Article history:Received 14 June 2012Received in revised form 12 July 2012Accepted 16 July 2012Available online 26 July 2012
Keywords:Branched alkanolViscositiesDeviations in viscosityCorrelating equations
Viscosities at T=(298.15 and 308.15) K and atmospheric pressure for binary liquid mixtures of cyclopentanewith 2-methyl-1-propanol, 3-methyl-1-butanol and 2-methyl-2-butanol over the whole composition rangehave been measured as a function of composition using an Ubbelohde viscometer. The deviations in viscosityand excess free energy of activation for viscous flow have been calculated from experimental data. The resultshave been compared with earlier results of n-alkanols. A number of approximations used to obtain deviationin viscosities have been analysed. Further, different correlations were used to correlate the viscosity data.
Among all physical properties, the knowledge of the viscosity ofliquid mixtures and their dependence on composition and tempera-ture is very important in designing the industrial processes. In thechemical industry, among the physical properties of fluids neededto design and optimise the industrial processes, viscosity is one ofthe most important. Indeed, viscosity is a fundamental characteristicof substances such as adhesives, lubricants, and paintings [1]. Viscos-ity is necessary for hydraulic calculations, fluid transport throughpipes and pore surfaces and in many mass and energy transfers [2].As a part of our ongoing research, in earlier studies we have reportedthe viscometric studies [3,4] of cyclopentane with 1-alkanol and2-alkanol at different temperatures. In the earlier study we shiftour focus on the branching in alkanol molecules. The main aim ofthe study is to see the effect of branching on the properties ofcyclopentane+alkanol binary liquid mixtures. So, in the presentstudy, we report experimental results on the viscosity, η, for the bi-nary liquid mixtures of cyclopentane (1)+2-methyl-1-propanol(2), +3-methyl-1-butanol (2), and +2-methyl-2-butanol (2) atT=(298.15 and 308.15) K over the whole composition range and at-mospheric pressure. Deviations in viscosity and excess free energy ofactivation of viscous flow have been calculated from experimental vis-cosities. Further, various correlating equations like that of McAllister,Heric, and Auslander have been used to correlate the experimental vis-cosities of the mixtures. For viscosity, which is an intensive property,the deviation from ideality should be referred to as such rather than
as excess quantities. Different ideal mixing laws have been chosen byvarious workers to calculate the deviations in viscosity, which giverise to different results and interpretations. In this paper, we havetried to examine the four expressions which are commonly used forthe calculation of deviations in viscosity.
2. Experimental
Cyclopentane, 2-methyl-1-propanol, 3-methyl-1-butanol and2-methyl-2-butanol that were obtained from SD Fine Chemicals,India have mass fraction purities greater than 0.995. Before usingthe components for study, these were stored in dark bottles over mo-lecular sieves (0.4 nm) to reduce water content and were degassedwith a vacuum pump. The details of chemicals, experimental viscosi-ties of pure components along with their corresponding literaturevalues [5–11] are reported in Table 1.
Viscosities of pure liquids and mixtures were measured with a cali-brated suspended level Ubbelohde viscometer [12] at (298.15 and308.15) K and atmospheric pressure. The viscometer was calibrated soas to determine the two constants A and B of the viscometer in equationη=(At−B/t)ρ obtained by measuring flow time t with thrice distilledwater, high purity benzene, toluene, and carbon tetrachloride. The vis-cometerwasfilledwith liquid or liquidmixtures and its limbswere closedwith teflon caps taking dueprecautions tominimise evaporation loses. Anaverage of three to four sets of flow times were taken for each liquid andliquid mixtures for the calculation of viscosity. The flow time mea-surements were made with an electronic stop watch having a preci-sion of ±0.01 s. The caps of the limbs were removed during flowtime measurements. The reproducibility of dynamic viscosity wasfound to be within ±1×10−4 mPa s and the uncertainty is less
Table 2Densities ρ, viscosities η and excess energies of activation for viscous flow ΔG*E forcyclopentane (1)+branched alkanol (2) systems at different temperatures.
x1 ρ×103 (kg m−3) η (mPa s) ΔG⁎E (J mol−1)
Cyclopentane (1)+2-methyl-1-propanol (2)T=298.15 K
64 H. Kumar et al. / Journal of Molecular Liquids 174 (2012) 63–69
than 5×10−3 mPa s. A thermostatically controlled well stirred waterbath, the temperature of which was controlled to ±0.01 K was usedfor all the measurements. The densities corresponding to pure compo-nents and binary liquid mixtures used in this study have been takenfrom our earlier measurements [13].
