z 6	37[27]
F 5	18[12]	z 7	38[28]
F 6	19[13]	Q 2	39[29]
F 7	20	ΔTlm 2	40[30]

We now used program fun.f to perform a structural analysis on this problem., First the completely specified 42 equations with 42 variables case is analyzed. The assignment of the equations for this problem is given in Table 2.2 and is indicated by the numbers in [ ]. The variable assignment is given in Table 2.3 with the numbers in [ ]. File mixer.matrix contains the input file necessary for running this problem. Again all files are available on the world wide web under http://optimal.colorado.edu/~ramirez/chen458Q.html. The original 42 by 42 functionality matrix is given in Figure 2.30.

Figure 2.30. Functionality Matrix for Mixer-Exchanger-Mixer Problem With All Design Variables Chosen

The results of running the equation ordering algorithm is the functionality matrix of Figure 2.31. There are two hierarchies which indicate that there are two iterative variables needed to solve this problem or that some of the equations must be solved simultaneously. The step equations will be located at rows 5 and 3 of the rearranged functionality matrix.

Figure 2.31. Equation Ordering Results for Mixer-Exchanger-Mixer Example

The final results of the equation ordering algorithm is given in Figure 2.32. There is one subgroup which starts at row 25 with two platform variables. This means that one section of the problem must be solved simultaneously. The simultaneous equation set starts at row 25 and goes through row 3. This is a set of 14 equations in 14 unknowns. Due to the presence of b elements (products), this is a set of nonlinear algebraic equations. All other equations can be solved explicitly with one equation in one unknown. We have reduced the computational complexity from one of 42 simultaneous nonlinear equations to one of 14 simultaneous nonlinear equations.

Figure 2.32. Final Rearranged Functionality Matrix for Mixer-Exchanger-Mixer Example

A new functionality matrix can be considered with the three design variables t ₁, t ₃, and w ₁ not specified. This problem now involves 45 variables and 42 equations. There are three degrees of freedom so that these design variables can be chosen using structural analysis to minimize computational complexity. File mix1.matrix gives the input file for this problem. The final rearranged functionality matrix for this problem is given in Figure 2.33. There is one small set of three nonlinear equations that must be solved simultaneously (equations 16 to 3). The loop starts with the platform variables associated with the first subgroup and ends with the step equation of equation 3 which is the location of the only hierarchy pointer. To make the rest of the problem acyclic, the design variables should be the platform variables 23 or 24 (z ₂ or z ₃), 33 or 44 (V ₃ or t ₃), and 32 or 43 (V ₁ or t ₁). Therefore, we can keep t ₁ and t ₃ as design variables and by substituting z ₂ or z ₃ for w ₁, we reduce the computational difficulty of the problem significantly (from 14 simultaneous nonlinear equations to 3 simultaneous nonlinear equations).

Figure 2.33. Rearranged Functionality Matrix with Three Degrees of Freedom

2.4.5.1 Nomenclature for Mixer–Exchanger–Mixer Design

xi,j	Mole fraction of component i in stream j
Fj	Molar flow rate of stream j (g mol/hr)
y i	Molar flow rate of component i (g mol/hr)
T j	Temperature of stream j (°C)
z j	Auxiliary state variable substitution for F j T j
Q k	Heat transferred in unit k (cal/hr)
Ak	Heat transfer area for unit k (m2)
vk	Volume of unit k (m3)
tk	Residence time for unit k (hr−1)
w 1	Auxiliary state variable substitution
v k	Volume of unit k (m3)
S k	Cost of unit k (dollars)
S	Total cost of installation (dollars)
C j	Molar heat capacity of stream j (cal/g mol °C)
U k	Overall heat transfer coefficient of unit k (cal/hr m2 °C)
fi	Row frequency of row i
ρ k	Molar density of fluid in unit k (g mol/cm3)
T j	Column frequency of column j
ΔTlmk	Log mean temperature difference for unit k (°C)

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