Canadian Mineralogist Vol. 19, pp. 531-534 (1981)

COMMENTSON THE CALCULATIONOF THEDEN$ITV OF MINERALS

J.A. MANDARINO Department ol Mineralogy & Geology, Royal Ontario Museum, 100 Queen's Park, Toronto, Ontario MSS 2C6

AssTRecr refraction and density (Mandarino 1979, l98l), it was necessary to calculate densities for nu- The densities of minerals are often calculated merous compounds. In many cases, densities incorrectly because wrong molecular weights of the had already been calculated, but as a routine formula units are used. If the empirical formula check they were recalculated, is derived on the basis of a given fixed number and many signifi- of oxygen ions (or other anions), this number cant discrepancieswere found. The discrepan- divided by the total anionic ratios is equal to a cies were commonly of such a magnitude that factor F. F is used to multiply the cationic and they could not be explained by arithmetic pro- anionic ratios to obtain formula subscripts; it also cesses such as premature "rounding off' of provides a simple means of calculating an accurate decimal places. The discrepancieshad to arise molecular weight. The analytical sum multiplied by from one (or more) of the quantities M, Z, F is the molecular weight of one formula unit. V or A used in the foregoing equation. After numerous checks, it became apparent that the Keywords: calculated density, molecular weight, empirical formula, anionic ratio method. main source of the discrepancies was the molec- ular weight of the formula unit. Furthermore, because the molecular weight of the formula SotvttuArns unit is directly related to the derivation of the formula, it is by the way On calcule souvent incorrectement la densit6 des affected in which the min6raux parce que l'on se sert de poids mol6- formula is derived. culaires erron6s pour les unit6s formulaires. Ot Ia formule empirique se rapporte i un nombre fixe DenryerroN oF THE Fontrur,e d'ions oxygBne (ou autre anion), ce nombre divi- s6 par le total des rapports anioniques donne le Throughoul the remainder of this paper, deri- facteur F. En multipliant par F les rapports ca- vations of empirical formulae are based on the tioniques et anioniques, on obtient les indices des underlying principle that all minerals can be 6l6ments de la formule; F fournit aussi une fagon considered as frameworks of anionic elements, simple de calculer le poids mol6culaire exact. La in the interstices of which are found the ca- par somme analytique multipli6e F devient le tions. It follows that for a particular mineral, poids mol6culaire d'une unit6 formulaire. the number of anions in a given unit-cell volume (Iraduit par la R6daction) (or multiple thereof) will be an integer. McCon- nell (1965) has discussed this principle, and Mots-cMsi densit6 calcul6e, poids moldculaire, for- no more need be said here. mule empirique, m6thode du rapport anionique. Various errors can be introduced during the calculation of the molecular wergbt of a for- INTRoDUCTIoN mula unit. Some are trivial and result from the use by different authors of slightly different The calculation of densities from unit-cell atomic weights. In general, this does not result parameter$ and chemical composition is a well- in large errors. The major error in M is pro- known operation in mineralogy; it is usually duced by the way it is calculated. An example done using an equation similar to D = MZ/ will serve to illustrate the problem. VA, wbere D is density, M is the molecular Table 1 shows the analytical data (average weight of one formula uni,/', Z is. the number of five analyses) for satterlyite given by Man- of formula units per unit cell, 7 is the unit-cell darino et al. (1978). To illustrate points to be volume and I is Avogadro's number. discussed later, the derivation of the empirical During research on the application of the formula from these analytical data is given in Gladstone-Dale relationship to the compatibi- detail. Column A lists the weight percentageof lity of chemical composition, mean index of each constituent. Each weight percentage is 531

