Monosaccharide and Disaccharide Isomerization over Lewis Acid Sites in Hydrophobic
and Hydrophilic Molecular Sieves
SUPPORTING INFORMATION
Rajamani Gounder and Mark E. Davis*
Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United
States
*Corresponding author. E-mail: mdavis@cheme.caltech.edu
S.1. X-ray diffractograms of zeolite samples
Powder X-ray diffraction patterns of the samples used in this study are shown in Fig. S.1.
Intensity (A.U.)
TiO2-SiO2
Ti-Beta-OH
Sn-Beta-F
Ti-Beta-F
4
8
12 16 20 24 28 32 36 40
2q²
Figure S.1. Powder X-ray diffraction patterns of Ti-Beta-F, Sn-Beta, Ti-Beta-OH and TiO2-SiO2
(bottom to top).
S.2. Single-component vapor-phase adsorption isotherms of zeolite samples
N2 (77 K), H2O (293 K) and CH3OH (293 K) adsorption isotherms are shown for TiBeta-F (Fig. S.2), Ti-Beta-OH (Fig. S.3) and Sn-Beta-F (Fig. S.4). Corresponding micropore
uptakes shown in Table 1 (main text).
200
180
160
3 g -1)
Vads
(cm
3 g-1 STP)
Vads (cm
140
120
100
80
60
40
20
0
0
0.2
0.4
0.6
0.8
1
P/P0
P/P0
Figure S.2. N2 (77 K) ( ), H2O (293 K) ( ) and CH3OH ( ) adsorption isotherms for Ti-BetaF.
800
700
3 g-1 3STP)
V
Vads
(cm
g-1)
ads (cm
600
500
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
P/P
P/P00
Figure S.3. N2 (77 K) ( ), H2O (293 K) ( ) and CH3OH ( ) adsorption isotherms for Ti-BetaOH.
180
160
120
3
Vads
g
Vads
(cm3(cm
g-1 STP)
-1
)
140
100
80
60
40
20
0
0
0.2
0.4
0.6
0.8
1
P/P0
P/P0
Figure S.4. N2 (77 K) ( ), H2O (293 K) ( ) and CH3OH ( ) adsorption isotherms for Sn-BetaF.
S.3. Deactivation studies of Ti-Beta-F during reactions of glucose in liquid water
The deactivation of Ti-Beta-F during conversion of a 1% (w/w) aqueous glucose solution
at 373 K (1:50 Ti:glucose ratio) was assessed by first measuring the evolution of fructose and
sorbose products as a function of batch reaction time for two hours (Figs. 5a and 5b). The
catalytic solids were then isolated from the reaction mixture by centrifugation; they retained a
yellow discoloration that reflected the presence of adsorbed sugars or sugar decomposition
products. The reaction of a fresh glucose reactant solution with the spent Ti-Beta-F catalyst led
to initial fructose and sorbose formation rates that were lower by factors of ~2.5 (Figs. S.5a and
S.5b), indicating that partial deactivation of Ti-Beta-F occurred during the first reaction cycle
and, in part, contributed to the approach of monosaccharide product concentrations to steadystate values at long reaction times.
4
(a)
(b)
6
Sorbose concentration
(mol m -3)
Fructose concentration
(mol m -3)
8
4
2
0
3
2
1
0
0
2
4
6
Time (ks)
8
10
0
2
4
6
8
10
Time (ks)
Figure S.5. Aqueous-phase (a) fructose and (b) sorbose concentrations as a function of reaction
time during the first ( ) and second ( ) reaction cycles of a 1% (w/w) aqueous glucose solution
with Ti-Beta-F (1:50 glucose:Ti molar ratio, 373 K).
