L. A. LAPLANCHE, H. 3.THOMPSON, AND M. T. ROGERS
1482
Chain Association Equilibria. A Nuclear Magnetic Resonance Study of the Hydrogen Bonding of N-Monosubstituted Amides'"
by L. A. LaPlanche,Ib H. B. Thompson, and M. T. Rogers Michigan State University, East Lanaing, Michigan
(Received June 27, 1964)
Equations which describe chain association equilibria are developed, and two special cases are considered, the self-associated solute in (1) an inert solvent and (2) a bonding solvent. Application of the theory to hydrogen bond equilibria is attained by relating the equilibrium parameters to the n.m.r. chemical shift expression. From a series of chemical shift measurements of the bonding proton at varying solute concentrations, the equilibrium constants for hydrogen bond formation may be determined, as well as the chemical shifts of the associating proton, free and bonded. The solutes studied are the N-monosubstituted acetamides, CH3CONHR, where R = CH3, CH(CH3)2,and C(CH3)3.
being greater than when only one monomer and a Introduction higher polymer unite. Studies of the thermodynamics of hydrogen bond forThe present treatment of associative equilibria is mation in molecules such as the N-monosubstituted similar to that of Redlich and Kister,za and Mave1,2b amides are complicated by the fact that these molecules with two important differences: (1) the theory has are highly self-associated, resulting in complex equibeen modified to include two equilibrium 'constants, libria among monomer, dimer, trimer, and higher K12 and R , and (2) the theory has been made more species. A theory for continuous association has general by allowing the consideration of solvents which been described by Redlich and Kister,)" assuming may also hydrogen bond to the self-associated species. that the equilibrium constant for the reaction nBy relating the equilibrium parameters to the n.m.r. mer monomer + (n 1)-mer does not depend on chemical shift expression, Klz and R can be calculated. the value of n. NIavel,2busing similar equations, has The equations have been programmed in Fortran for derived a relationship whereby the equilibrium contwo special cases of associated solutions: (1) the inert stant for continuous equilibria may be found from n.m.r. chemical shift data. However, other a ~ t h o r s ~ - ~solvent case and (2) the hydrogen-bonding solvent case. Both have been applied to the study of soluhave found that a single equilibrium constant is not tions of N-monosubstituted amides which are known sufficient to explain their experimental results. Coggeto self-associate by hydrogen bonding (NH. .OC) shall and Saier,3 in infrared studies of alcohols, and into long chains. Davies and tho ma^,^ in vapor pressure studies of amides, used two equilibrium constants, Klz for the monomer-dimer equilibrium and i? for the general equilibria among higher aggregates. The justifica(1) (a) This work was supported through a contract with the Nation for using at least two different equilibrium contional Science Foundation; (b) National Institutes of Health P r b stants comes from the statistical treatment of associated doctoral Fellow, 1961-1963. solutions by Sarolea-;Llathot,6 who predicts that (2) (a) 0. Redlich and A. T . Kister, J . Chem. Phya., 15, 849 (1947); (b) G.Mavel, Doctoral Thesis, University of Paris, 1961. should be larger than K12 by a factor of p , where p is (3) N. D. Coggeshall and E. L. Saier, J . A m . Chem. Soc., 73, 5414 the number of possible orientations of the monomer (1951). with equal energy. The difference in equilibrium (4) M.Davies and D. K. Thomas, J . Phys. Chem., 60, 767 (1956). constants is thus ascribed to an entropy factor, the (5) E.G.Hoffmann, Z . physik. Chem., 53, 179 (1943). loss of entropy when two monomer units form a dimer (6) L. Sarolea-Mathot, Trane. Faraday Soc., 49,8 (1953).
+
+
1
The Journal of Phyeical Chemistry
CHAINASSOCIATION EQUILIBRIA
1483
Theoretical. I. The General Case Equilibrium Expressions. The addition of a noninert solvent to a self-associated solute will, in general, produce a solut,ion composed of an equilibrium mixture of monomer, dimer, and higher aggregates of the solute molecules, some of which will be bonded to a solvent molecule. In the treatment to follow it has been assumed that the monomer-dimer equilibrium constants are distinct from the equilibrium constants of higher aggregates but that the equilibrium constants for n-mer monomer -+ (n 1)-mer, where n > 1 , are equal. In like manner, the monomer-solvent equilibrium constant is assumed to be different from the corresponding constant for solvation of higher polymers. It is further assumed that each n-mer molecule can associate with only one solvent molecule, by hydrogen bonding a t one end of the chain. Under these assumptions, equilibria of interest are
+
+
P = Z + S = l
(15)
These quantities may be expressed in terms of monomer and solvent mole fractions by making appropriate substitutions from eq. 5 , 6 , and 9.
r'
=
Y1
+ K1zYl2 + K1zRYl3 + . . . + K1zfiinYln+' (16)
When R Y , is less than unity, this may be expressed in closed form'
Kiz Yi2
P= Ylfl-Ryl By similar procedures
Kn
2 A z Az
Y = Y1
a
An
of the respective types of solute species, while X , Y , and Z are quantities that can be related to the stoichiometric concentration. Our definition of mole fraction is such that
+ AJ-An+i
n
>1
(2)
+ 2K1zYi2 + . . . + (n + 2)KlZRnYl"+' + . . . =
(18)
KS
A
+ B JcA * B a.
A,
+B Z A n . B
n
(3)
>1
(4)
where A represents solute monomer, A , is solute nmer, and B is the solvent. The equilibrium constants are then
Kiz = Yz/Yi2
R
=
Yn+l/Y,Yn K.
R,
=
=
(5) 12
>1
(7)
Zi/YiS
Zn/YnS
n
(6)
>1
(8)
where Y , = mole fraction of unsolvated n-mer; Z, = mole fraction of solvated n-mer; S = mole fraction of solvent. Quantities useful in the development of the theory are
P = CYn n
2
=
zz, n
A'
=
(1
+ SKs)Y1+ ( 1 +( 1 RsS)KizY1' - RY,)
(24)
(9) (10)
x= P+Z
(11)
Y
=
Cn n Y n
(12)
Z
=
CnZ,
(13)
n
By appropriate combinations
X = Y + Z (14) The quantities R, t,and 2 are effective mole fractions
andsinceX = 1 - S
S =
+ +
Y l Y R - Kid - Y l ( R 1) 1 (26) Y12(K12Rs- KSR) Y,(K. - X) 1
+
+
This relationship may be used to replace the solvent terms in eq. 25. (7) L. Jolley, "Summation of Series," 2nd Ed., Dover Publications, IC., New York, N. Y., 1961, p. 2.
Volume 69,Number 6 M a y 1966
1484
L. A. LAPLANCHE, H. B. THOMPSON, AND M. T. ROGERS
Basically, the problem of finding the equilibrium state of a system obeying this model is thus reduced to relating Y 1 or S to the stoichiometric concentration. Defining a concentration C
CE-
(number of moles of solute as monomer) (number of moles of solvent, free and associated with solute)
or
This can be simplified, using eq. 9,10,and 14,to
-
+
X(So - VD) = Y(VM- VD) Z(VC11. Special Case of an Inert SoJvent
VD)
(30)
Equilibrium Expressions. When the amount of hydrogen bonding of solute to solvent is small
K,S
