the bimolecular binding event

8/11/2019 The bimolecular binding event

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H + G H G

H S (H G) SG S+

in reality

K

[HG]

[H][G]Ka = [HG]

[H][G]

Kd =(M-1

) (M)

SS+

explicit solvent is not used because G for the association constant

reflects the stability of solvated H and G relative to solvated HG and released solvent

G= -RTln(Ka)assuming activity = concentra\ tion


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Kd (1/Ka) is a convenient reference point for estimating the amount of complex

Ka= 100 M-1

Kd= 1x10-2M = 10 mM

course

0,1 0,3 0,5 0,7 0,9Total concentration of A

0

20

40

60

80

100

%f

ormationrelativetoA

[H]0= [G]0= 20 mM [H] = [G] = 10 mM and [HG] = 10 mM

At Kd, [H] = [G] = [HG] = Kd (!! only in case [H]0= [G]0!!)


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G= -RTln(Ka)

Ka(M-1) G0(kJ.mol-1)

1 0

10 -5.8

100 -11.5

1000 -17.3

10000 -23.0

100000 -28.8

at T = 301 K (R = 8.314 J.mol-1.K-1)

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6

-G(kJmol-1)

log Ka


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H + G H GKa

how to determine the concentration of each species ?

if H, G, and HG are all known, then K is easily calculated

[HG]

[H][G]Ka =

regrettable this is hardly ever the case
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determination of the binding constant Ka

almost all experimental methods to measure binding constants rely on the analysis

of a binding isotherm.

A binding isotherm is the theoretical change in the concentration of one component

as a function of the concentration of another component at constant temperature.

The concentrations are experimentally determined (e.g. NMR, UV/vis, etc) and fitted to

the theoretical binding isotherm.

0,1

0 0,2 0,4 0,6 0,8 1,0Total concentration of A

0

20

40

60

80

100

%f

ormationrelativetoB Ka= 100 M

-1

[H]0

= 0.01Mexample


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Titration experiments

Typically a titration is performed holding the concentration of one species (H) constant

while varying the concentration of the other (G).

During the course of this titration, the physical changes in the system are monitored,usually spectroscopically, and this change is then plotted as a function of guest added

to host (equivalent guests).

The mathematical model used to obtain the association constant is usually developed

from realising that the physical change (Y, e.g., a NMR shift or a change in UV-Visabsorbance) observed is correlated to the concentration of the complex [HG] as Y =

f[HG], or in some cases, the free hostf[H] or the free guestf[G].

The physical change (Y) being monitored can usually be described as the aggregate of the

individual components according to eqn (7) as a function of concentration (e.g., for UV-Visspectroscopy) or eqn (8) as a function of mole fractions fX(fXdefined as: fX= [X]/[X]0) in

the special case of NMR.

Y = YH[H] + YG[G] + YHG[HG]

Y = YHfH+ YGfG+ YHGfHG


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[H] = [H]0 [HG]

[G] = [G]0 [HG]

[HG]

([H]0 [HG])([G]0 [HG])Ka =

[HG]

[H][G]K

a

=

Example

only HG has a particular absorption band in the UV/vis spectrum.

Thus, A = HG[HG]


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[HG]

([H]0 [HG])([G]0 [HG])Ka =

[HG]

[H]0[G]0 [HG][G]0- [HG][H]0+[HG]2

Ka =

Ka([H]0[G]0 [HG][G]0- [HG][H]0+[HG]2

) = [HG]

Ka[HG]2(Ka[G]0-Ka[H]0+1)[HG]+Ka[H]0[G]0= 0

thus

this expresses [HG] as a function of Ka, which is the only unknown !!


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Example, only HG has a particular absorption band in the UV/vis spectrum.

Since, A = HG[HG]

and

Then A =f(HG

, Ka

)

The power of this equation should not be understated as we can now start to developsolutions that require only the knowledge of the total (or initial) concentrations of the

host and guest ([H]0and [G]0) in addition to the association constant (Ka) and the physical

properties (Y) that are changing (Y) during the course of the titration.
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Y = YHfH+ YGfG+ YHGfHG

If we further assume that one of the components is silent e.g., a non-absorbing freeguest [G], we can simplify the equation to

which, since fHG = [HG]/[H]0 and fH= 1- fHG, can be further simplified to

Y=YH+([HG]/[H]0)(YHG- YH)

and, finally

Y = YHfH+ YHGfHG

in which Y = Y-YHand YHG= YHG-YH


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(Old-fashioned) Shortcuts to the binding constant

Older references and textbooks are full of examples on how some of the above

expressions and equations can be simplified or transformed to linear equations(y = a + bx) which could then be plotted by hand to obtain the Kaand other parameters

of interest by inspection of the slope and intercepts.

