C167H336 Is the Smallest Alkane with More Realizable Isomers than the Observed Universe Has "Particles" Robert E. Davies Departments of Animal Biology, Astronomy, and Astrophysics, University of Pennsylvania, Philadelphia, PA 19104 Peter J. Freyd Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
The precise number of structural and stereoisomers of C n H 2n +2 that can be written on paper has been calculated for all values of n up to 400. However, as n gets larger than 17, some, then many, then most, then virtually all of these isomers cannot exist because of overcrowding within the molecules. All previously published values in organic chemistry textbooks and in the Encyclopaedia Britannica for numbers of isomers that can exist are incorrect and examples of what chemists call "paper" chemistry (1). 'The number of stereoisomers of C 167 H 336 that can exist, though not all at the same time, is greater than the classical figure for the number of particles in the observed universe (...... 1()80). As part of the NASA Workshop on Exobiology in Earth Orbit the question arose about the complexity of extraterrestrial organic molecules and how complex such a molecule would have to be for it to have been made either within a living cell or by a chemist. It would be so unexpected to find some large molecules that their discovery outside or even on Earth would strongly point to the existence of life. Students of organic chemistry are routinely told that the number of different hydrocarbon molecules (isomers) with the formula CIl H 2"+2, i.e., the paraffins (alkanes), increases "astronomically" as n increases. 'Textbooks of organic chemistry discuss this and give the number of isomers for n = 1,5, 10,15, and 20 as 1, 3, 75, 4347, and 366, 319. Noller (2) in the Encyclopaedia Britannica records that for C40 H 82 the number is 6.25 X 1013 . Goncharov (3) stated that "the logarithm of the number of theoretically possible isomers of C"H2"+2 increases approximately parabolically for 5 < n < 15 and then increases linearly with n for n > 40. He extrapolated from CaoH62 to claim that there are ...... 1040 possible isomers for C lOoH w2. This number was misquoted by Rouvray (4) as ...... 1040 isomers for C4oo H 802. Cairns-Smith (5) extrapolated the results of Henze and Blair (6) to state that the number of particles in the observable universe (...... 1()80). This problem of the number of isomeric hydrocarbons of the methane series has concerned mathematicians and chemists since 1875 (see Henze and Blair (6)). Although there are general methods for such computations (7-9), they are laborious, and results have not been published for values of n greater than 10 by Nourse et al. (9), greater than 14 by Robinson et al. (8), or greater than 25 by Read (7). We have therefore developed direct methods that translate easily to rather short programs for large values of n that can be run in a few hours on a personal computer. As discussed later, they can also be used to determine which isomers cannot exist. Because carbon has four valences and hydrogen only one, the number T(n) ofstructural isomersofform C"Hz,,+2 is the number of trees (in the combinatorial sense) with 3n + 2 vertices, n of which (the carbons) are on four edges each, the rest (the hydrogens) on one edge each. Such a tree is uniquely reconstructable from the carbons alone: the tree on n vertices each of which is on four or fewer edges. It is easier first to compute R(n) the number of such "rooted" trees, the "root" defined for present purposes to be a designated vertex on three or fewer edges. Any rooted tree of n + 1 vertices 278
Journal of Chemical Education
is uniquely constructed by choosing three rooted trees on a total of n vertices and joining their roots using three new edges to a new root. This yields R(O) = 1
R(n+l)=
L
R(a)R(b)R{c)
L
+
o
Volume 66
Number 4
April 1989
-0-
279
that it is easier to compute Rlo(r + 1) the number of rooted trees of length at most r + 1. Because any such tree is either empty (r = 0) or uniquely consLructed by choosing three rooted trees each of length at most, and joining them to a new vertex, we obtain: R/ o (,+l)-J+ (
RI(r) + 3
2)
For Lrees of diameter 2r we redefine the centroid to be the common central arc of all paths of length 2r. Any tree of diameter 2r is uniquely constructed by joining woo rooted Lrees of length, by a new arc hence Ro(r)+J) 2 (
,
0
Ro(r) + 3) 4
l(RI(r - 1)+ 2) -Ro(r 3
.
,
3)
In order to count the number of stereoisomers the binomi· al coefficients (with one exception) are replaced with F: Rlo(r + 1) - 1 + F(R/o(r), 3) Ro(r
+ I) = RID(r + J) -
T o(2r) - (
For trees of diameter 2r + 1 the centroid is defined to be the common central vertex of all paths of length 2r + 1. Any SHE:LL
T o(2r+ I) = (
_ (RI(r -41) +
Ro(r + 1), the number of rooted Lrees of length precisely, + 1, is RIo(r + 1) - RIn(r).
T o(2r)-
tree of diameter 2, + 1is uniquely constructed by joining to a new vertex four rooted trees of length at most r, at least two of which are of length precisely r. This yields
T ll (2r
+ I) -
Rlo(r)
Ro(r)+J) 2
F(Ro(r), 4) - Rn(r)F(R/(r - 1),3) - F(Rf(r - 1),4)
,
•
IB
IA
FIgu'II 1. 2..()jrnensional fllpfllSllf118tlon of overcrowded moleeules ot type CJi1..+2 in 3--tp&c:e. A. Alllhll carbons in lhII ~lcal molIIcuIIIs arll dlspIayed on • pLanll.llIPP'"oxin\atety lhII same disl8llClllrom lhII centrel carbon II$Ihey woukl hllve in 3--space. The projected e-c disl8llCllls I LWliI.:. 1.3 A; lhII actual e-c (\i$.. IlWlCe Is 1.53 ;. [ _ (A) and (Sl). The CIIrtxIns have 10 til in lhIIlnterlor wILme; lhII hydrogeos have 10 Iii on lhII -'aa1.
