Physical Biochemistry

Kwan Hee Lee, Ph.D.

Handong Global University

Week 6

μA = μA0 + RTaA

ΔG = [(cμC + dμD) – (aμA + bμB)] = ΔG0 + RT ln(aC

caDd/aA

aaBb)

= ΔG0 + RT lnQ Activities are unitless numbers The activity aA of A is defined with respect to a

standard state. μA - μA

0 is a measurable quantity. Need to relate the activities to parameters

that have been widely used by chemists and biochemists, such as the partial pressure and concentrations.

Nonideal systems (Activity)

Standard states are chosen for convenience. The particular choices of standard states allow us

to relate the activities to well-known experimental parameters.

For ideal gases, the activity of a component is directly given its partial pressure, for ideal solutions, the activity of a component is given by its concentration.

For ideal gases and dilute solutions, the activity of a component remains a simple function of its partial pressure or concentration when convenient standard states are selected.

Standard states

For ideal gases: the standard state is the gas with a partial pressure equal to 1 atm: aA = PA/1 atm, where PA is a partial pressure of ideal gas.

Real gases

The activity of a real gas is a function of pressure: aA = γAPA/1 atm , where γA is termed the activity coefficient.

Standard states

All the gases become ideal at low pressure, and γA becomes 1 as the pressure of the system approaches zero. Then the real gases can be treated as ideal gases.

μA-μA0 = RT ln(γA PA/1 atm) = RT ln γA + RT

ln(PA/1 atm) for real gas Plot of μA-μA

0 as a function of ln(PA/1 atm) will give us a straight line.

If the straight line is extrapolated to PA =1, μA-μA

0 =0, so μA=μA0 , the chemical potential

of the gas at its standard state.

Standard states

Thus the standard state for a real gas is a hypothetical one: it is the extrapolated state where the partial pressure is 1 atm but the properties are those extrapolated from low pressure.

The standard state for a pure or liquid is the pure substance at 1 atm pressure. So the activity is equal to 1 for a pure solid or liquid at 1 atm.

The Gibbs free energy of a solid or liquid changes only slightly with pressure, so we can neglect the change and use 1 for the activity of a pure solid or liquid at any pressure.

Solution: homogeneous mixture of two or more substances.

The activity of each substance will depend on its concentration and the concentrations of everything else in the mixture.

Activity = (activity coefficient)(concentration)

The activity coefficient is not a constant, it incorporates all the complicated dependence of the activity of A on the concentrations of A, B, C, and so on.

Activity in solutions

The standard state that uses mole fraction as a concentration unit is often called the solvent standard state.

The solvent state for a component of a solution defines the pure component as the standard state.

aA = γAXA, XA is the mole fraction of A.

We choose the standard state so that the activity aA becomes equal to the mole fraction XA, as the mole fraction of A approaches unity. lim aA = XA as XA goes to 1.

Mole fraction and solvent standard state

lim γ = 1 as XA approaches 1. For a liquid solution, the standard state for

the solvent is defined as the pure liquid. The best example is dilute aqueous

solutions. For any concentration, the solvent has the

properties of the solvent in a dilute solution. To determine the activity coefficient and the

activity, we have to measure the Gibbs free energy of the solvent in the solution.

Solute standard state: the extrapolated state where the concentration is 1 molar(M) or 1 molal(m), but the properties are those extrapolated from very dilute solution.

The solute standard state is hypothetical state in the sense that it corresponds to a 1 m ideal solution.

For real solution, the ideal behavior is not approached except for concentrations that are much less than 1 M or 1 m.

Solute standard state

aB = γBcB

aB= activity of B

γB = activity coefficient of B on the molarity scale.

cB = concentration of B

lim aB = cB

lim γ = 1, as cB approaches to 0.

Molarity

To determine the activity coefficient and therefore the activity , we measure the free energy of component B in the solution.

It can be done indirectly through its effect on the vapor pressure of the solvent in solvent

Depending on the properties of B, various methods exist for measuring the free energy directly.