3. Results and discussions
3.1. Viscosity measurements
The experimental viscosities η, for binary mixtures of cyclopentanewith 2-methyl-1-propanol, 3-methyl-1-butanol and 2-methyl-2-butanolat (298.15 and 308.15) K and atmospheric pressure over the whole com-position range are reported in Table 2 and are graphically presented at298.15 K in Fig. 1.
The measured viscosity were fitted to polynomial of type
η ¼ ∑i¼0
Aixi1: ð1Þ
The coefficients Ai for the correlation of η composition data, evaluat-ed using the least squares method are given in Table 3 along with theresulting standard deviations σ.
From Fig. 1 and Table 2 it is observed that viscosities of all the binarymixtures decrease with an increase in concentration of cyclopentane atall temperatures. As temperature increases, the average speed of themolecules in a mixture increases and the average intermolecular forcesthat offer resistance to flow, decrease with an increase in temperature.The value of η for mixture of cyclopentane at both temperatures variesin sequence: 3-methyl-1-butanol>2-methyl-1-propanol>2-methyl-2-butanol. A further comparison of the data at different temperaturesshows that the temperature coefficient (∂η/∂T)P is decreasing in valuefor the mixtures with increasing temperature. A comparison of viscosi-ties of cyclopentane+branched alkanolsmeasured in the present studyhas beenmadewith earlier viscosity measurements on cyclopentane+1-alkanol [3] and+2-alkanol [4]. The graphical presentations of the com-parisonhave been given in Fig. 2. The comparisons have beenmade of bu-tanol isomers (1-butanol, 2-butanol and 2-methyl-1-propanol), pentanolisomers (1-pentanol, 3-methyl-1-pentanol, and 2-methyl-2-butanol)and methyl substituted n-alkanols. The comparison of butanol isomersas shown in Fig. 2(a) reveals that viscosity follows the sequence:2-methyl-1-propanol>1-butanol>2-butanol. In the case of pentanolisomers (Fig. 2(b)), the viscosities decrease in the following order:3-methyl-1-butanol>1-pentanol>2-methyl-2-butanol. Fig. 2(c) showsthe comparison of methyl substituted n-alkanols. It is observed from thefigure that on substitution of methyl groups in 1-propanol, 1-butanoland 2-butanol, the viscosity increases for 2-methyl-1-propanol, 3-methyl-1-butanol and 2-methyl-2-butanol.
Table 3Coefficients Ai and standard deviations σ for least-squares representations by Eq. (1) atvarious temperatures.
65H. Kumar et al. / Journal of Molecular Liquids 174 (2012) 63–69
3.2. Different expressions for deviations in viscosity
In an earlier study [3]we have analysed the different expressions usedfor the calculation of deviations in viscosity for cyclopentane+1-alkanolbinary liquid mixtures. These different expressions on ideal mixing lawfor viscosity have been chosen erroneously and often unknowingly havebeen used by various workers [14–21]. These different expressions giverise to different values of deviations in viscositywhichmay lead to the im-proper estimation of results which may not be correct. In the presentstudy, same expressions have been analysed for cyclopentane+branchedalkanols to see whether same results are obtained for binary liquid mix-tures containing cyclopentane and alkanol molecules.
The first expression for deviation in viscosity were calculated fromthe relationship [14,15]
Δη ¼ η− x1η1 þ x2η2� �
: ð2Þ
.
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
x1
x1
0
1
2
3
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
(a)
(b)
.
Fig. 1. Experimental viscosities for cyclopentane (1)+2-methyl-1-propanol (2) (ο);+3-methyl-1-butanol, (2) (Δ); +2-methyl-2-butanol (2) (□) at (a) T=298.15 Kand (b) T=308.15 K.
