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Relationship between diffusion constants and molecular weight
358
BIOCHIMICA ET BIOPHYSICA ACTA
RELATIONSHIP
BETWEEN
DIFFUSION
VOL. 5 (1950)
CONSTANTS AND
MOLECULAR WEIGHT by A. POLSON AND D. VAN DER REYDEN Section o/ Biophysics, Onderstepoort (Union o/South Alrica)
I t has recently been shown I that the relationship a D = M~
(I)
which was derived from the STOKES-EINST~-INequation, where D is the diffusion constant, M the molecular weight and a is an empirical constant, is applicable to substances of compact form and molecular weights ranging from those of the globular proteins to those of tannic acid (M = 17oo ) and Gramicidin S (M = 114o ). This equation, however, is not applicable to substances of molecular weights lower than 15oo. Below this value of M = x5oo, a shows an ever-increasing value and at M = 19 becomes roughly twice that which it has above M = I5OO. In an a t t e m p t to extend formula (I) to cover these cases an excellent fit over the entire range of the data from molecules of the size of water to African horsesickness virus was obtained b y the following empirical equation D
fO7
a
200
D =
b
+
c
+
(2)
w h e r e b a n d c are constants.
The first term in equation (2) resembles the relationship of PEDERSEN AND SY'NGEs
f~
k D = (MV)--------~ fOC
(3)
°
and in its derivation (eq. 2) V ½ was assumed to be constant as a first approximation, a in equation k (2) equals ~ - in equation (3).
o;
~25
2 1 o ,,;s - £ O i ~ 4 1 0 " 4 ' j
.... io"~'s Log M
Re/ere~es p. 36o.
Fig. z. Curve showing the relationship between log tool. weight and diffusion constant.
VOL. ~ (1950)
359
DIFFUSION CONSTANTS AND MOLECULAR WEIGHT
The equation was fitted b y an empirical process. After a was estimated, the first component calculated was subtracted from the observed data and then b was estimated from the residues obtained,.oxmTtmz/, c and so on. The goodness of fit 200 can be judged from Fig. x where, in order to show the whole range, the diffusion con- 160 stants are plotted against the logarithm of M. Table I below and Fig. 2 show the c o n t r i b u t i o n of each c o m p o n e n t of formula (2) to the e s t i m a t e d value of D. I t will be observed t h a t while a0 the c o n t r i b u t i o n s of the M % a n d M factors are v e r y i m p o r t a n t , below M 15oo t h e y are 40 almost negligible for m o l e c u l a r ',',~r ........ " . . . . . . . . . . . . ] weights above x5oo. ~'- i-~-_-_- = . . . . . . . . . . . . . . . . . . . . . . . . . .......... [ Since the d i m e n s i o n s of 300 700 1100 1500 1900 2JO0 M the second a n d t h i r d compoFig. 2. Contribution of each component in equation (2) to the diffusion constant n e n t s are those of surface a n d a b v o l u m e respectively, it is sugI=~-~; I I = ~c; III=M~ gested t h a t these c o m p o n e n t s
measure the effects of the m e d i u m on the diffusing particle. As a certain amount of slipping must take place when small molecules move amongst solvent molecules of similar size, the term related to the volume, or M, will most likely come into consideration. TABLE I Substance Exelsine Seram albumin Haemoglobin Dvalbumin Lactalbumin Myoglobin ryrocidine Tannic acid Gramicidin S Transvaalin Sucrose Citric acid rryptophane Pentaerythrite Proline Glycine Formic acid Deuterium oxide Re/evenees p. 36o.
a
M
Dob6" xo7
M%
294,000 70,10o 63.000 43,8oo 17,5oo 17,200 2,473 1,700 1,140 7OO 346 208 208 136 125 75 46 19
4.26 6.17
4.12 6.65
6.9o
6.89
7.76 xo.57
7.77 lO.55 to.6x 2o.26 22.96 26.23 3o.86 39.o3 46.24 46.24 53.28 54.80 64.97 76.46 lOO.94
II,25
21.3o 22.00 26.oo 32.9o 46.oo 6I.lO 62.oo 68.oo 80.20 95.00 134.oo 2oo.oo
I b
cl
~
0.037 0.o97 o.1o 4 o.133 0.245 0.248 0.902 1.158
0.006 0.024 0.027 0.039
1.512
1.49I 2.429 4.913 8.173 8.173 12.500 13.6oo 22.667 36.957 85.ooo
2.o93
3.348 4.700 4.7oo 6.239 6.6oo 9.278 12.852 22.394
D~.
lO~
4.16 6.77
0. I 0 0
7.02 7.94 lO.9O 1o.96
0.687
21.85
1.OOO
25.12 29.23 35.38 47.29 59.11 59.xx 72.02 75.oo 96.92 i26.28 208.33
O. I O O
I
Ref.
