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=Host-Guest Chemistry=

Introduction
In supramolecular chemistry, host-guest chemistry describes complexes that are composed of two or more molecules or ions held together in unique structural relationships by hydrogen bonding, ion pairing and/or by Van der Waals forces; forces other than those of full covalent bonds. Chemists have developed the concept of Host-guest chemistry when trying to measure the energy of these non-covalent interactions found throughout supramolecular chemistry.

The host component is defined as an organic molecule or ion whose binding sites converge in the complex and the guest component is defined as any molecule or ion whose binding sites diverge in the complex.
 * $$H + G \rightleftharpoons\ HG$$

H ="host", G ="guest" , HG ="host-guest complex"

Host-guest chemistry encompasses the idea of molecular recognition and interactions through non-covalent bonding. The thermodynamic benefits of host guest chemistry are derived from the idea that there is a lower overall Gibbs free energy due to the interaction between host and guest molecules. On the basis of host-guest chemistry, a new field of science designated supramolecular chemistry was born and has become the next great frontier of modern science. During the 1990’s, progress in supramolecular chemistry led to the fascinating new field of macromolecular architecture, which aims to mimic the host-guest interactions within nature. ) These interactions are being studied by various techniques such as NMR spectroscopy, Raman spectroscopy, isothermal titration caliorimetry, and UV-Vis Spectroscopy. Their are also many biological applications of Host-Guest chemistry (see below). Physical organic chemists are trying to mimic binding events of real biological systems.  They are focusing on the energy exchange of different binding interactions, the fundamental origins of these interactions, and trying to develop scientific experiments to quantify and understand the fundamental origins of these interactions.  The experimental data is quantified and explained through analysis of binding constants Ka, Gibbs free energy ΔGo, Enthalpy ΔHo, and entropy ΔSo.

Association and Dissociation constants

 * $$H + G \rightleftharpoons\ HG$$



$$K_a = \frac{[HG]}{[H][G]}$$

The association constant, K_a, is equal to the concentration of the Host-Guest complex divided by the product of the concentrations of the individual Host and Guest molecules

The equilibrium that has been established between the Host-Guest complex and free molecules can also be defined by a dissociation constant, Kod.
 * $$K_d = \frac{[H][G]}{[HG]} = \frac{1}{K_a}$$

A Large Koa value can be equated to a small Kod, both values indicate a strong complexation between host and guest molecules to form the host-guest complex.

Units of Ka and Kd
The units of the equilibrium constants varies depending on the the concentrations of species it is equal to. The concentrations of the species in the equations are molarity, M. Like terms cancel to provide the units of the equilibrium constant specific for that host-guest interaction.

$$K_a = \frac{[HG]}{[H][G]}$$, $$K_a = \frac{1}{M}$$

$$K_d = \frac{[H][G]}{[HG]}$$, $$K_d = M$$

$$M = Molar$$

Gibbs Free Energy dependency on K
The Change in Gibbs Free Energy, ΔG, is a function of the equilibrium constant, Ka

$$\Delta G = RTlnK_a$$

One can see that by knowing the association constant, you can solve for the Gibbs free energy of the reaction.


 * $$\Delta G = \Delta H - T \Delta S \,$$,

We can see the dependency of Gibbs Free energy on the marcroscopic terms of Enthalpy, ΔH and Entropy, ΔS.

Experimental Determination of [HG] using Steady-State Kinetics
In order to determine the concentration of the HG complex, one needs to write the equilibrium equation in terms that are able to be measured in the laboratory. Experimentally, one can determine [HG] as a function of the initial concentration of the guest molecules, [G]o.



Their occurs a problem because our association constant, which one wants to relate to the Gibbs Free Energy of the system, is a function of of the concentration of host, [H]o and guest,[G]eq, molecules at equilibrium. In order to overcome this obstacle, one can make the steady-state assumption that says that the initial concentration of host molecules is equivalent to the concentration of both $$[HG]_ eq.$$ and $$[H]_eq.$$

$$[H]_o = [HG]_eq. + [H]_eq.$$

Solving for [H] yields:

$$[H] = [H]_o - [HG]$$

This can be substituted into the original equilibrium equation

$$K_a = \frac{[HG]}{[G]([H]_o - [HG])}$$

In solving for [HG], one can determine that:

$$[HG] = \frac{K_a[G][H]_o}{1 + K_a[G]}$$

Another assumption that is made in the determination of [HG] is that by setting the initial concentration of the Guest molecules, [G]o, much greater than the initial concentration of Host molecules, [H]o, then one can say that:

because: $$[G]_o>>>[H]_o$$ then:

$$[G]_o = [G]$$

With this final substitution, one can solve for the concentration of the Host-Guest complex [HG] as a function of the initial concentrations of the Host and Guest Molecules:

$$[HG] = \frac{K_a[G]_o[H]_o}{1 + K_a[G]_o}$$

The final equation follows the form of the graph seen above. At low concentrations of [G], (1>>>[G]o) the plot become linear as a function of [G]o.

$$[HG] = \frac{K_a[G]_o[H]_o}{1}$$

At high concentrations of [G]o, ([G]o>>>1) then [HG] becomes independent of [G]o and the plot becomes linear (as seen at the top region of the plot)

This saturation curve is one type of experimental data that can be analyzed and quantified by various spectroscopic imaging techniques.