S.4. Derivation of batch reactor concentration profiles for glucose and lactose isomerization
S.4.1. Concentration profiles in a batch reactor for a reversible first-order reaction
Here, we provide an abridged derivation of the concentration profiles in an ideal, constant
volume, batch stirred tank reactor for the following reversible reaction:
A
K𝑅
B
⇄
(S.1)
where KR is the equilibrium constant for the stoichiometric reaction. The mole balances for A and
B (Eq. (S.1)) in a constant volume, stirred batch tank reactor can be expressed in terms of their
concentrations (cA(l), cB(l)) according to:
−
𝑑c𝐴(𝑙) (𝑡)
𝑑𝑡
=
𝑑c𝐵(𝑙) (𝑡)
𝑑𝑡
𝑀
= 𝑟𝑛𝑒𝑡 𝑉
(S.2)
where rnet is the net rate (per metal site) of the reaction shown in Eq. (S.1), M is the total number
of metal sites in the reactor, and V is the reactor volume. The assumption that the forward and
reverse rates are first-order in cA(l) and cB(l), respectively, leads to the following expression for the
net rate of reaction:
𝑟𝑛𝑒𝑡 = 𝑘𝑓 c𝐴(𝑙) − 𝑘𝑟 c𝐵(𝑙)
(S.3)
where kf and kr are the effective first-order forward and reverse rate constants for the reaction
(Eq. (S.1)). These two effective rate constants are related by the equilibrium constant for the
stoichiometric reaction (Eq. (S.1)):
𝐾𝑅 =
𝑘𝑓
𝑘𝑟
(S.4)
At any point in time, the following relationship must hold between the concentrations of A and
B:
𝑐𝐴0(𝑙) = c𝐴(𝑙) + c𝐵(𝑙)
(S.5)
where cA0(l) is the initial concentration of A in the reactor.
Combining Eqs. (S.2)-(S.5) leads to the following expression:
𝑑c𝐵(𝑙) (𝑡)
𝑑𝑡
𝑀
𝑘
𝑀
= 𝑘𝑓 (𝑐𝐴0(𝑙) − c𝐵(𝑙) ) 𝑉 − 𝐾𝑓 (c𝐵(𝑙) ) 𝑉
𝑅
(S.6)
Rearrangement and separation of variables enables Eq. (S.6) to be integrated with respect to
time, with the appropriate limits, to give the following expression for cB(l)(t):
c𝐵(𝑙) (𝑡) = c𝐵(𝑙),𝑒𝑞 (1 − 𝑒 −𝑡/𝜏 )
(S.7)
where c𝐵(𝑙),𝑒𝑞 is the concentration of B at equilibrium and is given by:
𝑐𝐴0(𝑙)
c𝐵(𝑙),𝑒𝑞 = 1+1⁄𝐾
(S.8)
𝑅
In Eq. (S.7),  is an effective first-order time constant given by:
𝜏 = [𝑘𝑓
𝑀
1
(1 + 𝐾 )]
𝑉
−1
𝑅
(S.9)
Combining Eqs. (S.5), (S.7) and (S.8) gives the following expression for CA(l)(t):
c𝐴(𝑙) (𝑡) = c𝐴(𝑙),𝑒𝑞 − (c𝐴(𝑙),𝑒𝑞 − c𝐴0(𝑙) )𝑒 −𝑡/𝜏
(S.10)
where c𝐴(𝑙),𝑒𝑞 is the concentration of A at equilibrium. The net rate of reaction at time t can be
evaluated from the derivative of Eq. (S.7) with respect to time (according to Eq. (S.2)) and the
initial rate of reaction can be determined by evaluating this expression at t = 0.