--,-.... of """""
""" • , 0
"
'''''''''' C,Ho CsH12
2
C,,",,
3
CW"!'OI
•, • • 7
C,.,Hn • C~n
C,.!,H 2t1l C.,,,H,,.. C,),I,H al ••
~
..,.., 2.'
3.' .., ,.,
..,
., 7.'
B. This Is II tertiary butyl group.
280
Journal of Chemical Education
_.
... _. -' -' - -' -' ...... m
.....~of sphere .1
VoI.ot of
Vol./
-...
C.~
0.524 14.14 65.45 179.59 381.70 696.90 1150.4 1767.2 2572.4
0.524 2.83 3.85 3.39 2.37 1.44 0.790 0.404 0.196
.......
' " " of
Hlllom
,
2
3
•, 6 7
• 9
'"'"
H.~
ofH.lofM
12.57 SO.26 113.10 201.06 314.16 452.39 815.75 804.25 1017.8
3.14 4.19 3.14 1.86 0.970 0.465 0.211 0.0919 0.0388
Table 2. Numbers 01 Isomers 01 Type C"H1o>+2 by Combinatorial Diameters (d)
, , 2 3
•5 •,
• "" .."" "" """ " 9
20 22
ALL structural
t.OOOOOOOooOooOOE + 00000 1.0000000000oo00E + 00000 3.00000000000000E + 00000 6.0000oo00000000E + 00000 5.30000000000000E + 00001 4.96000000000000E + 00002 8. 10960000000000E + 00004 3.52926010000000E + 00007 2. I 1275732504203E + 00014 5.01307895213134E + 00021 4.18849484190550E + 00042 1.39981725808730E + 00064 3.26581392673175E + 00127 3.04769513089384E + 00191 1.54807426847900E + 00382 3. 14537227353680E + 00573 1.64889445652234E + 01146 3.45759127035638E + 01719 1.99248956S47307E + 03438 4.59280968524144E + 05157 3.51565013415047E+ 10314 1.07644746573067E+ 15472
1,OOOOOOOOOOOOOOE + 1.00000000000000E + 3.00000000000000E + 6.00000000000000E + 5.80000000000000E + 8.61 OOOOOOOOOOOOE + 3.73141000000000E + 5.25901096000000E + 9.27091020759361E + 6.5 1902919397758E + 1.41659140384744E + 1.23130676935343E + 5.05372120085193E + 8.29690816671 144E + 2.29462283756128E + 2.53843666139747E + 2. 14788689464154E + 7.26969978386003E + 1.76161783158309E + 1.70752436926125E + 9.71879823879171E+ 2.21267797770886E+
OK slereo ALL stereo
OK stereo
All. stereo
00000 00000 00000 00000 00001 00002 00005 00008 00016 00025 00051 00077 00153 00230 00461 00692 01384 02076 04153 06230 12459 18690
t.OOOOOOOOooOOOOE + 1.00000000000000E + 3.00000000000000E + 6.00000oo0oo0000E + 5.70000000000000E + 8.380000000oo0ooE + 3. 19924000000000E + 3.56308890000000E + 1. 13254237548448E + 8.97709503121433E + 7.39415700852559E + 1.37818326829106E + 5.7709879958 1490E + 5.43732592414286E + 2.39465292603472E + 2.37291S67792362E + 2.53563761565923E + 6.09641585664883E + 6.41267442498107E + 1.S1770133562471E + 2.09361879309914E + 6.49817522254778E+
00000 00000 00000 00000 00001 00002 00005 00008 00016 00023 00044 00066 00124 00183 00347 00511 00965 01419 02680 03942 07447 10952
1.00oo00oo0oo0ooE + 1.00000000000000E + 1.000000oo000000E + 1.000000oo0oo000E+ 9.82758620689655E9.73286875725900E8.57380990027898E6.77520721500835E1.22160861244976E1.37706010666551E-
0000 0000 0000 0000 0001 0001 0001 0001 0001 0002 5.2198822S166633E~ 0007 1.1 1928505762602E - 0011 1.141928445685 14E - 0029 6.55343631011657E- 0048 1.0435932593S237E- 0114 9.34794125064804E - 0182 1.18052660127714E- 0419 8.38606275074399E- 0658 3.64021884316512E- 1473 8.88831435113043E - 2289 2.15419513982979E- 5013 2.93679210803028E-7738
d _ The combinatoO].. Fo< any 91""" value '" d, !he """,Ilest mleCule with comI:Iinato.-ial diameter d;s the stralght-chaln paraffin C....,...,. The largest is !he ...,;que "",.imaHy bfanched parallin Cc'.'I>'- ,H'I"'I>'. _ e d. 2T + jan(! j . 0 0.- 1 (ac
