Research: Science and Education www.JCE.DivCHED.org • Vol. 82 No. 3 March 2005 • Journal of Chemical Education 473 A Brief History of Lanthanides In the history of chemistry, one can hardly find a more intricate field than the discovery and classification of the rare earth elements. Their unusual similarity of properties, as well as their scarcity, which prevented the exact determination of these properties, caused considerable difficulties; it is even probable that the development of the periodic table would have been significantly delayed had these elements been known in the middle of the 19th century. It has been stated that “the group of rare earths forms a little periodic system of its own, where all the relations of the main system are re- produced on a small scale” (1). After failing to classify the rare earth elements as homo- logues of other elements, the solution was to group the lan- thanides together in a separate series (2). The atomic model of Bohr made it possible to settle the number of lanthanides at 14, but the attempt to find relations within the group of rare earth elements never ceased. The chapter on lanthanides in the book by van Spronsen (3) gives a historically com- plete account, so only some of these relationships will be dis- cussed here. An early classification of the rare earth elements (the ox- ides of lanthanides) in relation to their separation from ores, was in the ceritic earth elements (the oxides from lanthanum to samarium) and yttric earth elements (from europium to lutetium, but also scandium and yttrium). A further refine- ment of analytical methods made it possible to split the yttric earth elements into terbic (europium, gadolinium, terbium), erbic (dysprosium, holmium, erbium, thulium), and ytterbic (ytterbium and lutetium) earth elements, along with yttrium oxide and scandium oxide. In the “arena” system of Clark (4), some of the lan- thanides are correlated with other groups of elements (Fig- ure 1) while 7 lanthanides remain without analogues. Scheele (5) suggested that lanthanides could be classed, as could ac- tinides, in new kinds of groups, c, as follows: Ic: thulium (as a homologue of rubidium (Ia) and silver (Ib); IIc: ytterbium; IIIc: lutetium; IVc: cerium, praseodymium, neodymium (as homologues of zirconium); Vc: promethium; VIc: samarium; VIIc: europium; VIIIc: gadolinium, terbium, dysprosium, holmium, erbium (as homologues of ruthenium, rhodium and palladium). Ternstrom (6) built up this idea and divided lanthanides into two rows after gadolinium, and distributed them in c-groups (Figure 2). Another problem still under discussion is the identity of the element that should be considered the homologue of yttrium in group 3. This role is usually assigned to lantha- num, with the series of lanthanides going from cerium to lutetium. However, there are also reasons to put lutetium into group 3, as the homologue of yttrium, and thus begin the series of lanthanides with lanthanum (7). Since the problem of a rational classification for the rare earth elements, based on their properties, cannot be consid- ered as resolved, a real challenge exists to make use of the methods of multidimensional analysis to establish some cor- relations between these elements and possibly to group them into distinct classes, and at the same time find correlations among the properties used to characterize them. We applied principal component analysis and cluster analysis in the study of 84 chemical elements (including the lanthanides, except promethium), characterized by 10 physical properties or a combination of 10 physical, chemical, and structural char- acteristics (8, 9). The lanthanides formed, as a rule, clusters distinct from other elements. Only europium and ytterbium (the lanthanides with relatively stable electronic configura- tions, f 7 and f 14 , respectively) appear as being close to the elements of the main groups of the periodic table. Also, a certain analogy between these lanthanides and calcium and lead could be observed. Quite similar results were found in a fuzzy classification of chemical elements (9, 10). As a conse- quence, the processing and rational interpretation of the reg- istered data concerning the lanthanides appeared to be better performed by using multivariate analysis (8–18). Cluster Analysis The objective of cluster analysis (CA) is to sort samples (elements) into groups of similar characteristics (properties or descriptors) (8, 9, 12–16). Because of its unsupervised character, CA can be used to perform a preliminary data scan and to uncover the structure residing in the data set. If m characteristics have been determined for each of n elements, then these elements could be represented by n points in an Characterization and Classification of Lanthanides by Multivariate-Analysis Methods Ossi Horovitz and Costel Sârbu* Department of Chemistry and Chemical Engineering, Babes-Bolyai University, 400028 Cluj-Napoca, Romania; *[email protected]Y Zr Nb Mo Tc Rb Sr IIIb IVb Vb VIb VIIb Ia IIa La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Lu Yb Figure 1. Clark’s system of organizing the lanthanides (4). Figure 2. Ternstrom’s system of organizing the lanthanides (6). Sc Y La Ce Pr Nd Pm Sm Eu Gd B3 C4 Group C5 C6 C2 C3 Tb Dy Ho Er Tm Yb Lu
Transcript
Research: Science and Education
www.JCE.DivCHED.org • Vol. 82 No. 3 March 2005 • Journal of Chemical Education 473
A Brief History of Lanthanides
In the history of chemistry, one can hardly find a moreintricate field than the discovery and classification of the rareearth elements. Their unusual similarity of properties, as wellas their scarcity, which prevented the exact determination ofthese properties, caused considerable difficulties; it is evenprobable that the development of the periodic table wouldhave been significantly delayed had these elements beenknown in the middle of the 19th century. It has been statedthat “the group of rare earths forms a little periodic systemof its own, where all the relations of the main system are re-produced on a small scale” (1).