S.4.2. Concentration profiles for parallel glucose-fructose and glucose-sorbose isomerization
The conversion of glucose (G) to fructose (F) and sorbose (S) in parallel reactions is
reflected in the following:
G
K 𝑅1
F
⇄
(S.11)
G
K 𝑅2
S
⇄
(S.12)
The full solution of the concentration profiles requires solving a system of coupled differential
equations, but for illustrative purposes we simply provide the following expressions, which
describe their functional dependence on reaction time:
c𝐹(𝑙) (𝑡) = c𝐹(𝑙),𝑒𝑞 (1 − 𝑒 −𝑡/𝜏1 )
(S.13)
c𝑆(𝑙) (𝑡) = c𝑆(𝑙),𝑒𝑞 (1 − 𝑒 −𝑡/𝜏2 )
(S.14)
The concentration of glucose is given by:
c𝐺(𝑙) (𝑡) = c𝐺0(𝑙),𝑒𝑞 − c𝐹(𝑙) (𝑡) − c𝑆(𝑙) (𝑡)
(S.15)
The glucose conversion (X) and fructose-to-sorbose ratios are given by:
𝑋(𝑡) = 1 −
c𝐺(𝑙) (𝑡)
c𝐺0(𝑙)
c𝐹(𝑙) (𝑡)
c𝐹(𝑙),𝑒𝑞 (1−𝑒 −𝑡/𝜏1 )
c𝑆(𝑙)
c𝑆(𝑙),𝑒𝑞 (1−𝑒 −𝑡/𝜏2 )
=
(𝑡)
(S.16)
(S.17)
The experimental data are modeled accurately by Eqs. (S.13) and Eqs. (S.14), evident in the
parity plots shown in Figures S.6a and S.6b of measured fructose and sorbose concentrations
(Fig. 2a and 2b of the main text) from reaction of a 1% (w/w) aqueous glucose solution with TiBeta-F (1:50 glucose:Ti molar ratio, 373 K), plotted against the predicted fructose and sorbose
concentrations using Eqs. (S.13) and Eqs. (S.14).
We note that the functional dependence of the fructose and sorbose concentrations on
reaction time given in Eqs. (S.13) and (S.14), respectively, is the same as expected for reaction
with a first-order deactivation process, evidence for which is provided for glucose isomerization
on Ti-Beta-F in liquid water in Section S.3. Effective rate constants for glucose isomerization
change as a function of batch reaction time because of the reversibility of isomerization steps and
because of first-order deactivation processes. As a result, the values of forward rate constants are
not determined from interpretation of fitted parameters in Eqs. (S.13) and (S.14). Instead, they
are estimated by extrapolation of concentration profiles to zero reaction time to determine initial
turnover rates, which are subsequently normalized by initial liquid-phase glucose concentration.
This method gives identical values, within experimental error, to those estimated from
differential batch reactors (<5% conversion) as well as from the dependence of initial turnover
rates on liquid-phase glucose concentration (Section 3.2, main text of the manuscript).
8
4
(b)
Measured sorbose concentration
(mol m -3)
Measured fructose concentration
(mol m-3)
(a)
6
4
2
0
3
2
1
0
0
2
4
6
0
8
Predicted fructose concentration
(mol m -3)
1
2
3
4
Predicted sorbose concentration
(mol m -3 )
Figure S.6. Measured (a) fructose and (b) sorbose concentrations from reaction of a 1% (w/w)
aqueous glucose solution with Ti-Beta-F (1:50 glucose:Ti molar ratio, 373 K) against values
predicted using Eqs. (S.13) and (S.14).
S.4.3. Concentration profiles for lactose isomerization to lactulose
The conversion of lactose (L) to lactulose (L’) is reflected in the following:
L
K 𝑅3
L′
⇄
(S.18)
A derivation similar to that presented in Section S.4.1 gives the following expressions for the
concentrations of lactose and lactulose:
c𝐿(𝑙) (𝑡) = c𝐿(𝑙),𝑒𝑞 − (c𝐿(𝑙),𝑒𝑞 − c𝐿0(𝑙) )𝑒 −𝑡/𝜏3
(S.19)
c𝐿′(𝑙) (𝑡) = c𝐿′(𝑙),𝑒𝑞 (1 − 𝑒 −𝑡/𝜏3 )
(S.20)
where
𝑐𝐿0(𝑙)
c𝐿′(𝑙),𝑒𝑞 = 1+1⁄𝐾
(S.21)
𝑅3
Experimentally measured disaccharide concentrations are modeled accurately by Eqs. (S.19) and
Eqs. (S.20). A parity plot is shown in Figure S.7 of measured lactulose concentrations (Fig. 6 of
the main text) from reaction of a 1% (w/w) aqueous lactose solution with Sn-Beta-F (1:20
Sn:lactose molar ratio, 373 K), plotted against the predicted lactulose concentrations using Eq.