Molarity

The free energy in the standard state must be measured to obtain aB, so we must understand the definition of the standard state for the molarity scale.

It is seen in the figure in the right.

Extrapolation.

To obtain the the free energy in the standard state, we measure μB as a function of ln cB in a dilute solution and extrapolate linearly to ln cB = 0(cB = 1); we thus obtain μB(figure in the right)

The solute standard state is thus the state that has the properties of a very dilute solution extrapolated to a 1-M concentration.

It is a hypothetical one.

Molarity

Strong electrolyte dissociates completely into its component ions.

When we speak of the partial molar Gibbs free energy or chemical potential of NaCl in aqueous solution, we use the sum rule for the chemical potential.

μNaCl ≡ μNa+ + μCl

-

= μ°Na+ + μ°Cl

- + RT lnaNa+ + RT lnaCl

-

= μ°Na+ + μ°Cl

- + RT ln(aNa+ ㆍaCl

-) μNaCl = μ°NaCl + RT lna NaCl

Molarity in a strong electrolyte

Molality : moles of solute per kilogram of solvent. aB = γBmB

In dilute solution or ideal solution: aB = mB

Real solution: aB = γBmB

The standard state is an extrapolated state;μB is obtained by linearly extrapolating μB measured in dilute solution to ln mB=0 (mB=1).

Molality is used instead of molarity for the most accurate thermodynamic measurement, because it is defined by weight which does not change much against temperature change.

Molality

For a molecule that dissociates in solution, it is difficult to determine the concentration of each species in a system.

For example, a reaction may involve H2PO4

-, which have many ionic species including H3PO4, H2PO4

-, HPO42- .

The distribution of these species will depend on pH, so the concentration of H2PO4

- may be difficult to specify.

Biochemist’s standard state

To simplify this situation, biochemists have chosen pH 7.0 as their standard condition for aH

+.

This means that aH+= 1 for a concentration of H+

= 10-7 M.

The activity of each other molecule is set equal to the total concentration of all species of that molecule at pH=7.0.

Knowledge of ionization constants is thus not needed, nor is necessary to specify the concentration of each of the actual species involved in the reaction.

Biochemist’s standard state

ΔG° is the free energy change for a reaction at pH 7 when each product and reactant has a total 1-M concentration, but the solution is ideal.

So the biochemist’s standard state is then a practical and useful choice.

The standard free energy and the equilibrium constant can be used directly from the tables at or near pH 7.

Biochemist’s standard state

To simplify thermodynamic calculations, we often set activity coefficients to 1.

But for multivalent ions such as PO43-, the activity

coefficients may not be 1, even in very low concentration.

In ionic solutions, the total number of positive charges is always equal to the total number of negative charges.

Therefore, we can separately measure the activity coefficients of a positively or negatively charged ion; we can measure only the mean activity coefficient of an ion and its counterion.

For 1-1, 2-2 electrolyte, the mean ionic activity coefficients is γ± = (γ+ γ-)

1/2

Activity coefficients of ions

For H2SO4 or any 1-2 electrolyte, γ± = (γ2+ γ-

)1/3

For LaCl3 and 1-3 electrolyte, γ± = (γ+ γ3-)

1/4

To understand why activity coefficients are so different from 1 for electrolytes and why the ions with the larger charges have the smaller activity coefficients, we need to consider the interactions between the solutes.

Activity coefficients of ions

Activities are effective concentrations. In an HCl solution, the effective concentration of

the H+ ions is reduced by the surrounding Cl- ions; the effective concentration of the Cl- ions is also reduced by the surrounding H+ ions.

This kind of decrease in effective concentrations of the ions is greater in H2SO4 solution, which has a doubly charged ion, SO4

2-. Coulomb proposed an equation to calculate it in

very dilute solution: for ions in water at 25 °C, log γi= -0.509Zi

2I1/2, where Zi=charge on ion (±1, ±2 etc) and I = ionic strength = ½ ΣiciZi

2.

Activity coefficients of ions