BenesiHildebrand plot

(determination of binding constants based on absorbance)

assumingthat [G]0>> [H]0(and thus AH> [H]0one can assume that [G]=[G]0and this

with = HG- G


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[H] = [H]0 [HG]

[HG][H][G]

Ka = [HG]

[G]([H]0-[HG])Ka = and Ka([G]([H]0-[HG]))-[HG]=0

which is

Ka[G][H]0-Ka[G][HG]-[HG]=0

or

The binding isotherm can be rewritten as

Together this gives

and finally

(assuming that [G] = [G]0)


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intercept

slope1

b[H]0Ka

Other examples include Lineweaver-Burke plots, Scatchard plots, etc.
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There are two key problems associated with using these linear transformations that

make their use highly questionable:

(i) they violate some of the fundamental assumption of linear regression by

distorting the experimental error

(ii) they frequently involve assumptions and shortcuts (such as assuming that [G]0

>> [H]0 or that YHG= Y at the end of titration (i.e., the complex is fully formed atthe end of titration - which would then help to give YHG). These assumptions

are often not valid and distort the results.

The non-linear regression approach with exact solutions of the quadratic equation

(see before) produces the most accurate results. This approach is not difficult with

modern computer technology and there is no real excuse for using old-fashion lineartransformations anymore!

Limitations


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Scientist


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0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02

Example: NMR titration

[G0]

o

bs

(ppm)


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6.8

7

7.2

7.4

7.6

7.8

8

8.2

0 0.005 0.01 0.015 0.02

Serie1

Serie2

K 1938.702 78

DH 7.00624 0.004DHG 8.000867 6.00E-03

M-1

ppmppm

[G0]

obs

(ppm)


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Experimental conditions

When a supramolecular titration study is carried out one has to first make a decisionon what technique is going to be used to follow the physical changes (Y) in the

system during the course of experiment. The two key concerns here should be:

i. The expected association constant(s).

ii. The expected physical changes (Y) upon association.

The expected association constant determines what concentration should be chosen for the

host system which in turn will have an influence on the choice of technique.

Wilcox, using a parameter defined as probability of binding (p), showed that it is vital to

collect as many data point as possible within the range: 0.2


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[HG]

([H]0 [HG])([G]0 [HG])Ka =

gives

Ka =p

[G]0 ([H]0+ [G]0)p+[H]0p2

p

for [H]0=[G]0=0.001 M


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nearp =0 and p = 1, small errors inp (experimental error in determination of the

concentrations !!= results in large variations in Ka.

The best results are obtained in the region 0.2


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EXAMPLE

p

for [H]0=0.001 M

Ka= 2000 M-1

0.00

0.20

0.40

0.60

0.80

1.00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

log [G]0


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0.00

0.20

0.40

0.60

0.80

1.00

0.00E+00 2.00E-03 4.00E-03 6.00E-03 8.00E-03 1.00E-02

[G]0

for [H]0=0.001 M

Ka= 2000 M-1

p


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Using

it is possible to calculatepfor a range of [H]0, [G]0and Kavalues.When the results are plotted for a fixed [H]0concentration (here 10

-5M) as a

function of Kaand [G]0/[H]0 (equivalents of guest added) typically employed in

UV-Vis spectroscopy studies a revealing pattern appears with the shaded areas

indicatingpin the range of 0.20.8 (note that this applies only to 1 : 1 binding

systems).


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if Kd> [H]0(hence Kafairly low) then a relatively large excess of [G]0is required to obtain

goodp-values. In this situation it would be advisable to collect several data points in the

range of 150 equivalents of G added.

If Kd> [H]0(hence Kafairly high) the only data points with goodp-values are within the

range of [G]0o [H]0. In other words, it is essential to obtain as many points as possible

between 01 equivalents of G added.

If Kd [H]0, goodp-values are obtained almost anywhere within the range of 0 to >10

equivalent of G added. Note that when Kd= [H]0= [G]0, then p = 0.38.


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The fourth scenario to consider is when Kd1000 , it is clear that there is very little

information content in the isotherms.

When binding occurs under saturation

conditions, this implies that the experimental

conditions are NOT adequate for

determination of K.