(S.20).
Measured lactulose concenration
(mol m -3)
6
5
4
3
2
1
0
0
1
2
3
4
5
6
Predicted lactulose concentration
(mol m -3)
Figure S.7. Measured lactulose concentrations from reaction of a 1% (w/w) aqueous lactose
solution with Sn-Beta-F (1:20 glucose:Ti molar ratio, 373 K) against values predicted using Eq.
(S.20).
S.5. 13C NMR spectra of products formed from reaction of glucose-13C-C1 with TiO2-SiO2
in methanol
The
13
C NMR spectrum of the total reactor contents (without fractionation of individual
products) after reaction of glucose-13C-C1 with TiO2-SiO2 in methanol is shown in Fig. S.8. 13C
resonances appeared for the C1 position in unreacted glucose reactants ( = 95.8 and 92.0 ppm)
and for the C2 position in mannose products ( = 71.1 and 70.5 ppm), consistent with glucosemannose epimerization by an intramolecular C2-C1 shift of C3 carbon centers known as the
Bilik reaction. A resonance at  = 76.6 ppm is also present in Fig. S.8, but in trace amounts (<3%
of total
13
C content; <7% of product
but does not appear in
13
C content); this resonance currently remains unassigned,
13
C NMR spectra for hexose sugars (glucose, fructose, mannose or
sorbose) observed from glucose reactions with other solid Lewis acids.
105
100
95
90
85
80
75
70
65
60
Chemical shift (ppm)
Figure S.8. 13C NMR spectrum of the total reactor contents obtained after reaction of a 1%
(w/w) glucose-13C-C1 solution in methanol with TiO2-SiO2 at 373 K for 4 h.
S.6. Glucose isomerization rate constants on Ti-Beta-F samples of varying Si/Ti ratio
Glucose-fructose and glucose-sorbose isomerization rate constants (per total Ti; 373 K)
on Ti-Beta-F samples of varying Si/Ti ratio (66-207) in liquid water and methanol are shown in
Figs. S.9a and S.9b, respectively. Across all Ti-Beta-F samples, rate constants for glucosefructose isomerization were within factors of ~1.7 and ~1.8 in water and methanol, respectively,
and for glucose-sorbose isomerization were within factors of ~1.6 and ~1.3 in water and
methanol, respectively. Strong intrazeolitic mass transfer limitations would lead to a systematic
increase in isomerization rate constants (per total Ti) with decreasing Ti/Si ratio, as appears for
glucose-sorbose isomerization in methanol (Fig. S.9b); yet, the lack of a similar dependence on
Ti/Si ratio for parallel glucose-fructose isomerization (Fig. S.9b) suggest that intrazeolitic
glucose diffusion does not influence isomerization turnover rates, and that small rate constant
differences (within factors of ~1.3-1.8) may reflect active site heterogeneities among samples.
40
(a)
Isomerization rate constant (373 K)
(/10-6 mol [(mol Ti)*s*((mol glucose) m -3)]-1 )
Isomerization rate constant (373 K)
(/10-6 mol [(mol Ti)*s*((mol glucose) m -3)]-1)
40
30
20
10
0
0.000
0.005
0.010
0.015
Ti/Si atomic ratio
0.020
(b)
30
20
10
0
0.000
0.005
0.010
0.015
0.020
Ti/Si atomic ratio
Figure S.9. Dependence of measured first-order glucose-fructose ( ) and glucose-sorbose ( )
isomerization rate constants (373 K) on Ti/Si atomic ratio for different Ti-Beta-F samples in (a)
water and (b) methanol solvent.