After failing to classify the rare earth elements as homo-logues of other elements, the solution was to group the lan-thanides together in a separate series (2). The atomic modelof Bohr made it possible to settle the number of lanthanidesat 14, but the attempt to find relations within the group ofrare earth elements never ceased. The chapter on lanthanidesin the book by van Spronsen (3) gives a historically com-plete account, so only some of these relationships will be dis-cussed here.
An early classification of the rare earth elements (the ox-ides of lanthanides) in relation to their separation from ores,was in the ceritic earth elements (the oxides from lanthanumto samarium) and yttric earth elements (from europium tolutetium, but also scandium and yttrium). A further refine-ment of analytical methods made it possible to split the yttricearth elements into terbic (europium, gadolinium, terbium),erbic (dysprosium, holmium, erbium, thulium), and ytterbic(ytterbium and lutetium) earth elements, along with yttriumoxide and scandium oxide.
In the “arena” system of Clark (4), some of the lan-thanides are correlated with other groups of elements (Fig-ure 1) while 7 lanthanides remain without analogues. Scheele(5) suggested that lanthanides could be classed, as could ac-tinides, in new kinds of groups, c, as follows: Ic: thulium (asa homologue of rubidium (Ia) and silver (Ib); IIc: ytterbium;IIIc: lutetium; IVc: cerium, praseodymium, neodymium (ashomologues of zirconium); Vc: promethium; VIc: samarium;VIIc: europium; VIIIc: gadolinium, terbium, dysprosium,holmium, erbium (as homologues of ruthenium, rhodiumand palladium). Ternstrom (6) built up this idea and dividedlanthanides into two rows after gadolinium, and distributedthem in c-groups (Figure 2).
Another problem still under discussion is the identityof the element that should be considered the homologue ofyttrium in group 3. This role is usually assigned to lantha-num, with the series of lanthanides going from cerium tolutetium. However, there are also reasons to put lutetium intogroup 3, as the homologue of yttrium, and thus begin theseries of lanthanides with lanthanum (7).
Since the problem of a rational classification for the rareearth elements, based on their properties, cannot be consid-ered as resolved, a real challenge exists to make use of themethods of multidimensional analysis to establish some cor-relations between these elements and possibly to group theminto distinct classes, and at the same time find correlationsamong the properties used to characterize them. We appliedprincipal component analysis and cluster analysis in the studyof 84 chemical elements (including the lanthanides, exceptpromethium), characterized by 10 physical properties or acombination of 10 physical, chemical, and structural char-acteristics (8, 9). The lanthanides formed, as a rule, clustersdistinct from other elements. Only europium and ytterbium(the lanthanides with relatively stable electronic configura-tions, f 7 and f 14, respectively) appear as being close to theelements of the main groups of the periodic table. Also, acertain analogy between these lanthanides and calcium andlead could be observed. Quite similar results were found in afuzzy classification of chemical elements (9, 10). As a conse-quence, the processing and rational interpretation of the reg-istered data concerning the lanthanides appeared to be betterperformed by using multivariate analysis (8–18).
Cluster Analysis
The objective of cluster analysis (CA) is to sort samples(elements) into groups of similar characteristics (propertiesor descriptors) (8, 9, 12–16). Because of its unsupervisedcharacter, CA can be used to perform a preliminary data scanand to uncover the structure residing in the data set. If mcharacteristics have been determined for each of n elements,then these elements could be represented by n points in an
Characterization and Classification of Lanthanidesby Multivariate-Analysis MethodsOssi Horovitz and Costel Sârbu*Department of Chemistry and Chemical Engineering, Babes-Bolyai University, 400028 Cluj-Napoca, Romania;*[email protected]
Y Zr Nb Mo Tc Rb Sr
IIIb IVb Vb VIb VIIb Ia IIa
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm
Lu
Yb
Figure 1. Clark’s system of organizing the lanthanides (4).
Figure 2. Ternstrom’s system of organizing the lanthanides (6).