The deviations in the viscosity have been expressed in the form ofexpression [16,17]
Δ lnη or lnΔηð Þ ¼ lnη−X2i¼1
xi lnηi
!: ð3Þ
Sometimes the deviations in viscosity [18,19] are given by expres-sion
Δη ¼ η− ln−1 X2i¼1
xi lnηi
!ð4Þ
0
0.5
0 0.2 0.4 0.6 0.8 1
00.5
11.5
22.5
33.5
44.5
0 0.2 0.4 0.6 0.8 1
(mP
a s)
1-pentanol
3-methyl-1-butanol
2-methyl-2-butanol
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1
(mP
a s)
x1
1-propanol
2-methyl-1-propanol
1-butanol
3-methyl-1-butanol
x1
x1
Fig. 2. Comparison of viscosities of cyclopentane+alkanol mixtures at T=298.15 K.[1-propanol and 1-pentanol (Ref. 3); and 1-butanol and 2-butanol (Ref. 4)].
66 H. Kumar et al. / Journal of Molecular Liquids 174 (2012) 63–69
where η are the dynamic viscosities of the mixtures and x1, x2 and η1,η2 are the mole fractions and viscosities of the pure components,respectively.
Lorenzi et al. [20,21] have expressed the deviation in viscosity asgiven by equation
Δη ¼ η−ηid ð5Þ
where
lnηidMρid
!¼X2i¼1
xi lnηiMi
ρi
� �ð6Þ
ρid ¼ ϕ1ρ1 þ ϕ2ρ2 ð7Þ
where Mi, ρi and ϕi are the molar mass, density and volume fractionsof ith component, respectively.
Deviations in viscosity values calculated from Eqs. (2)–(5) are givenin Table 4 and are graphically represented in Fig. 3 at 298.15 K. It is ob-served from the table and figures that different sets of values of devia-tions in viscosity are obtained which may lead to improper estimatesof results which may not be correct. The deviations in viscosity values
Table 4Deviation in viscosities Δη or lnΔη for cyclopentane (1)+branched alkanol (2) systems at
obtained from Eq. (2) shows deviations from other sets of values calcu-lated from Eqs. (3)–(5). Almost identical results were obtained in thecase of cyclopentane+n-alkanol binary liquid mixtures [3] whereEq. (2) shows deviations from other sets of values which are havingclose values to each other.
3.3. Deviations in viscosity
The magnitude of deviations in viscosity, Δη and excess free energyof activation for viscous flow, ΔG*E is indicative of the nature of interac-tions present between the components of binary liquid mixtures. Theexcess energy of activation for viscous flow ΔG*E were obtained withthe expression [22]
ΔG�E ¼ RT ln ηV=η2V2� �
−x1 ln η1V1=η2V2� �� � ð8Þ
where V is molar volume of the mixture, and R and T have their usualmeanings. The calculated values of ΔG*E at both temperatures arereported in Table 2. The values of Δη (Eq. (2)) at 308.15 K and ΔG*E at298.15 K are graphically represented in Figs. 4 and 5, respectively.
Fig. 3. Deviations in viscosity, Δη or ln Δη, for cylopentane (1) +2-methyl-1-propanol(2) (a); +3-methyl-1-butanol, (2) (b); and +2-methyl-2-butanol (2) (c) at T=298.15 K. [― Eq. (2) - - - - Eq. (3) -⋅-⋅-⋅- Eq. (4) ⋅⋅⋅⋅⋅ Eq. (5)].
-800
-600
-400
-200
0
200
0 0.2 0.4 0.6 0.8 1
G*E
/ (J
mol
-1)
x1
Fig. 5. Excess free energy of activation for viscous flow for cyclopentane (1)+2-methyl-1-propanol (2) (ο); +3-methyl-1-butanol, (2) (Δ); and +2-methyl-2-butanol (2)(□) at T=298.15 K.