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TABI-E1. SATTERLYIIE:AMLYTICAL DATA F so that the product is 5. At{DDERIVATION OT FONilULA ratios by a factor Solving for this factor gave F = 2.0156. Each DE of the cationic ratios in Column D was multi- ut, % Molecular Anlonic Catlonic Nunberof Ions Ratlos Ratlos Ratios (Subscrlpts) plied by F to give formula subscripts for those l'{S0 7.r 0.L762 0.1762 0.1762 l'19 0.36 cations. The subscripts are given only to the I'ln0 It 0.0183 0.01f|:l 0.0183 l'|n-- 0.04 place Column E, but could Fe0 43.1 0.5999 0.5999 Fez. L,zL second decimal in Fe203 0.0470 0.1409 0.0939 Fe3' 0.19 have been expressed to additional places. If Na20 tq 0.0242 0.0242 0.0484 Na 0.10 Pzos 34.8 0.2452 L,2259 0.4903 P 0.99 halide or 52- ions are present, an appropriate 5t02 0.2 0.0033 0.0067 0.0033 si 0.01 Hzo 0.2886 0.2886 0.5771 H 1.16 amount of the oxygen. anionic ratio must be I 100.7 2.4807 0 5.00 subtracted from the sum of the anionic ratios particular Derivatlon of factor F: total anlonlc ratio (2.4807)x F " 5; before F can be calculated for a F - 2.0i56. Calculatlonof mlecular welght:t l'lt. t (100.7)x F number of total anions. An example of this is " 202.97, given later in this PaPer.

CeLcuuerton or Moreculen WrrcHt divided by the molecular weight of the appro- priate constituent, and the resulting molecular Table 2 lists several ways in which the molec- ratios are listed in Column B. Column C con- ular weight of a formula unit of satterlyite may tains the anionic ratios which. in this case be derived. The first deals with the empirical (where the only anionic element is oxygen), formula which. in this case, is (Fe2+r.:tMgo.au are simply the molecular ratios multiplied by Fe"*0.r, Ho.ru Nao.,o Mno.or):r.ou (Po."t Sio.ot)rt.oo the number of oxygens in each constituent. The gr.oo (OH)r.ro. The molecular weight of the cationic ratios glyen in Column D are the result mineral representedby this formula can be sal- of multiplying the molecular ratios by the num- culated by multiplying each elemenfs atomic ber of cations in each constituent. Column E weight by the appropriate subscript from the lists the number of ions (= subscripts of chem- formula and adding these products together. An ical symbols) in a formula unit of the mineral. alternative method is to use molecular weights It was decided to calculate the formula sub- of oxide constituents, multiply these by the scripts on the basis of five oxygen ions in the appropriate factors and add the results. It must formula by multiplying the sum of the anionic be rememberedthat in the caseof RnO, consti-

TABL,E2. I'IOLECULARWEIGHTS 0F F0RI''IULAUNITS DERMD IN VARIOUSI'IAYS ANDTHE RESULTING VALUES OF DENSITY

Irlethodof calculatlng the molecular l.lol ecular cal cul ated welght of one fomula unit welght of density formulaunlt (O/ctnll Empir'lcalfoYmula (Fef]rrmso. (Po.ggsl (oH) rure3]rg*0.ru*to. roilno. oq)re. oo o.or)rt.0004. o0 1.00 a. subscrlptsto seconddeclmal place 203.87 3.604 b. subscrlptsto third decimalplace 202.94 3.588 c. subscr'lptsto fourth decimalplace 202.96 3.588 Norma'llzed enplrical formula (ref+rrltso.ruFe3laHo. roNuo. roMno. oa) rz. oo(Po. ggsi o. ot )rr. oo0*.oo(0H) t. oo a. subscripts to second decimal place 199,79 3.532 b. subscripts to thlrd decimal place 200,2L 3.540

Normalizedideal fornula (Fei+54Mso. 46)r2. OOPO4(oH) a. subscripts to seconddeclmal place ?09,L7 3.698 b. subscripts to third decimal place 209.36 ? 7n1