S.7. Derivation of mechanism-based rate expressions for glucose isomerization and lactose
isomerization on Lewis acid zeolites
S.7.1. Glucose-fructose and glucose-sorbose isomerization in parallel via different adsorbed
glucose precursors
A reaction sequence for the formation of fructose and sorbose in parallel reactions of
glucose, mediated by different adsorbed glucose precursors, is given in Scheme S.1. In this
reaction scheme, G, F and S represent glucose, fructose and sorbose in the liquid-phase, ∗
represents a Lewis acid site, G∗’ and G∗’’ represent the different bound glucose precursors
respectively leading to fructose and sorbose, and F∗ and S∗ respectively refer to bound fructose
and sorbose. The sequential adsorption of two solvent molecules (B) onto a Lewis site forms
bound intermediates represented by B∗ and 2B∗ (Scheme S.1).
K1a
k2a
G*’
k-2a
K3a
F*
F + *
G + *
k2b
K1b
G*’’
k-2b
K3b
S*
S + *
K4
B + *
B*
K5
B + B*
2B*
Scheme S.1. Plausible reaction mechanism for glucose-fructose isomerization (Steps 1a, 2a, 3a)
and glucose-sorbose isomerization (Steps 1b, 2b, 3b) on a Lewis acid site (∗). Quasi-equilibrated
adsorption (Steps 1a, 1b) of glucose from the liquid phase (G) to active sites to form bound
precursors (G∗’ and G∗’’) that isomerize to fructose (F∗) and sorbose (S∗), respectively, in
kinetically-relevant and reversible steps (Steps 2a, 2b), followed by quasi-equilibrated desorption
of fructose and sorbose into the liquid phase (F, S). Quasi-equilibrated sequential adsorption of
two solvent (B) molecules at Lewis acid sites shown in Steps 4 and 5.
In Scheme S.1, glucose isomerization to fructose and to sorbose occur in sequences a and
b, respectively. Net rates of isomerization to fructose (risom,F) and to sorbose (risom,S) are given by:
risom,F = r2a − r−2a
(S.22)
risom,S = r2b − r−2b
(S.23)
Reaction rates for the elementary steps in Eqs. (S.22) and (S.23) are given by the law of mass
action, and are proportional to rate constants and reactant thermodynamic activities, allowing net
rates to be written as:
risom,F = k 2a aG∗′ − k −2a aF∗
(S.24)
risom,S = k 2b aG∗′′ − k −2b aS∗
(S.25)
Eqs. (S.24) and (S.25) can be rewritten as:
risom,F = k 2a aG∗′ (1 −
k−2a aF∗
k2a aG∗′
k
aS∗
risom,S = k 2b aG∗′′ (1 − k−2ba
2b G∗′′
) = k 2a aG∗′ (1 − K
aF∗
2a aG∗′
) = k 2b aG∗′′ (1 − K
)
(S.26)
)
(S.27)
aS∗
2b aG∗′′
or:
risom,F = k 2a aG∗′ (1 − η2a )
(S.28)
risom,S = k 2b aG∗′′ (1 − η2b )
(S.29)
where 2a and 2b are the approach-to-equilibrium terms for Steps 2a and 2b (Scheme S.1),
respectively.