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NMR concentrations UV-Vis concentrations

fluorescence concentrations

The suitable analytical technique fordetermination of Kadepends on its

value.

limit: [H]0/Kd> 100


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analytical techniques: scope and limitations


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The most informative technique in most situations is 1H NMR. Other forms (13C, 19F etc.)

of NMR are also applicable. Apart from the quantitative information that an

NMR titration can yield, the relative shifts and changes in symmetry can often give

valuable information about how the host and guest(s) are interacting and thestoichiometry of interaction. This information can be of significant benefit even

in situations where complete quantitative data cannot be obtained from the NMR

titration.

Classical approaches for data analysis of NMR titrations assume that the resonance () of

interest is the weighted average of the free host (H) and the bound host in the complex(HG) in the experiment for a simple 1 : 1 system

obs= HH+ HGHG

since H = 1 - HG

obs- H= HG(HG HG)

NMR spectroscopy

since

this gives


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NMR spectroscopy

With modern NMR instruments it is possible to obtain good quality spectra with sub-

millimolar concentrations (routinely now as low as 10-4M), suggesting that NMR is

suitable for Kaup to and even above 106M-1. Many literature references will statethat 105M-1is the limit for NMR titration experiments.

With NMR, one has also to take into account the relative exchange rates within the

hostguest the relationship between the equilibrium association constant and the

kinetics on/off rates (Ka=k1/k-1) and the timescale of the NMR experiment. The reallimiting factor for NMR titrations is therefore whether the system of interest is in the

fast or slow exchange region under the conditions used.

It may be tempting to think that in the (very) slow exchange region of NMR, one could

obtain an association constant directly from the relative ratios of the free and bound

host, however, can be difficult in practice due to complications that arise in theintermediate-to-slow region with the size (amplitude) of the observed resonances and

the usual limitation of obtaining accurate (quantitative) integration from NMR

experiments.


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UV-Vis spectroscopy

The second most common method for supramolecular titration experiment is probably

UV-Vis spectroscopy. With the right chromophore, host concentration in the sub-

micromolar (10-7

M) can be applied, making the determination of association constantsas high as 109M-1in simple 1 : 1 systems possible (albeit difficult) with Kd/[H]0= 100 as

discussed above.

advantages

rapid. absorption and related phenomenon (fluorescence) occur on ps time scale or

faster in all cases faster than complex dissociation rate slow exchange

straightforward to correlate signal intensity to concentration (linear regime;

Lambert-Beer)

sensitive

disadvantage titration by UV-Vis spectroscopy is particularly vulnerable to dilution and temperature

effects (all supramolecular titration experiments need some temperature control) and

the presence of impurities in either host or guest solutions.

requires chromophoric hosts or guests


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fluorescence spectroscopy

The phenomenal sensitivity of this technique makes routine measurements in the sub-micromolar,

even nanomolar (nM) range possible and hence, fluorescence spectroscopy is ideal for the

determination of very large association constants (Ka> 1010M-1).

Fluorescence is a particularly useful technique in the case when only one of the species in solution

is fluorescently active, i.e. when either the free host or guest is fluorescent silent or inactive and

the fluorescence of the remaining species is either turned off (quenched) or on upon

complexation.

If quenching plays a role, it is necessary to differentiate between static and dynamic (collisional)quenching, with only the former of real significance for supramolecular binding studies.

Dynamic quenching is usually measured by plotting the ratio of the initial (F0) and measured (F)

fluorescence intensity ratio (F0/F) against the concentration of the quencher [Q] according to the

SternVolmer relation F0/F = 1 + KSV[Q], with KSV= the SternVolmer constant.

Unfortunately, pure 1 : 1 static quenching follows a nearly identical relation: F0/F = 1 + Ka[Q], with [Q]

= the free concentration of the quencher (guest) and Kais the association constant of interest in

supramolecular binding studies.

In many cases the observed quenching is a mixture of both static and dynamic quenching which can

lead to some complication in the analysis of the titration data.


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Swager et al.J. Am. Chem. Soc., 1995, 7017-7018

control

Stern-Volmer plots (F0/F=1+K[PQ2+] )

Small molecule sensing

2

Mean molecular weight (PDI)

6: 31 100 (1.6)

7: 65 400 (1.6)

8: 122 500 (1.8)
http://upload.wikimedia.org/wikipedia/commons/3/3d/Paraquat.png


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+

+

++

+

++

+

+

++

+

+

++

+

- --2

0

250

500

750

1000

0.0 1.0 2.0 3.0 4.0 5.0

FI(a.u.)