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m dimensional space. It is then possible to define the(dis)similarity between these elements in terms of variousgeometric parameters such as the distance between the pointsin the space or the angles between the vectors that could bedrawn from the origin of these n points. Using these inter-elements (dis)similarity measures, groups of “close” elementscan be found that form a class or a cluster. There are a num-ber of ways in which the differences between elements canbe expressed and how to group elements based on these vari-ous distance measures. Thus, the Euclidean distance xkl be-tween two points (elements ) k and l in m-dimensional spaceis given by the formula
d d2 (xkl kj l jj
= −=11
2m
k l k l∑ = −( ) ⋅ −( )) d d d d T
where (dk − dl ) is the difference vector between the patternvectors dk and dl for elements k and l, respectively, while (dk− dl )T is its transpose.
Principal Component AnalysisPrincipal component analysis (PCA) is also known as
eigenvector analysis, eigenvector decomposition, orKarhunen–Loéve expansion. Many problems from chemis-try and other technical fields are strongly related to PCA (8,
9, 13–18). The main purpose of PCA is to represent in aneconomic way the location of the elements in a reduced co-ordinate system where instead of m axes (corresponding tom characteristics) only p axes ( p < m) can usually be used todescribe the data set with maximum possible information.
PCA practically transforms the original data matrix(Xnxm) into a product of two matrices, one of which containsthe information about the elements (Snxm) and the other aboutthe properties (variables) (Vmxm). The S matrix contains thescores of the n elements on m principal components (thescores are the projection of the elements on principal com-ponents). The V matrix is a square matrix and contains theloadings of the original properties on the principal compo-nents (the loadings are the weights of the original propertiesin each principal component). Moreover, it may well turnout that usually two or three principal components providea good summary of all the original variables. Loadings andscores plots are very useful as a display tool for examiningthe relationships between properties and between elements,and looking for trends, grouping, or outliers.
Properties of Lanthanides Considered in This StudyWe applied PCA and CA methods to characterize lan-
thanum, the 14 lanthanides, and the other group 3b elements,scandium and yttrium, with the idea of clarifying the rela-
bY 040.371 79.6 491 02.1 4.306 8.4711 9.6142 7901 7641
uL 769.471 48.9 371 72.1 5.325 1.1431 2.2202 6391 8663
Note: Ma is the atomic mass; d is the density; ra is the atomic radius; χ is the electronegativity (Pauling); I1 is the first ionization energy; I2 is the second ionizationenergy; I3 is the third ionization energy; Tm is the melting point; Tb is the boiling point.
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tions of the latter two with the lanthanides. The characteris-tics selected as a basis for the classification were almost ex-clusively physical properties. Properties were selected forwhich the values were known for all (or nearly all) the ele-ments under consideration. The data were taken from dif-ferent tables or textbooks (for example, refs 19, 20). Sincethe values given in various sources often differ significantly,we tried to select the most probable values to make up a con-sistent data set. A special case is that of promethium; datafor this element are sometimes scarce and uncertain. For someproperties there are no reliable experimental data in the lit-erature, but the values are obtained by interpolation fromthose of the neighboring elements. Table 1a and Table 1bshow the 18 selected properties. Table 1a includes: atomicmass (Ma), density (d ), atomic radius (r a), first (I1), second(I2) and third (I3) ionization energy, Pauling electronegativ-ity (χ), melting point (Tm), and boiling point (Tb). Table 1bshows: enthalpy of fusion (∆fusH ), vaporization (∆vapH ), at-omization (∆atmH ), standard entropy (S ), specific heat ca-pacity (Cp), surface tension at the melting point (σm),electrical resistivity (ρ), heat conductivity (λ), and Gibbs en-ergy of formation of the chloride LnCl3 per mol of Cl (∆fG ),as a characteristic of chemical reactivity.
The electronic configurations of the elements studiedand of their tripositive ions are shown in Table 2. Eu and Yb
Note: ∆fusH is the enthalpy of fusion; ∆vapH is the enthalpy of vaporization; ∆atmH is the enthalpy of atomization; S is the entropy; Cp is the specific heat capacity, σm isthe surface tension at the melting point; ρ is the electrical resistivity; λ is the heat conductivity; ∆fG is the Gibbs energy of formation of the chloride LnCl3 per mol of Cl.
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have relatively stable electron configurations and the ions,Gd3+ and Lu3+ should be the most stable, as well as Ce4+,Tb4+, Eu2+, and Yb2+. La, Gd, and Lu have configurationsanalogous to those of Sc and Y.
Correlation and Classification of Properties
The results obtained from the initial data set (17 ele-ments and 18 properties) using the well known Statistica soft-ware package (21) are presented in three tables. The data
statistics are shown in Table 3. These results and the correla-tion data in Table 4 confirm that the lanthanides’ propertiesare related to each other and so could be reduced. The eigen-vectors associated with each of the first six principal compo-nents are displayed in Table 5. Table 5 also lists the eigenvaluesof the correlation matrix, ordered from largest to smallest andshows also the proportion and cumulative proportion for eachcomponent.