67H. Kumar et al. / Journal of Molecular Liquids 174 (2012) 63–69
The calculated values of Δη and ΔG*E of the binary mixtures werefitted to a Redlich–Kister [23] type polynomial equation
Y xð Þ ¼ x1x2Xni¼o
Ai x1−x2ð Þi ð9Þ
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
x1
Fig. 4. Deviations in viscosity for cyclopentane (1)+2-methyl-1-propanol (2) (ο);+3-methyl-1-butanol, (2) (Δ); and +2-methyl-2-butanol (2) (□) at T=308.15 K.
where Y(x)=Δη /mPa s or ΔG*E / J mol−1. The coefficients Ai for thecorrelation of Y(x) — composition data were obtained by fitting theequations to the experimental values with least squares method.There values are given in Table 5 along with standard deviation σ.The standard deviation was calculated by
σ ¼Xn1
Y xð Þ−Y xð Þcalcd� �2
= n−mð Þ" #1=2
ð10Þ
where n and m are the number of experimental points and parame-ters, respectively.
All the mixtures show negative deviation in viscosity frommole frac-tion linearity at all temperatures. The values of Δη as calculated fromEq. (2) at both temperatures decrease in the sequence: 2-methyl-1-propanol>3-methyl-1-butanol>2-methyl-2-butanol. The values ofΔG*E are negative for all mixtures except for cyclopentane+2-methyl-1-propanol mixture where it shows positive deviations at lowmole frac-tion range at both temperatures. Positive deviations are also observed forcyclopentane+3-methyl-1-butanol mixtures at 308.15 K. The negativedeviation in viscosity may be due to the combined effect of breaking thehydrogen bonding as well as destruction of orientational order presentin the cyclopentane. The positiveΔη andΔG*E are indicative of specific in-teractions [24–26]. The negative Δη values for mixture indicate that thedispersion forces are dominant and furthermore the existence of disper-sion forces indicated that the componentmolecules have differentmolec-ular sizes and shapes. The negative Δη values for three binary mixturespoint out the inclusion of smallermolecules in the structure of largermol-ecules, where there are no specific interactions between the componentsof binary liquid mixtures. This also reveals that viscosity behaviour or de-viations in viscosity of liquid mixtures is not only influenced by thestrength of specific interactions but the molecular size and shape of theparticipating molecule also play an equally important role. Further, thesign of the values of the excess energies of activation of viscous flowΔG*E can be considered as a reliable criterion of detecting or excludingthe presence of interactions between unlike molecules. The negativeΔG*E values also indicate the presence of dispersive forces in thesemixtures.
3.4. Comparison of n-alkanols and branched alkanols
For comparison, we have plotted the deviations in viscosity valuesfor butanol isomers (1-butanol, 2-butanol and 2-methyl-1-propanol),pentanol isomers (1-pentanol, 3-methyl-1-pentanol, and 2-methyl-2-butanol) and methyl substituted n-alkanols in Fig. 6. Fig. 6(a) showsthe comparison of Δη values for butanol isomers which follows theorder 1-butanol>2-methyl-1-propanol>2-butanol. A complex behav-iour is observed in the case of pentanol isomers (Fig. 6(b)) The
Table 5Coefficients Ai and standard deviations σ for least-squares representations by Eq. (9) at different temperatures.
68 H. Kumar et al. / Journal of Molecular Liquids 174 (2012) 63–69
overlapping of curves ofΔη values shows the complexity of themixtures.2-methyl-2-butanol is a secondary alkanol. The presence of methyl andethyl group creates hindrance in the formation of hydrogen bonding
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
1-pentanol
3-methyl-1-butanol
2-methyl-2-butanol
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
1-butanol
2-butanol
2-methyl-1-propanol
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
/ (m
Pa
s)
x1
1-propanol
2-methyl-1-propanol
1-butanol
3-methyl-1-butanol
(a)
(b)
(c)
x1
x1
Fig. 6. Comparison of deviations in viscosities of cyclopentane+alkanol mixtures at T=298.15 K. [1-propanol and 1-pentanol (Ref. 3); and 1-butanol and 2-butanol (Ref. 4)].
