Lecture 9: Chemical Potential and Chemical Reactions



Lecture 9: Chemical Potential and Chemical Reactions

Review

Non-bonded Interactions

• Ionic

• Hydrogen Bond

• Hydrophobic Interactions

Today

• Variation of chemical potential

• Pressure

• Composition

• Relation between chemical potential and chemical reactivity.

• Notion of activity and standard states for solutions

o Mole fraction

o Concentration

• Activity coefficients

o Ionic species

Variation of chemical potential with Pressure

We have shown that at constant temperature G varies with pressure as:

[pic]

Note in the above equation, P is dimensionless since it is the ratio of P to 1atm. Now consider an ideal gas that is mixture of two components made in a following way. We start with components 1 and 2 at the same constant pressure, P; confine them in two separate chambers. Then we open the connection between the two chambers so that the gases can mix. Since the gases are ideal the final pressure will be P. This pressure is simply sum of the partial pressures of components 1 and 2, i.e. P1+P2. Thus for this isolated system we can calculate Gibbs free energy by considering initial and final states:

[pic]

Chemical Reactions

Consider following reaction and associated change in free energy. We can show:

[pic]

However, in chemical reaction confined to a closed system, the moles of A,B,C,D do not vary independently. That is C and D production occurs at the expense of A and B. We can quantitatively say:

[pic] [pic]

[pic] In other words stoichiometry decides the rate at which a component disappears. But the normalized rate of the reaction, dα is constant. Substituting in the above expression:

[pic][pic]

[pic]As discussed before the reaction will occur spontaneously, (at constant T and P) if dG is negative, that is sum of the chemical potentials of stoichiometrically weighed reactants and products are positive. Now at equilibrium dG is zero. Considering case of gaseous reaction of N2 and H2 to produce NH3.

Fritz-Haber Process: Nitrogen fixation

[pic][pic]

[pic]Where K is the equilibrium constant for the reaction. Note it is dimensionless since each of the partial pressures used in above equation are dimensionless. This is one of the most common usage of thermodynamics in biology. That is if we know the equilibrium constant we can evaluate free energy accompanying the reaction and vice versa.

[pic]This equation can be used in the non-equilibrium situation also, since:

[pic][pic]

How do we apply above equation for real gases and solutions?

Concept of activity.

Recall, our basic equation for chemical potential:

[pic]

In this equation, we used P1 to be 1atm for standard state of a gas. Also, we used the ideal gas approximation to get this relation. For real gases, which do not obey ideal gas equation, we generalize the concept of pressure with “activity”.

[pic]

where gamma is the activity coefficient. The notion of activity coefficient to correct for pressure inside the real gas is also useful in treatment of solutions. But we still need to define a reference state. For low pressures and high temperature most gases behave like ideal gases. Thus γ approaches 1 in this limit. In general, at 1atm for most gases γ≈1 so it is common use that as the standard state

This arbitrariness in selecting standard state does not affect the chemical potential of the component; only μ0 gets affected. The choices of reference states are many and are dictated by the type of problem. For a chemical potential of solvent used in dilute solution, mole fraction is used. For solutes, it is common to used molar/molar concentrations. Colligative properties are used for determination of activity coefficients.

Concept of Activity

Consider a case of solvent in solution. We choose its standard state as pure solvent i.e. when it’s mole fraction is unity. In this limit, the activity and activity coefficient is unity. In other words:

Very dilute Solutions : aSolvent = 1

Dilute Solution : aSolvent = Xsolvent

Nonideal Solutions : aSolvent = γsolventXsolvent

For solutes we use a different standard state. We prefer to work with molar/molal solutions so we define:

aSolute=γSolutec

[pic]

Where c is the molar/molal concentration. So we define the standard state of solute corresponding to 1 molar/molal solutions. However, for most solutes 1 molar solution behaves nonideally, hence a virtual state is defined by extrapolating the solute activity measured in dilute solution to 1 molar concentration. See pages 130-137 for a through discussion of activity and standard states.

Thus although the chemical potential or free energy of system is fixed quantity the activity or activity related quantities can vary with choice of the reference state.

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