The first component explains only 34.21% of the totalvariance and the second component explains 27.40%; a two
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component model, for example, thus accounts only for61.61% of the total variance (Table 5). The column corre-sponding to the first eigenvector obtained in this case indi-cates that the contribution to the first component is verydifferent (Table 5). The greatest contribution is given by theσm (0.940), the next highest is the ∆atmH (0.788), r a (0.766),∆vapH (0.741), and S (0.718). It is interesting to note thatthe contribution to the first principal component of Tm, Tb,and Cp are similar and the smallest contribution was obtainedfor ρ and ∆fG . Similar contribution were also obtained for∆vapH (0.741), and ∆atmH (0.788); the contribution of r a isalso high but negative (0.766). It is interesting to note thatr a and S are similar but the values are negative, which sug-gests that the values of these characteristics decrease when,for example, ∆vapH and ∆atmH increase (negative correlation).The second component describes the remaining variance af-ter the first component is removed from the data. In this casethe highest contribution to PC2 was obtained for I3 (0.807),the second being χ (0.752), and the third ∆fG (0.743).The lowest contribution to PC2 was given by Ma (0.023), d(0.102), and r a (0.132). The major contribution to PC3 wasobtained for d (0.921) and Ma (0.705). We note that theseresults are more or less in agreement with the correlation datain Table 4, and also observe the position of σm as an outlier.This statement is well supported by a two-dimensional (i.e.,PC1 versus PC2) and a three-dimensional (i.e., PC1 versusPC2 versus PC3) representations of the loadings in Figure 3and 4.
The results from the PCA of the 15 lanthanides (Table6) are different from the results from the PCA of the 17 lan-thanides. We can see that, for example, the first principalcomponent explains 38.08% of the total variance and thesecond component explains 32.74%: a two component modelthus accounts for 70.82% of the total variance (as comparedto 61.61% for PCA of the 17 elements) and a three compo-nents model accounts for 84.68% (as compared to 79.83%for PCA of 17 elements), for the PCA of 15 elements (Table6). Hence, the first PCA-derived components for 15 elementsaccount for significantly more of the variance than the PCA
Figure 3. Projection of variables on the plane defined by PC1 andPC2.
1.0
0.5
0.0
0.5
1.0
1.00.50.00.51.0
S
ra
Ma d
λ
I3
σm
Tb∆vapH
∆atmHρ
Cp
Tm
χ
∆fusH
I1
∆fGI2
PC1: 34.21%
PC
2: 2
7.40
%
Figure 4. Projection of variables in the space defined by PC1, PC2,and PC3.
478 Journal of Chemical Education • Vol. 82 No. 3 March 2005 • www.JCE.DivCHED.org
of 17 elements counterparts. The data in Table 6 correspond-ing to the first eigenvector illustrate that the greatest contri-bution to the first component is obtained for Ma (0.946),∆fG (0.837), χ (0.814), I2 (0.703), ∆fusH (0.711), andI3 (0.692). A less significant contribution is obtained fromr a (0.055) and S (0.038). With respect to the second princi-pal component it is easy to observe that the highest contri-bution is obtained for σm (0.908), r a (0.820), ∆atmH(0.765), Tm (0.752), Tb (0.728), d (0.707), and ∆vapH(0.697) (Table 6). The third component appears to be wellcorrelated with ρ (0.851), S (0.813) and λ (0.598), respec-tively. These statements are also confirmed by two-dimen-sional and three-dimensional representations of the loadingsin Figure 5 and 6.
Examining all the pairs of variables for relationships be-tween them, rather strong correlations are found (Table 4).Nevertheless, these relations should be considered with cau-tion, as concerns their generalization for other elements. Lan-thanides are a rather peculiar group of elements as theirproperties are very similar; that is, the variation of a givenproperty in the series of these elements is rather limited andinferences to all the elements are doubtful. Keeping this inmind, we may discuss the closest correlations found here.
The highest correlation coefficient (r = 0.98) is betweenenthalpy of vaporization and standard enthalpy of atomiza-tion (Table 4). For metals, the latter one is actually the stan-dard sublimation enthalpy and is given, according to Hess’law, by the sum of the standard enthalpies of fusion and va-
porization. The enthalpies of vaporization in Table 1 are thoseat the boiling temperature and therefore lower than the stan-dard ones. Other very close correlations are those betweenthe boiling point and the enthalpy of atomization (r = 0.96)or the enthalpy of vaporization (r = 0.93). This is in agree-ment with the well-known rule of Pictet and Trouton thatstates that the ratio of the enthalpy of vaporization at theboiling point to the boiling temperature is a constant. For93 chemical elements a linear relation with a correlation co-efficient of 0.96 was also obtained (9).
