Physical Biochemistry

Physical Biochemistry

Kwan Hee Lee, Ph.D.

Handong Global University

Week 6

 μ_A = μ_A⁰ + RTa_A

 ΔG = [(cμ_C + dμ_D) – (aμ_A + bμ_B)] = ΔG⁰ + RT ln(a_C^ca_D^d/a_A^aa_B^b)

= ΔG⁰ + RT lnQ

 Activities are unitless numbers

 The activity a_A of A is defined with respect to a standard state.

 μ_A - μ_A⁰ 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:

a

= P

/1 atm, where P

is a partial pressure of ideal gas.



Real gases



The activity of a real gas is a function of pressure: a

= γ

P

/1 atm , where γ

is termed the activity coefficient.

Standard states



All the gases become ideal at low pressure, and γ

becomes 1 as the pressure of the

system approaches zero. Then the real gases can be treated as ideal gases.



μ

-μ

= RT ln(γ

P

/1 atm) = RT ln γ

+ RT ln(P

/1 atm) for real gas



Plot of μ

-μ

as a function of ln(P

/1 atm) will give us a straight line.



If the straight line is extrapolated to P

=1, μ

-μ

=0, so μ

=μ

, 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.

 a_A = γ_AX_A, X_A is the mole fraction of A.

 We choose the standard state so that the

activity a_A becomes equal to the mole fraction X_A, as the mole fraction of A approaches unity.

lim a_A = X_A as X_A goes to 1.

Mole fraction and solvent standard

state



lim γ = 1 as X

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



a

= γ

c



a

= activity of B



γ

= activity coefficient of B on the molarity scale.



c

= concentration of B



lim a

= c



lim γ = 1, as c

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 μ_B as a function of ln c_B in a dilute solution and extrapolate linearly to ln c_B = 0(c_B = 1); we thus obtain μ_B (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.



μ

≡ μ

+ μ

= μ°

+ μ°

+ RT ln a

+ RT ln a

= μ°

+ μ°

+ RT ln( a

ㆍ a

)



μ

= μ°

+ RT ln a

Molarity in a strong electrolyte

 Molality : moles of solute per kilogram of solvent.

 a_B = γ_Bm_B

 In dilute solution or ideal solution: a_B = m_B

 Real solution: a_B = γ_Bm_B

 The standard state is an extrapolated state;μ_B is obtained by linearly extrapolating μ_B measured in dilute solution to ln m_B=0 (m_B=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

H

PO

, which have many ionic species including H

PO

, H

PO

, HPO

.



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

PO

may be difficult to specify.

Biochemist’s standard state

 To simplify this situation, biochemists have chosen pH 7.0 as their standard condition for a_H⁺_.

 This means that a_H⁺= 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 PO₄^3-, 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 H

SO

or any 1-2 electrolyte, γ

= (γ

γ

)



For LaCl3 and 1-3 electrolyte, γ

= (γ

γ

)



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 γ_i= -0.509Z_i²I^1/2, where Z_i=charge on ion (±1,

±2 etc) and I = ionic strength = ½ Σ_ic_iZ_i².

Activity coefficients of ions