360
A. P O L S O N , D. V A N
DER
REYDEN
VOL. $
(195o)
Th e surface effect or M ~ will m o s t p r o b a b l y be r e l a t e d to t h e e x t e n t t h a t a m o l e c u l e is solvated~ T h e n u m e r i c a l values of t h e coefficients of t he e m p e r i c a l e q u a t i o n were found to be a ---- 2.74. IO -s, b = 1.65- IO -~ a n d c = 17.oo. Io -5. I t is, however, possible in p r a c t i c e to change these values s o m e w h a t w i t h o u t sensibly affecting the goodness of fit. I t would be v e r y i n t e r e s t i n g to deduce this relationship f r o m t h eo r et i cal considerations.
SUMMARY The diffusion constants of electrically neutral organic compounds of compact structure can be presented by an empirical equation of the form a
b
c
a, b and c axe constants and M is the molecular weight. F r o m this equation it is suggested that the diffusion constant is a function of the radius of the molecule, its area and its volume.
RI~SUMI~ Les constantes de diffusion de composds organiques dlectriquement neutres et de structure compacte peuvent ~tre reprdsentdes pax une dquatiou empirique de la forme a
b
c
o~I a, bet c sout des constautes et M le poids mol4culaire. Cette 4quation sugg~re que la constante de diffusion est fonction du rayon, de la surface et du
volume de la moldcule. ZUSAMMENFASSUNG Die Diffusionskonstanten organischer, elektrisch neutraler Verbindungen mit kompakter Struktur k6nnen durch eine empirische Gleichung yon der Form
ausgedriickt werden, wo a, b und ¢ Konstanten und M das Molekulaxgewicht bedeuten. Aus dieser Gleichung geht hervor, dass die Diffusionskonstante eine Funktion des Radius, der OberflAche uud des Volumens der Molekel ist. REFERENCES * A. POLSON, Forthcoming publication in J. Phys. Coll. Chemistry. t K. O. PEDERSEN AND R. L. M. SYNGE, Acta Chem. Stand., 2 (I948) 408. 3 T. SVEDBERG, K. O. PEDERSEN, The Ultracentri/uge, Oxford, Clarendon Press (x94o). • P. G. J. Louw, Nature, I63 (I949) 3O. A. POLSON, Bioch. J., 3z (x937) I9o3. 6 0 . LA~aM, Thesis, Upsala (1937). R e c e i v e d J a n u a r y r6th, I95O
BIOCHIMICA ET BIOPHYSICA ACTA
RELATIONSHIP
BETWEEN
DIFFUSION
VOL. 5 (1950)
CONSTANTS AND
MOLECULAR WEIGHT by A. POLSON AND D. VAN DER REYDEN Section o/ Biophysics, Onderstepoort (Union o/South Alrica)
I t has recently been shown I that the relationship a D = M~
(I)
which was derived from the STOKES-EINST~-INequation, where D is the diffusion constant, M the molecular weight and a is an empirical constant, is applicable to substances of compact form and molecular weights ranging from those of the globular proteins to those of tannic acid (M = 17oo ) and Gramicidin S (M = 114o ). This equation, however, is not applicable to substances of molecular weights lower than 15oo. Below this value of M = x5oo, a shows an ever-increasing value and at M = 19 becomes roughly twice that which it has above M = I5OO. In an a t t e m p t to extend formula (I) to cover these cases an excellent fit over the entire range of the data from molecules of the size of water to African horsesickness virus was obtained b y the following empirical equation D
fO7
a
200
D =
b
+
c
+
(2)
w h e r e b a n d c are constants.
The first term in equation (2) resembles the relationship of PEDERSEN AND SY'NGEs
f~
k D = (MV)--------~ fOC
(3)
°
and in its derivation (eq. 2) V ½ was assumed to be constant as a first approximation, a in equation k (2) equals ~ - in equation (3).
o;
~25
2 1 o ,,;s - £ O i ~ 4 1 0 " 4 ' j
.... io"~'s Log M
Re/ere~es p. 36o.
Fig. z. Curve showing the relationship between log tool. weight and diffusion constant.