Ideal formula re!+ron{oH) 223.68 3.954

Surmationx Factor 202,97 ? Rnn

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tuents, the molecular weights should be multi- should not be used to calculate density. In plied by haU the subscript in the formula. Thus, this regard, it should also be remembered that for Feu* the molecular weight of FerO, should departures from ideal stoichiometry are not at be multiplied by 0.19/2. In the following dis- all unusual in minerals. cussion of molecular weight calculations, the On the other hand, the unit-cell volume is oxide molecular weights listed by Mandarino calculated from cell parameters determined, (1976) are used. ideally, from the same material used to deter- Let us examine the molecular weight cal- mine chemical composition. Therefore, only the culated from the empirical formula. Three empirical formula should be considered in cal- values of the molecular weight and the cor- culating density. Referring back to the first responding density values are given for tbe entries in Table 2. it can be seen that the use empirical formula. These are based on subscripts of subscripts given to two decimal places gave expressedto two, three and four decimal places. a slightly different density than that derived The density calsulated in the first case (3.6O4 from subscripts given to three or four decimal g/cm") is significantly different from those places. It would seem necessary,then, to cal- calculated in the other two cases (each 3.588 culate the formula subscripts to at least three g/cmu). decimal places to obtain the proper molecular The next entries in Table 2 deal with cal, weight and density. The following section pro- culations from the "normalizedo' empirical for- poses a simple method that enables the molec- mula, where each cation (and anion) is recal- ular weight of a formula unit to be deter- culated so that the sum of each cationic (and mined without using formula subscripts. anionic) group is a whole number. Thus, the normalized empirical formula of satterlyite is Tun ANrorsrc RATIo MetHop or (Fes*r.rzMgo.6 Fet*o.r, IIo..u Nfu.ro Mno.or):z.oo CeLcuLerrxc MorecuI-en WercHT (Po.* Sio.o,)rt.ooOa.oo(OH)r.*. If the subscripts of this formula are canied to two decimal Referring to Table l, we see that the factor places, the calculated density is 3.532 g/cm8; F used to determine the formula subscripts of subscripts to three decimal places give a cal- satterlyite is 2.o156. If this factor is multiplied culated density of 3.540 g/cm'. by the analytical sum (l : 100.7 wt. Vo), the Another type of formula that many authors result is the molecular weight of the formula seem to prefer can be salled the "normalized" unit. In the example, the molecular weight cal- ideal formula, in which only the major consti- culated in this manner is 2A2.97 and the density tuents are considered. For satterlyite, this for- is 3.588 g/cmg. These compare favorably with mula becomes (Fe2+t.- Mgo..r)"r.ooPO.(OH). 202.96 and 3.588 g/cms calculated from the Calculation of the density from this formula empirical formula, using subscripts calculated to using subscriptsof two and three decimal places the fourth decimal place. gives, respectively, 3.698 and 3.7O1 g/cm'. Finally, many authors calculate density on MrNeners CoNterNrNc Her,rpn on Sunnn [oNs the basis of the theoretical pure mineral and use the "ideal" formula. In thp case of satter- The discussion up to this point has dealt lyite, this is Fe2*rPOa(OH),and the calculated with minerals that contain only oxygen in their density is 3.954 g/cm". _ Obviously, the ideal formula should not be TABLE3. TRIPLITE:AMLYTICAL DATA used to calculate density, because for a min- ANDDERIVATION OF FORMIjLA eral that exhibits significant solid-solution effeca the density value would be meaningless. Not ABCDE only would 'tt. % llolecular Anlonlc Catlonic Nmber of Ions the calculated density be wrong for Ratlos Ratlos Ratlos (Subscrlpts) the particular mineral studied, but it would also Cao 2.1,7 0.0387 0.0387 0.0387 Ca 0.09 be .wrong for the theoretical pure end-member, 1490 0.31 0.0077 0.0077 0.0077 llq 0.o2 Feo 6.68 0.0930 0.0930 0.0930 te. 0.21 as it is highly unlikely that the volume of the l'1n0 53.77 0.7580 0.7580 0.7580 Mn 1.69 unit cell would be unaffected by compositional Fe203 0.40 0.0025 0.0075 0.0050 Fe3' 0.01 Pz0s 32.20 0.2269 1.1343 0.4537 P 1.01 changes. The same argument holds for the F 7.58 0.3989 0.3989 F 0,89 calculation of density from the normalized ideal 'lesstotal 103.U 2.4381 0 4.11 formula. The unit-cell parameters will not be 0.1994 [email protected] the same for a mineral that does not contain t-di@ . all the constituents reported in a good analysis. r 0 1.8398 Similarly, the normalized empirical formula does Derivation of factor F: total anionlc ratio (2,2387)x F = 5; F = 2.2334. Calculatlon of nolecu'larweight: r llt.7, (99.92) X F not accurately reflect the analytical data and - 28.16.