Figure 5. Projection of variables on the plane defined by PC1 andPC2 (15 lanthanides).
Ma
d
1.0
0.5
0.0
0.5
1.0
1.00.50.00.51.0
PC1: 38.08%
PC
2: 3
2.74
%
ra
χ
I1
I3
I2
TmTb
∆fusH
∆vapH
∆atmH
S
Cp
σm
ρ
λ
∆fG
Figure 6. Projection of variables in the space defined by PC1, PC2,and PC3 (15 lanthanides).
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A significant correlation, but with a negative correlationcoefficient (r = 0.93), appears between the specific heat ca-pacity and the atomic mass. For lanthanides, as for most met-als, the rule of Dulong and Petit is applicable (inverseproportionality between the two quantities). The correlationbetween the enthalpy of fusion and the melting point is looser(r = 0.87), but better than for the ensemble of all chemicalelements: for 95 elements, r = 0.775 (9).
The negative correlations between the third ionizationenergy and the enthalpy of vaporization (r = 0.87), the en-thalpy of atomization (r = 0.83), or the boiling point (r =0.84) seem rather fortuitous. The correlation of density andatomic mass (r = 0.86) is particular to lanthanides; the den-sity increases with the atomic number (with some exceptions,such as Eu, Yb), while the atomic volume is nearly constant(actually it decreases slightly—the contraction of lanthanides).On the other hand the correlations of surface tension withthe enthalpy of atomization (r = 0.84), the boiling tempera-ture (r = 0.78), and the enthalpy of vaporization (r = 0.77)are well-known general relations. Surface tension representsthe free energy necessary for the creation of a new interfaceand is therefore related to the energy spent in the splitting ofbonds, the same entity that is implied in vaporization or at-omization.
In the second step of data analysis, a study of the struc-ture of data by cluster analysis was carried out. The searchfor natural groupings among the properties of lanthanides isone preliminary way to study the data structure. Clusteranalysis describes the nearness between properties. In this case,a matrix consisting of the squared Euclidean distance was usedas similarity matrix. The data were autoscaled to eliminatethe effect of the different sizes of the variables. Thus, a simi-larity matrix S18×18 was constructed from the autoscaled data;the values of this matrix were the squared Euclidean distanceof one property from the rest. To obtain clusters, a hierar-chical agglomerative method was employed, the Wardmethod. This procedure considers, in each step, the hetero-geneity of deviance (sum of squares of the distance of a prop-erty from the barycenter of the cluster) of every possiblecluster that can be created by linking two existing clusters(22). The results obtained, presented as a dendrogram (Fig-ures 7 and 8), showed the presence of clusters of properties.
One can easily see (Figure 7) that the highest similarityis that of the enthalpy of vaporization and atomization andthe boiling point; the surface tension is soon joined to them.All these properties are related to the splitting of the bondsbetween the particles of the substance. The distances calcu-lated for their representative points in the space of the firstsix factors are correspondingly the lowest: ∆vapH –∆atmH :0.138; ∆atmH –Tb: 0.197; ∆vapH –Tb: 0.256; σm–∆atmH : 0.49;σm–∆vapH : 0.59; σm–Tb: 0.606. The order is much the sameas that of the correlation coefficients discussed earlier. Thecluster of these four properties is the last to be linked withthe other properties.
Two other pairs to be linked early are composed by themelting temperature with the enthalpy of fusion (distance0.671) and the second enthalpy of ionization with electrone-gativity (0.517). These four variables constitute later a clus-ter together with the pair first ionization energy–specific heat(distance 0.916). The correlation of electronegativity and theionization energies is not surprising, since the Pauling elec-
tronegativity correlates well with those in the Mulliken scale.Here electronegativity is proportional to the difference be-tween the first ionization energy and the electron affinity (thelast being nearly constant for considered elements).
The pair density–atomic mass (distance 0.694) includestwo properties strongly correlated for the lanthanides (as wasexplained earlier); the entropy (which increases for heavieratoms) and the electrical resistance. The remaining four vari-ables make up a cluster, which subsequently joins to the pre-cedent. Among them, the closest pair is the third ionizationenergy and the Gibbs energy of formation of the chloride(distance 0.733). A correlation of these characteristics appearsreasonable, since GEF refers to the formation of chlorideLnCl3. A much looser pair is that of atomic radius and ther-mal conductibility (distance 1.13).
As a rule, strongly correlated variables have representa-tive points in the space of factors near to each other. How-ever, this is not the case for negatively correlated variables,for instance, inversely proportional ones. Atomic mass andspecific heat are strongly correlated (r = 0.93), but their rep-resentative points are far away (distance 0.967) and there-fore they are put in different clusters. Thermal and electrical
Figure 7. Dendrogram corresponding to 18 properties and 17 ele-ments.