between alkanol–alkanol molecules as compared to 1-pentanol and3-methyl-1-butanol where hydrogen bonding between alkanol–alkanolmolecules occurs easily. The presence of cyclopentane molecule in themixture breaks down the hydrogen bonding between alkanol–alkanolmolecules i.e. the self association of alkanols molecule breakdown dueto the cyclopentane molecules. The more negative deviations in viscosityin the case of cyclopentane+2-methyl-2-butanol or cyclopentane+3-methyl-1-butanol as compared to cyclopentane+1-pentanol are indica-tive of dispersion interactions between mixing components. In the caseof cyclopentanewith 1-propanol [3], 1-butanol [4], 2-mehtyl-1-propanol,2-methyl-2-butanol as shown in Fig. 6(c), the Δη values are more nega-tive for cyclopentane+2-methyl-1-propanol and cyclopentane+2-methyl-2-butanol mixtures as compared to correspondingcyclopentane+1-propanol and cyclopentane+2-butanol mixtures.The Δη values become more negative as branching in alkanol moleculesincreases. The negative Δη values for mixture indicate the presence ofdispersion forces. The dispersion interactionmay arise owing to breakingof cohesive forces acting in alkanol–alkanol molecules. These values areattributed to the breaking of hydrogen bonded structure of alkanolsdue to addition of cyclopentane which is not compensated by weak in-teractions between unlike molecules. This behaviour suggests that pack-ing is not efficient in the case of branched alkanols due to sterichindrance as that of 1-propanol and 1-butanol.
3.5. Correlating equations
Experimental data on viscosity of liquid mixtures have been corre-lated by using classic laws of mixing representations of the viscosityof binary mixtures with three empirical adjustable parameters. Wehave considered the correlating equations given by McAllister [27](four body interaction model), Heric [28] and Auslander [29].
Table 6Parameter values determined from Eqs. (11)–(13) and percentage standard deviations in correlating viscosities for the cyclopentane (1)+branched alkanol (2) systems at bothtemperatures.
T (K) McAllister Heric Auslander
Z1112 Z1122 Z2221 σ (%) a b c σ (%) B12 B21 A21 σ (%)
where B12, A21, and B21 are the parameters representing binary 1–2interactions.
The correlating ability of all equations was tested by calculatingthe percentage standard deviation σ(%) between the experimentaland calculated values. The values of the parameter for cyclopentane(1)+branched alkanol (2) mixtures at both temperatures are listedin Table 6 along with the percentage standard deviations. It is ob-served from the table that the highest value of percentage standarddeviations is observed in the case of McAllister and the lowest valueis observed in the case of Heric equation. Overall it can be seen thatHeric shows a good correlation with experimental data as evidentfrom less values of percentage standard deviations as compared toMcAllister and Auslander. Same observations were made in our earli-er study [3] on cyclopentane+1-alkanol binary mixtures for threeparameter empirical equations.
4. Conclusions
In the present study the viscosities of binary liquid mixtures ofcyclopentane with ranched alkanols have been reported at differenttemperatures. The study is carried out to see the effect of branchingon the properties of cyclopentane+alkanol binary liquid mixtures.The different expressions on ideal mixing law for viscosity havebeen analysedwhich give rise to different values of deviations in viscos-ity. These different values lead to the improper estimation of resultswhich may not be correct. The Δη values become more negative asbranching in alkanol molecules increases The negative Δη values formixture indicate the presence of dispersion forces. The dispersion inter-actionmay arise owing to breaking of cohesive forces acting in alkanol–alkanol molecules. These values are attributed to the breaking of hydro-gen bonded structure of alkanols due to addition of cyclopentanewhichis not compensated by weak interactions between unlike molecules.This behaviour suggests that packing is not efficient in the case ofbranched alkanols due to steric hindrance as that of n-alkanols. Experi-mental data on viscosity of liquid mixtures correlated by using classiclaws of mixing representations of the viscosity of binary mixtureswith three empirical adjustable parameters reveals that Heric shows agood correlation with experimental data as compared to McAllisterand Auslander.
Acknowledgements
Authors are thankful to The Director and Head, Department of Chem-istry for providing necessary facilities and support.
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