The assumption of quasi-equilibrium on Steps 1a, 1b, 3a, 3b, 4 and 5 (Scheme S.1) give
the following equilibrium expressions relating the thermodynamic activities (ai) of reactant and
product species:
a ′
K1a = a G∗a
G ∗
(S.30)
a ′′
K1b = a G∗a
(S.31)
G ∗
K 3a =
K 3b =
aF a∗
(S.32)
aF∗
aS a∗
(S.33)
aS∗
a
K 4 = a B∗a
(S.34)
B ∗
a
K 5 = a 2B∗
a
(S.35)
B B∗
Combining Eqs. (S.28)-(S.31) allows the isomerization rates to be expressed as:
risom,F = k 2a K1a aG a∗ (1 − η2a )
(S.36)
risom,S = k 2b K1b aG a∗ (1 − η2b )
(S.37)
The activities of each species appearing in Eqs. (S.36) and (S.37) can be written as the product of
their activity coefficients (i) and concentrations (ci):
risom,F = k 2a K1a γG γ∗ cG c∗ (1 − η2a )
(S.38)
repim,S = k 2b K1b γG γ∗ cG c∗ (1 − η2b )
(S.39)
The concentration of total Lewis acid sites (c∗,tot ) is related to the concentration of unoccupied
sites (c∗ ) and those of sites containing the bound adsorbates in Scheme S.1 according to the
following site balance:
c∗,tot = c∗ + cG∗′ + cG∗′′ + cF∗ + cS∗ + cB∗ + c2B∗
(S.40)
Combining Eqs. (S.30)-(S.35) and (S.40) allows the Lewis acid site balance to be written as:
c∗,tot = c∗ +
K1a aG γ∗ c∗
𝛾G∗′
+
K1b aG γ∗ c∗
𝛾G∗′′
a γ∗ c ∗
+ KF
3a γF∗
a γ∗ c ∗
+ KS
3b γS∗
+
K4 aB γ∗ c∗
𝛾B∗
+
K4 K5 aB 2 γ∗ c∗
𝛾2B∗
(S.41)
Factoring out the c∗ term in the right-hand side of Eq. (S.41) leads to the following equation
c∗,tot = c∗ (1 +
K1a aG γ∗
𝛾G∗′
+
K1b aG γ∗
𝛾G∗′′
aF γ∗
+K
3a γF∗
aS γ∗
+K
3b γS∗
+
K4 aB γ∗
𝛾B∗
+
K 4 K 5 aB 2 γ ∗
𝛾2B∗
)
(S.42)
in which the argument in parentheses represents the fractional coverage (i) of each bound
intermediate:
c∗,tot = c∗ (𝜃𝑖 + 𝜃G∗′ + 𝜃G∗′′ + 𝜃F∗ + 𝜃S∗ + 𝜃B∗ + 𝜃2B∗ )
(S.43)
and is located in the denominator of the isomerization rate expressions before assumption of the
most abundant surface intermediate.
The assumption that the Lewis acid site with two bound solvent molecules is the most
abundant surface intermediate allows Eq. (S.41) to be reduced to:
c∗,tot =
K4 K5 γB 2 cB 2 γ∗
𝛾2B∗
c∗
(S.44)
Substitution of Eq. (S.44) into Eqs. (S.38) and (S.39) give expressions for the isomerization
turnover rates (per total Lewis site):
risom,F
c∗,tot
risom,S
c∗,tot
= (K
= (K
𝛾2B∗
4 K5 γB
) k 2a K1a γG γ∗ cG (1 − η2a )
(S.45)
) k 2b K1b γG γ∗ cG (1 − η2b )
(S.46)
2c 2γ
B ∗
𝛾2B∗
2 2
4 K5 γB c B γ∗
The assumption that only a fraction of all Lewis sites (xisom,F) are able to mediate glucosefructose isomerization or bind glucose in the configuration required for isomerization to fructose
(G∗’), and that a separate fraction (xisom,S) are able to mediate glucose-sorbose isomerization or