100
(Dlin
k/D
max
)
100
80
60
40
20
0
Tm I 2 χ I 3 λ r a ρ S d Ma
σm Cp I 1Tb
∆va
pH
∆fu
sH ∆fG
∆at
mH
Figure 8. Dendrogram corresponding to 18 properties of 15 lan-thanides.
480 Journal of Chemical Education • Vol. 82 No. 3 March 2005 • www.JCE.DivCHED.org
Figure 12. Projection of elements in the space defined by PC1,PC2, and PC3.
0-2
-4-6
24
6
0
-1
-2
-3
1
2
3
4
54
32
10
-1-2
-3-4
-5
Sc
Y
LuGd
La
Tb
ErCe
Ho
Pr Nd
Pm
Dy
Tm
Sm
Yb
Eu
PC
3
PC2PC1
Figure 9. Projections of the elements on the plane defined by PC1and PC2.
0
0
-1
-2
-3
-4
-5
-6
-7
1
2
3
4
5
6
-2-4-6-8 4 6 8 102
PC1: 34.21%
PC
2: 2
7.40
%
Sc
Y
LaCe
PrNd
Pm
SmEu
Gd
Tb
Dy
Ho Er
Tm
Yb
Lu
Figure 10. Projections of the elements on the plane defined by PC1and PC3.
0
0
-1
-2
-3
-4
-5
1
2
3
4
5
-2-4-6-8 4 6 8 102
PC1: 34.21%
PC
3: 1
8.22
%
Sc
Y
La
Ce
PrNd
Pm
Sm
Eu
GdTb
Dy
HoErTm
Yb
Lu
Figure 11. Projections of the elements on the plane defined by PC2and PC3.
PC2: 27.40%
PC
3: 1
8.22
%
Sc
Y
La
CePrNd
PmSm
Eu
GdTb
DyHoEr
Tm
Yb
Lu
0-1-2-3-4-5-6-7 2 3 4 5 61
0
-1
-2
-3
-4
-5
1
2
3
4
5
conductivity for metals are both related to the delocalizedelectrons and are roughly proportional (the law ofWiedemann and Frantz). Consequently, heat conductivityand electrical resistivity (the inverse value of electrical con-ductivity) should be negatively correlated and so they are (r= 0.72), but their representative points are quite far awayand they fall in different clusters.
The dendrogram shown in Figure 8 considering only the15 lanthanides, seems to be somewhat different from the firstdendrogram taking into account 17 elements (Figure 7). Thecluster of the enthalpy of vaporization and atomization, boil-ing point, and surface tension remains the same, very com-pact and clear-cut. Here we find the lowest distances betweenrepresentative points and the same order of these distances.Yet the remainder of the dendrogram is different.
To the pair density–atomic mass the Gibbs energy of for-mation of the chloride is added and this cluster is joined tothe cluster of electronegativity–melting point–enthalpy offusion–second ionization energy (present also in the previ-ous classification). The cluster first and third ionization en-ergy–thermal conductivity–atomic radius differs from asimilar cluster in the dendrogram with 17 elements by thepresence of the first ionization energy instead of GEF. A lastcluster, joined to this, is formed from entropy–specific heatand electrical resistivity.
These changes are explicable by the fact that removingSc and Y, elements with properties rather dissimilar to thoseof the lanthanides, the extreme values from most of the vari-ables were removed and the variance of the remaining valuesis quite restricted. This accounts for the high susceptibilityof the classification to this modification.
Characterization and Classificationof the Lanthanides
Principal components or eigenvectors that were calcu-lated are orthogonal and each of them is a linear combina-
www.JCE.DivCHED.org • Vol. 82 No. 3 March 2005 • Journal of Chemical Education 481
Figure 15. Projections of the lanthanides on the plane defined byPC1 and PC3.
0-2-4-6-8 4 6 82
PC1: 38.08%
PC
3: 1
3.86
%
0
-1
-2
-3
-4
-5
1
2
3
4
La
CePr
NdPm
Sm
Eu
Gd
TbDy
HoEr
Tm
YbLu
Figure 13. Dendrogram corresponding to the 17 elements.
Yb Eu Sm Pm Nd Pr Ce La Lu Tm Er Ho Dy Tb Gd Y Sc
100
80
60
40
20
0
100
(Dlin
k/D
max
)
Figure 16. Projections of the lanthanides on the plane defined byPC2 and PC3.
PC2: 32.74%
PC
3: 1
3.86
%
0
-1
-2
-3
-4
-5
1
2
3
4
La
CePr
NdPm
Sm Eu
Gd Tb Dy
HoEr
Tm
YbLu
Figure 14. Projections of the lanthanides on the plane defined byPC1 and PC2.