[ATPF] (M)

ATPF(ex= 305 nm, em= 370 nm)

2.5 M

~ 18

[TACN Zn(II)] = 10 M

pH = 7.0, T = 25 C.

[TACN Zn(II)] = 10 M

pH = 7.0, T = 25 C.


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G= -RTln(Ka)

From K to G and H and S (Van t Hoff analysis)

G= H- TSln(Ka) = -H/RT+ S/R

slope = -H/R

intercept = + S/R

problem: often small temperature interval possible + inversed temperature
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Isothermal calorimetry (ITC)

Q = VH[HG]

or

dq= VH [HG]

fitting provides K and H
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Determination of stoichiometry

The other aim of a supramolecular titration experiment is the determination of the

stoichiometry of the system.

Methods

(i) The method of continuous variations (Jobs method).

(ii) Consistency with the host structure and available information on the host guest complex structure.

(iii) Specific experimental evidence such as isosbestic point(s).

(iv) Constancy of stability concentration as the concentration is varied, that is, the

success of a stoichiometric model to account for the data.


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Jobs method

The idea behind it is simple; the concentration of a HmGn([HmGn]) complex is at

maximum when the [H]/[G] ratio is equal to m/n.

To do this, the mole fraction (fG) of the guest is varied while keeping the total

concentration of the host and guest constant ([H]0+[G]0=constant). The concentration of

the hostguest complex [HmGn] is then plotted against the mole fractionfGyielding a

curve with a maxima atfG= n/(m + n), which in the case of m = n (e.g., 1 : 1) appears at fG

= 0.5
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When there is more than one complex present, the Jobs method becomes

unreliable. This includes many situations with m/n = 1 : 2 or 2 : 1 as these usuallyinclude two forms of complexes (e.g., HG and HG2) that have different physical

properties, hence the assumption that the physical property of interest (e.g., obs)

is linearly dependent may not be valid. For similar reasons, the Jobs method is

likely to fail when either the host or guest aggregates in solution.

Limitations


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This method is perhaps the simplest but often the most effective of all the approachesavailable to determine the stoichiometry in hostguest complexes. In modern

supramolecular chemistry it is now rare not to have detailed information through X-ray

crystallography, 2D-NMR and Molecular Modelling about the structure of the host and

guest and, in some cases, even the hostguest complex itself. This structural information

can make the prediction of stoichiometry quite straightforward and accurate.

(ii) Consistency with the host structure and available information on the

hostguest complex structure.


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This relies on specific evidence such as isosbestic points which can be used to confirm

that more than one type of complex is present and hence that simple 1 : 1 complexationis not appropriate to describe the system if more than one isosbestic point is observed.

The converse is not necessarily true, i.e. the absence of more than one isosbestic point

cannot be used to rule out more complex stoichiometry such as 1 : 2 complex formation,

especially in cases where the cooperative (positive or negative) processes play a

significant role.

(iii) Specific experimental evidence such as isosbestic point(s).

(i ) C t f t bilit t ti th t ti i i d th t i th
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(iv) Constancy of stability concentration as the concentration is varied, that is, the

success of a stoichiometric model to account for the data.

This method is probably the most generally applicable method for determining

stoichiometry.

Firstly, if anything other than 1 : 1 stoichiometry is suspected, the data should be fitted

to other plausible models (e.g., 1 : 2) and the quality of fit of the different models

compared in details, taking into account factors such as the increase in parameters in

the fitting process.

Secondly, and more importantly, it is strongly advisable to carry out the titration at

different concentrations and even with different techniques (e.g., NMR and UV-Vis). If a

particular model is successful at explaining the data at different concentrations then it

can be taken as very strong evidence for that model.

8.2


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6.8

7

7.2

7.4

7.6

7.8

8

0 0.005 0.01 0.015 0.02

Serie1

Serie2

[G0]

obs

(ppm)

1:1 model

K 1938.702 78

DH 7.00624 0.004

DHG 8.000867 6.00E-03

M-1ppmppm

6.8

7

7.2

7.4

7.6

7.8

8

8.2

0 0.005 0.01 0.015 0.02

Serie1

Serie2

1:2 model

[G0]

obs(p

pm)

K 2615630 5.00E+05

DH 6.98578 0.02

DHG 8.169739 0.04

M-1ppmppm

[G0]

obs

(ppm)

2:1 modelHHG

K 3566244 1.20E+06

DH 7.05578 0.02

DHG 7.92317 0.03

M-1ppmppm

6.8

7

7.2

7.4

7.6

7.8

8

8.2

0 0.005 0.01 0.015 0.02

S

S

HGG


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48/48

Setting up a titration experiment

weigh H stocksolution H

weigh G

(concentrated solution)

use to fill

stocksolution G

(contains H at the same concentration !)