bind glucose in the configuration required for isomerization to sorbose (G∗’’), is reflected in the
following relations:
c∗,isom,F = xisom,F c∗,tot
(S.47)
c∗,isom,S = xisom,S c∗,tot
(S.48)
The rate equations given in Eqs. (S.45) and (S.46) are rigorously correct only if, in the left-hand
side of the equations, isomerization rates (risom,F, risom,S) are normalized by the number of sites
able to catalyze their respective reactions. Thus, isomerization rates per total Lewis sites are
rigorously expressed only after incorporation of the relations in Eqs. (S.47) and Eqs. (S.48),
respectively:
risom,F
c∗,tot
risom,S
c∗,tot
= xisom,F (K
= xisom,S (K
𝛾2B∗
4 K5 γB
𝛾2B∗
4 K5 γB
) k 2a K1a γG γ∗ cG (1 − η2a )
(S.49)
) k 2b K1b γG γ∗ cG (1 − η2b )
(S.50)
2 c 2γ
B ∗
2c 2γ
B ∗
Eqs. (S.49) and (S.50) can be rearranged to give:
risom,F
c∗,tot
risom,S
c∗,tot
= (γ
= (γ
γG
B
2c 2
B
γG
2 2
B cB
)(
)(
𝛾2B∗ K1a
) xisom,F k 2a cG (1 − η2a )
(S.51)
) xisom,S k 2b cG (1 − η2b )
(S.52)
K4 K5
𝛾2B∗ K1b
K4 K5
or:
risom,F
c∗,tot
risom,S
c∗,tot
= k isom,F cG (1 − η2a )
(S.53)
= k isom,S cG (1 − η2b )
(S.54)
where kisom,S and kisom,S are measured first-order glucose-fructose and glucose-sorbose
isomerization rate constants given by:
k isom,F = (γ
k isom,S = (γ
γG
K1a k2a
2 2
B cB
γG
B
2c 2
B
)(
K4 K5
) 𝛾2B∗ xisom,F
K1b k2b
)(
K4 K5
) 𝛾2B∗ xisom,S
(S.55)
(S.56)
The rate and equilibrium constants that appear in Eqs. (S.55) and (S.56), expressed in terms of
free energy differences between transition states, reactants and products for the steps in Scheme
S.1:
⁄𝑅𝑇 )
K1a = e(−(∆𝐺°G∗′ −∆𝐺°G −∆𝐺°∗)
K1b = e(−(∆𝐺°G∗′′ −∆𝐺°G −∆𝐺°∗)
k 2a =
⁄𝑅𝑇)
𝑘𝐵 𝑇 (−(∆𝐺°‡,2a −∆𝐺° ′ )⁄𝑅𝑇 )
G∗
e
ℎ
(S.57)
(S.58)
(S.59)
k 2b =
𝑘𝐵 𝑇 (−(∆𝐺°‡,2b −∆𝐺° ′′ )⁄𝑅𝑇)
G∗
e
ℎ
(S.60)
K 4 = e(−(∆𝐺°B∗−∆𝐺°B −∆𝐺°∗)⁄𝑅𝑇)
(S.61)
K 5 = e(−(∆𝐺°2B∗−∆𝐺°B −∆𝐺°B∗)⁄𝑅𝑇)
(S.62)
Combining Eqs. (S.55), (S.57), (S.59), (S.61) and (S.62) give the following expression for the
product of rate and equilibrium constants that appear in the measured glucose-fructose
isomerization rate constant (Eq. (S.55)):
K1a k2a
K 4 K5
=
𝑘𝐵 𝑇 (−((∆𝐺°‡,2a +2∆𝐺°B )−(∆𝐺°G +∆𝐺°2B∗ ))⁄𝑅𝑇 )
e
ℎ
(S.63)
Thus, measured glucose-fructose isomerization rate constants depend on the free energy of
between one bound isomerization transition state and two solvent molecules in the liquid phase,
relative to two bound solvent molecules and one glucose molecule in the liquid phase (Scheme 2,
main text), according to the following relation:
G + 2𝐵 ∗ ⇄ ‡𝑖𝑠𝑜𝑚,𝐹 ∗ + 2𝐵
(S.64)
with an effective equilibrium constant given by K1ak2aK4-1K5-1.