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anionic frameworks. If halide or sulfide ions the empirical formula should be used to cal- also are presentothe procedure for deriving tle culate the molecular weight of the formula chemical formula and for calculating the mole- unit. Ideally, the unit-cell parameters and for- cular weight musr be modified. A triplite from mula should be derived from the same material. Bagdad,Arizona, describedby Hurlbut (1936), An accurate molecular weight can be calculated has been chosen to illustrate the procedure. using the multiplying factor F derived from the Richmond (1940) determined the unit-cell sum of the anionic ratios. This factor multiplied parameters of this analyzed triplite, and his by the sum of the analysis gives the molecular data (converted from kX to A) give a unit-cell weight of one formula unit. volume of 754.30 A'8. The analytical data for this mineral are listed in Table 3. Note that in Column A. there is an AcKNowLEDGEMENTS entry' of 3.19 wt. Vo" which represents the amount of excess oxygen introduced into the Dr. F.J. Wicks. Mr. B.D. Sturman and Mr. analytical data because of the method of re- P.J. Dunn kindly read the manuscript and of- porting the cation contents as oxides. This fered suggestions for its improvement. Miss excess oxygen must be subtracted from the Helen Driver and Mrs. Gloria Nelson typed total to give the true analfiical sum. In this the manuscript. particular case, the figure of 3.19 is found by multiplying the wt. 7o of fluorine by the factor REFERENcEs 8.00/19.00, where 8.,00is half the atomic weight of oxygen and 19.00 is the atomic weight of HURLBUT, C.S., Jn. (1936): A new phosphate, fluorine. bermanite, occurring with triplite in Arizona. In Column B, the excessoxygen (or oxygen- Amer. Mineral. 2L, 656-661. equivalent of fluorine) is divided by the atomic weight of oxygen to give a molecular ratio of MeNoenrNo, J.A. (1976): The Gladstone-Dale - 0.1994. This figure is repeated in Column C relationship Part I: Derivation of new con- stants. Can. Mineral. L4. 498-502. as an anionic ratio with a negative sign to indicate that it must be subtracted from the total (1979): The Gladstone-Dale relationship. anionic ratio to give the true total. It is this Part III: Some general applications, Can. Min- final anionic ralio (2.2387) that is used to eral. L7. 7l-16. determine the factor F. In the case of triplite, the total number of anions (oxygen plus fluorine (1981) : The Gladstone*Dale relationship: and possibly hydroxyl) in a formula unit is 5; Part IV. The compatibility concept and its ap- thus the factor F is 2.2334, and the molecular plication. Can. Mineral. L9, 441-450. weight (l wt. Vo x F) is 223J6. The density (1978): the SrunuaN, B.D. & Conrerr, M.I. calculated from this molecular weight and phos- g/cm"; Satterlyite, a new hydroxyl-bearing ferrous unit-cell volume is 3.930 the measured phate from the Big Fish River area. Yukon Terri- density given by Hurlbut (1936) is 3.84 g/cmu, tory. Can. Mineral, 16, 411-413. and the densitycalculated by Richmond (1940) is 3.94 g/cm". McCoNNELL, D. (1965): Deriving the formula of Note that the anionic ratio of oxygen is a mineral from its chemical analysis. Mineral. given at the bottom of Column C in Table 3 Mag. 35, 552-554. as l0 : 1.8398. This representsthe difference between the true total anionic ratio and the RrcHMoND,W.E. (1940): Crystal chemistry of the anionic ratio of the fluorine (i.e.,2.23874.3989 phosphates, arsenates and vanadates of the type AIXOI(Z). Anrer. Mineral. 2t, 441-479. - 1.8398). This figure must be used to deter- mine the oxygen subscript.

CoNcrusroNs Received March 1981, revised mawrscript accepted In calculating the density of a mineral, only luly 1981.

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