1) measure initial spectrum

2) add aliquots of G3) measure after each addition
http://www.google.it/url?sa=i&rct=j&q=cuvette&source=images&cd=&cad=rja&docid=cBCZfR4BN3TBGM&tbnid=6swS0mw_4Nx1_M:&ved=0CAUQjRw&url=http://www.precisioncells.com/products/Spectrophotometer-Cuvettes/Tandem-Divided-Cuvettes/106/106/Precision-Cells-Type-25A-Cuvette-Lightpath-10mm&ei=-INIUeTgGsGwO5OrgbgF&bvm=bv.43828540,d.ZGU&psig=AFQjCNFPBnJU7FAnMOaN5JSOOy49BgPsYQ&ust=1363793257542606http://www.google.it/url?sa=i&rct=j&q=syringe&source=images&cd=&cad=rja&docid=GRf47xt7UMqrSM&tbnid=K2xxIUmryFPatM:&ved=0CAUQjRw&url=http://www.oocities.org/medical_supplies_equipment/needles-syringes.html&ei=R4RIUcz9HMurPI3LgfAB&bvm=bv.43828540,d.ZGU&psig=AFQjCNHvPJ9r_Z039E41RNnfIoClDG9z0g&ust=1363793347475099http://www.google.it/url?sa=i&rct=j&q=cuvette&source=images&cd=&cad=rja&docid=cBCZfR4BN3TBGM&tbnid=6swS0mw_4Nx1_M:&ved=0CAUQjRw&url=http://www.precisioncells.com/products/Spectrophotometer-Cuvettes/Tandem-Divided-Cuvettes/106/106/Precision-Cells-Type-25A-Cuvette-Lightpath-10mm&ei=-INIUeTgGsGwO5OrgbgF&bvm=bv.43828540,d.ZGU&psig=AFQjCNFPBnJU7FAnMOaN5JSOOy49BgPsYQ&ust=1363793257542606http://www.google.it/url?sa=i&rct=j&q=volumetric+flask&source=images&cd=&cad=rja&docid=x8KBhsJmSGDNQM&tbnid=PzARIkUSmRrzGM:&ved=0CAUQjRw&url=http://www.ctechglass.com/glassware-cylinders-c-46_22.html&ei=EYFIUd3REcSxPLfkgeAK&bvm=bv.43828540,d.ZGU&psig=AFQjCNEdhWtcVyQMtdFSQ-B0m9_5sVN6LQ&ust=1363792501966910http://www.google.it/url?sa=i&rct=j&q=volumetric+flask&source=images&cd=&cad=rja&docid=x8KBhsJmSGDNQM&tbnid=PzARIkUSmRrzGM:&ved=0CAUQjRw&url=http://www.ctechglass.com/glassware-cylinders-c-46_22.html&ei=EYFIUd3REcSxPLfkgeAK&bvm=bv.43828540,d.ZGU&psig=AFQjCNEdhWtcVyQMtdFSQ-B0m9_5sVN6LQ&ust=1363792501966910http://www.google.it/url?sa=i&rct=j&q=volumetric+flask&source=images&cd=&cad=rja&docid=x8KBhsJmSGDNQM&tbnid=PzARIkUSmRrzGM:&ved=0CAUQjRw&url=http://www.ctechglass.com/glassware-cylinders-c-46_22.html&ei=EYFIUd3REcSxPLfkgeAK&bvm=bv.43828540,d.ZGU&psig=AFQjCNEdhWtcVyQMtdFSQ-B0m9_5sVN6LQ&ust=1363792501966910http://www.google.it/url?sa=i&rct=j&q=volumetric+flask&source=images&cd=&cad=rja&docid=x8KBhsJmSGDNQM&tbnid=PzARIkUSmRrzGM:&ved=0CAUQjRw&url=http://www.ctechglass.com/glassware-cylinders-c-46_22.html&ei=EYFIUd3REcSxPLfkgeAK&bvm=bv.43828540,d.ZGU&psig=AFQjCNEdhWtcVyQMtdFSQ-B0m9_5sVN6LQ&ust=1363792501966910

the bimolecular binding event

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