0-2-4-6-8 4 6 82
PC1: 38.08%
PC
2: 3
2.74
%
0
-2
-4
-6
-8
2
4
6
8
LaCePr
NdPm
Sm
Eu
Gd
TbDyHoEr
Tm
Yb
Lu
tion of the original variables (properties). As can be seen againin Tables 4 and 5 the first three eigenvectors represented79.83% and 84.68% of the total variability of the data for17 and 15 elements, respectively. Thus, by reducing the num-ber of features from 18 original properties (manifest variables)to three principal components (latent variables), the preservedinformation is enough to permit a primary examination ofthe elements according their origin in a two-dimensional anda three-dimensional plot, respectively. By plotting the ele-ments in the plane described by two of the first three eigen-vectors and in the space defined by the first three principalcomponents, interesting results were afforded.
From the plot of the 17 elements on the plane describedby PC1 and PC2 (Figure 9), PC1 and PC3 (Figure 10), andPC2 and PC3 (Figure 11), as well as in the space defined bythe first three eigenvectors (Figure 12), the representativepoints of Sc and Y appear clearly as outliers. However, Yband Lu are also rather far from the other elements. There isalso an evident grouping in two classes: on the one hand thelight lanthanides (La, Ce, Pr, Nd, Pm, Sm—the elementsfrom the ceritic earth elements) with Eu and also Yb some-what farther, in the left side in Figures 7 or 10, and the other(Gd, Tb, Dy, Ho, Er, Tm, and farther Lu, but also Sc andY—on the right side.
The dendrogram (Figure 13) confirms also the conclu-sions above: Sc and Y constitute a distinct cluster, but theirdistance to each other is considerable (5.97). This cluster joinsonly late to the group of heavier lanthanides. The results ofcluster analysis seem therefore to support the assigning of lu-tetium rather then lanthanum into the third group of theperiodic table as homologue of scandium and yttrium.
To make more distinct the representative points of thelanthanides, we repeated the multidimensional analysiswith the 14 lanthanides and La only. In this way, thegrouping of elements in the dendrogram remains exactlythe same and the relative positions of the elements in thescatterplots were only slightly modified. The two-dimen-sional scatterplots of various pairs of the first three fac-tors are given in Figures 14–16 and the three-dimensional
482 Journal of Chemical Education • Vol. 82 No. 3 March 2005 • www.JCE.DivCHED.org
scatterplot in Figure 17. The outlier positions of Lu andYb appear clearly in Figure 17 and a relevant classifica-tion is evident in Figure 14.
By careful visual examination of the results obtained bycluster analysis it is easy to observe from the dendrogram (Fig-ure 18) that the first separation groups together the “light”lanthanides (La, Ce, Pr, Nd, Pm, Sm, Eu) and Yb. In Figure14, for example, they are situated in the upper half of thediagram, with the elements from La to Eu placed in the firstquadrant and Yb in the second (Sm and Eu are on the bound-ary line between the quadrants). Yb is added to the lighterlanthanides according to its similarity with europium. In theelectron configurations of these elements, the 4f level is lessthan half full, except Eu (half-complete level, f 7) and Yb(complete level, f 14), the elements with relatively stable con-figurations. The second group is that of the “heavier” lan-thanides, situated in the lower half of the diagram (Figure12), with Gd and Tb in the fourth quadrant and Dy, Ho,Er, Tm in the third, together with Lu–placed at a higher dis-tance from the other.
The cluster of “light lanthanides” is further divided intoone cluster including the first five elements (from La to Pm)and another containing the elements with relative stable elec-tron configuration (Yb, Eu) and Sm, which has one electronfewer than such a configuration. From the “heavier lan-thanides”, the dendrogram differentiates the pair Gd–Tbfrom the cluster Dy, Ho, Er, Tm, and from Lu.
As a rule, the highest similarity degrees appear for adja-cent elements (in the sequence of atomic numbers). The clos-est pair is that of Nd and Pm (distance 0.4); the first elementto join to them is Pr (distance Nd–Pm: 1.04). The similarityof Nd and Pr is well known; for a long time they were evenconsidered as a single element, didymium. The next element
in the dendrogram is Ce (Pr–Ce: 1.61) and then La (Ce–La:3.4). Ce, Pr, and Nd share also the chemical property to givecompounds in the valence state 4, besides those in the usualvalence 3.
In the cluster Sm–Eu–Yb the distances are greater: Sm–Eu 3.22 and Eu–Yb 6.65. It is interesting to observe that allthese elements give also compounds in the valence state 2.