Combining Eqs. (S.56), (S.58), (S.60), (S.61) and (S.62) give the following expression
for the product of rate and equilibrium constants that appear in the measured glucose-sorbose
isomerization rate constant (Eq. (S.56)):
K1b k2b
K 4 K5
=
𝑘𝐵 𝑇 (−((∆𝐺°‡,2b +2∆𝐺°B )−(∆𝐺°G +∆𝐺°2B∗ ))⁄𝑅𝑇 )
e
ℎ
(S.65)
Thus, measured glucose-sorbose isomerization rate constants depend on the free energy of
between one bound isomerization transition state and two solvent molecules in the liquid phase,
relative to two bound solvent molecules and one glucose molecule in the liquid phase, according
to the following relation:
G + 2𝐵 ∗ ⇄ ‡𝑖𝑠𝑜𝑚,𝑆 ∗ + 2𝐵
(S.66)
with an effective equilibrium constant given by K1bk2bK4-1K5-1.
Separation of the zeolite-dependent and zeolite-independent terms in Eqs. (S.63) and
(S.65) gives:
K1a k2a
K4 K5
K1b k2b
K4 K5
𝑘𝐵 𝑇 (−(2∆𝐺° −∆𝐺° )⁄𝑅𝑇 )
B
G
e
) (e(−(∆𝐺°‡,2a −∆𝐺°2B∗)⁄𝑅𝑇) )
ℎ
(S.67)
𝑘𝐵 𝑇 (−(2∆𝐺° −∆𝐺° )⁄𝑅𝑇 )
B
G
e
) (e(−(∆𝐺°‡,2b −∆𝐺°2B∗)⁄𝑅𝑇) )
ℎ
(S.68)
=(
=(
S.7.2. Lactose isomerization to lactulose on Lewis acid sites
A plausible reaction mechanism for the isomerization of lactose (L) to lactulose (L’),
which involves the isomerization of a glucose moiety within a galactose-glucose sugar dimer, on
Lewis acid centers (*) coordinated tetrahedrally within pure-silica zeolite frameworks (e.g., SnBeta, Ti-Beta) is shown in Scheme S.2. This mechanism, based on that for glucose isomerization
(Scheme S.1), involves the adsorption of lactose from the liquid phase (L(l)) onto Lewis acid
centers (L*) (Step 1), subsequent isomerization to form lactulose bound to Lewis acid centers
(L’*) (Step 2), and desorption into the aqueous phase (L’(l)) (Step 3) to complete the catalytic
cycle (Scheme 5). Scheme 5 also includes steps for the sequential adsorption of two water (B)
molecules at Lewis acid centers (Steps 4 and 5). The assumption of quasi-equilibrated lactose
and lactulose adsorption (Steps 1 and 3) and water adsorption (Steps 4 and 5), together with two
bound water molecules (2B*) as the MASI, by extension of the assumptions made for glucose
isomerization, can be used to derive an expression for the lactose isomerization turnover rate (per
LA) with the following functional form (analogous to the derivation for glucose isomerization
presented in Section S.7.1):
risom
c∗,tot
= k isom cL(l) (1 − η2 )
(S.69)
K1
L(l) + *
L*
k2
L’*
L*
k-2
K3
L’*
L’(l) + *
K4
B(l) + *
B*
K5
B(l) + B*
2B*
Scheme S.2. Plausible reaction mechanism for lactose isomerization (Steps 1-3) on a Lewis acid
site (*), based on the mechanism of glucose isomerization (Scheme S.1). Quasi-equilibrated
adsorption (Step 1) of lactose from the liquid phase (L(l)) to active sites to form bound precursors
(L*) that isomerize to lactulose (L’*) in the kinetically-relevant and reversible step (Step 2),
followed by quasi-equilibrated desorption of lactulose into the liquid phase (L’(l)). Quasiequilibrated sequential adsorption of two solvent (B(l)) molecules at Lewis acid sites shown in
Steps 4 and 5.