VOL. ~ (1950)
359
DIFFUSION CONSTANTS AND MOLECULAR WEIGHT
The equation was fitted b y an empirical process. After a was estimated, the first component calculated was subtracted from the observed data and then b was estimated from the residues obtained,.oxmTtmz/, c and so on. The goodness of fit 200 can be judged from Fig. x where, in order to show the whole range, the diffusion con- 160 stants are plotted against the logarithm of M. Table I below and Fig. 2 show the c o n t r i b u t i o n of each c o m p o n e n t of formula (2) to the e s t i m a t e d value of D. I t will be observed t h a t while a0 the c o n t r i b u t i o n s of the M % a n d M factors are v e r y i m p o r t a n t , below M 15oo t h e y are 40 almost negligible for m o l e c u l a r ',',~r ........ " . . . . . . . . . . . . ] weights above x5oo. ~'- i-~-_-_- = . . . . . . . . . . . . . . . . . . . . . . . . . .......... [ Since the d i m e n s i o n s of 300 700 1100 1500 1900 2JO0 M the second a n d t h i r d compoFig. 2. Contribution of each component in equation (2) to the diffusion constant n e n t s are those of surface a n d a b v o l u m e respectively, it is sugI=~-~; I I = ~c; III=M~ gested t h a t these c o m p o n e n t s
measure the effects of the m e d i u m on the diffusing particle. As a certain amount of slipping must take place when small molecules move amongst solvent molecules of similar size, the term related to the volume, or M, will most likely come into consideration. TABLE I Substance Exelsine Seram albumin Haemoglobin Dvalbumin Lactalbumin Myoglobin ryrocidine Tannic acid Gramicidin S Transvaalin Sucrose Citric acid rryptophane Pentaerythrite Proline Glycine Formic acid Deuterium oxide Re/evenees p. 36o.
a
M
Dob6" xo7
M%
294,000 70,10o 63.000 43,8oo 17,5oo 17,200 2,473 1,700 1,140 7OO 346 208 208 136 125 75 46 19
4.26 6.17
4.12 6.65
6.9o
6.89
7.76 xo.57
7.77 lO.55 to.6x 2o.26 22.96 26.23 3o.86 39.o3 46.24 46.24 53.28 54.80 64.97 76.46 lOO.94
II,25
21.3o 22.00 26.oo 32.9o 46.oo 6I.lO 62.oo 68.oo 80.20 95.00 134.oo 2oo.oo
I b
cl
~
0.037 0.o97 o.1o 4 o.133 0.245 0.248 0.902 1.158
0.006 0.024 0.027 0.039
1.512
1.49I 2.429 4.913 8.173 8.173 12.500 13.6oo 22.667 36.957 85.ooo
2.o93
3.348 4.700 4.7oo 6.239 6.6oo 9.278 12.852 22.394
D~.
lO~
4.16 6.77
0. I 0 0
7.02 7.94 lO.9O 1o.96
0.687
21.85
1.OOO
25.12 29.23 35.38 47.29 59.11 59.xx 72.02 75.oo 96.92 i26.28 208.33
O. I O O
I
Ref.
360
A. P O L S O N , D. V A N
DER
REYDEN
VOL. $
(195o)
Th e surface effect or M ~ will m o s t p r o b a b l y be r e l a t e d to t h e e x t e n t t h a t a m o l e c u l e is solvated~ T h e n u m e r i c a l values of t h e coefficients of t he e m p e r i c a l e q u a t i o n were found to be a ---- 2.74. IO -s, b = 1.65- IO -~ a n d c = 17.oo. Io -5. I t is, however, possible in p r a c t i c e to change these values s o m e w h a t w i t h o u t sensibly affecting the goodness of fit. I t would be v e r y i n t e r e s t i n g to deduce this relationship f r o m t h eo r et i cal considerations.
SUMMARY The diffusion constants of electrically neutral organic compounds of compact structure can be presented by an empirical equation of the form a
b
c
a, b and c axe constants and M is the molecular weight. F r o m this equation it is suggested that the diffusion constant is a function of the radius of the molecule, its area and its volume.
RI~SUMI~ Les constantes de diffusion de composds organiques dlectriquement neutres et de structure compacte peuvent ~tre reprdsentdes pax une dquatiou empirique de la forme a
b
c
o~I a, bet c sout des constautes et M le poids mol4culaire. Cette 4quation sugg~re que la constante de diffusion est fonction du rayon, de la surface et du
volume de la moldcule. ZUSAMMENFASSUNG Die Diffusionskonstanten organischer, elektrisch neutraler Verbindungen mit kompakter Struktur k6nnen durch eine empirische Gleichung yon der Form
ausgedriickt werden, wo a, b und ¢ Konstanten und M das Molekulaxgewicht bedeuten. Aus dieser Gleichung geht hervor, dass die Diffusionskonstante eine Funktion des Radius, der OberflAche uud des Volumens der Molekel ist. REFERENCES * A. POLSON, Forthcoming publication in J. Phys. Coll. Chemistry. t K. O. PEDERSEN AND R. L. M. SYNGE, Acta Chem. Stand., 2 (I948) 408. 3 T. SVEDBERG, K. O. PEDERSEN, The Ultracentri/uge, Oxford, Clarendon Press (x94o). • P. G. J. Louw, Nature, I63 (I949) 3O. A. POLSON, Bioch. J., 3z (x937) I9o3. 6 0 . LA~aM, Thesis, Upsala (1937). R e c e i v e d J a n u a r y r6th, I95O