The distance Eu–Gd is high (7.78); the nearest neigh-bor for Gd is Tb (distance 2.60); so they form a pair insidethe group of “heavier lanthanides”. In this group, the closestpair of elements is Er–Ho (distance 0.81); in the same clus-ter we find Dy (nearest element, Ho, at 1.16) and Tm (near-est element, Er, at 1.56). The distance Tm–Yb is much higher(6.46). For Lu, the nearest lanthanides are Er (5.16), Ho(5.51), and Tm (6.00); the distance to Yb is much greater(9.74). The outlier position of lutetium among the lan-thanides (unlike La, which is much closer to its neighbors)assigns it as a homologue of Y in the third group of the peri-odic table.
The sum of the distances from an element to all the other14, as a measure for the position of the representative pointof this element in the factor-space is the lowest for Pm (57.2),Dy (57.7), Tb (59.8), Nd (59.8), and Ho (61.8). Therefore,they hold central positions into the group of elements andcould be considered as typical lanthanides. The results forpromethium should be considered with caution, since experi-mental data for its properties are uncertain and some prop-erties were estimated by interpolation.
The outliers are Yb (114.2) and Lu (104.6), followedby Eu (94.05), and La (90.2). Yb and Eu are the elementswith relatively stable electron configuration and many of theirproperties deviate from the regularities observed for the otherlanthanides. As a matter of fact, multidimensional and fuzzy
Lu Tm Er Ho Dy Tb Gd Yb Eu Sm Pm Nd Pr Ce La
100
80
60
40
20
0
100(
Dlin
k/D
max
)
Figure 18. Dendrogram corresponding to the 15 lanthanides.Figure 17. Projection of lanthanides in the space defined by PC1,PC2, and PC3.
www.JCE.DivCHED.org • Vol. 82 No. 3 March 2005 • Journal of Chemical Education 483
analysis of the chemical elements (9–11) suggested some simi-larities of Eu and Yb with main group elements. As for Luand La, in their electron configuration with one electron inthe d subshell with the f subshell empty (La) or completelyoccupied (Lu), unlike the other lanthanides.
Conclusions and Remarks
The variables selected as a basis for this classification weremainly physical properties, with only the Gibbs energy offormation of the chloride as a chemical property; no struc-tural features, such as electron configuration, or chemicalproperties, such as the oxidation state, were introduced. Nev-ertheless, the resulting classification discriminates betweengroups of lanthanides with different structural and chemicalcharacters. Elements with oxidation state 2+ are grouped to-gether and so are elements with oxidation state 4+. Elementswith similar peculiarities of electron configuration are alsoput in the same cluster.
Based on this analysis and of the electron configuration,we may propose a “periodic system” of the lanthanides (Fig-ure 19). Setting lutetium as the homologue of yttrium, theseries of lanthanides begins with lanthanum and ends withytterbium. It is broken into two rows, after europium, so thatelements with electron configurations up to f 7 are placed inthe first row and those with the f subshell more than half-full in the second. There is always a difference of seven f elec-trons between the homologous elements in the two rows. Forthis reason, these elements have the same number of addi-tional electrons as compared with the relative stable configu-rations [Xe]6s2 (first row) and [Xe]4f 76s2 (second row) orthe same number of electrons less than the relative stable con-figurations [Xe]4f 76s2 (first row) and [Xe]4f 146s2 (secondrow).
A broken line separates the two classes of “light” lan-thanides, including also Yb from the “heavier” lanthanides,together with Sc and Y. The clusters are framed by solid bor-ders. Going along the series of lanthanides in the directionof the arrows we find all the clusters given by the cluster analy-sis. In comparison with the traditional splitting of the lan-thanide series (for instance, ref 7 ), the main difference is theplacement of Gd in the second row, while the ions Gd3+ andLu3+ have the most stable configurations ([Xe]4f 7 and[Xe]4f 14, respectively). Among the neutral atoms the moststable configurations are those of Eu, [Xe]4f 76s2, and Yb,
La Ce Pr Nd Pm Sm Eu
Gd
Lu
Y
Sc
Tb Dy Ho Er Tm Yb
Figure 19. The “periodic system” of lanthanides.
[Xe]4f 146s2. Therefore these two elements should play thepart of the “noble gases” in the system of lanthanides, andnot Gd and Lu. On the other hand, the electron configura-tion of gadolinium [Xe]4f 75d16s2 makes this element thenatural homologue of lanthanum [Xe]5d16s2.
Comparing with the c groups of Scheele (5), we find thathis IVc group contains the elements of the same cluster (Ce,Pr, Nd) and in his VIIIc-group we find together the elementsof two neighboring clusters: Gd and Tb on the one hand andDy, Ho, and Er on the other. Obviously, the chemical intu-ition of the chemists who attempted to classify the lanthanidesled them to groupings similar in many respects to those foundby the methods of multidimensional analysis.
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