Biological systems are open

Biological systems are open

• Develop our understanding of the utility of G

• Introduce thermodynamic equilibrium (helps us to determine G°´)

• Effect of T on thermodynamic equilibrium

• Develop our understanding of the utility of G

• Introduce thermodynamic equilibrium (helps us to determine G°´)

• Effect of T on thermodynamic equilibrium

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Thermodynamics of open systems (reaction mixes)

We need to handle systems that contain more than one component, the concentrations of which can vary (e.g. a solution containing a number of dissolved reactants). These are called open systems.

Consider: A + B <===> C + D

There are four solutes (reactants or products) and the concentrations of A, B, C and D are affected by this and other reactions. How do we calculate G for this reaction?


For one component (A): GA = GA°´ + nART.ln[A]

And for 1 mole of A:

where the units are free energy per mole (J mol-1). This quantity is also known as the chemical potential (µA) and we write:

µA = µA°´ + RT.ln[A]

Previously we had for the general case: dG = Vdp - SdT

For open, multicomponent systems, we write:

dG = Vdp - SdT + i µidni

G A G Ao 'RT ln[A]


In biological systems (constant p and T):

dG = i µidni or G = i µini

We can now calculate G for a specific reaction:

aA + bB <===> cC + dD

where the reactants are A and B, the products are C and D. a, b, c and d represent the number of moles of each that participate in the reaction. For this reaction:

G = Gproducts - Greactants

= (cµc + dµd) - (aµa + bµb)


But µA = µA°´ + RT.ln[A] etc. so:

G = [(cµc°´ + dµd°´) - (aµa°´ + bµb°´) + RTln([C]c[D]d/[A]a[B]b)

which we express as:

G = G°´ + RTln([C]c[D]d/[A]a[B]b) (Joules)

To find the free energy change per mole, note that a, b, c and d will reflect the stoichiometry of the reaction (the numbers of each type of molecule involved in a single reaction). For example, a single reaction might involve the following numbers of molecules:

2A + 1B <----> 1C + 2D

which is the same as: A + A + B <----> C + D + D


We now have an equation which allows us to calculate G in practice:

(J mol-1)

a, b, c and d are the stoichiometric coefficients.

G°´ is the standard free energy change per mole.

G Go RT lncC dD aA bB

Standard free energy change per mole is the free energy change that occurs when reactants at 1 M are completely converted to products at 1 M at standard p, T and pH.

Thermodynamic equilibrium

We calculate G so that we can determine the spontaneous direction of a reaction (favourable or unfavourable). To do that we must determine:

G°´• the concentration of each component • the stoichiometry of the reaction.

We need G < 0 for a favourable forward reaction. We can drive the reaction forward either by:

• increasing [A] and/or [B]• decreasing [C] and/or [D]

Living cells can (sometimes) control reactant/product concentrations to ensure that G < 0 for desired reactions.

G G o 'RT lncC dD aA bB


In many cases there is a dynamic steady state: new reactants are made and products consumed in other reactions in order to keep concentrations steady. So G holds constant and, if it is negative, the reaction keeps going.

If the cell dies, the reaction will reach thermodynamic equilibrium and come to a halt. Let’s see what happens:

We have:

If G < 0 at time zero and the system is isolated, then G becomes less negative as the concentrations of C and D build up (and those of A and B decline). Eventually equilibrium is reached when G =0.

G G o 'RT lnC c

D d

A aB b


At equilibrium,

and define the equilibrium constant K:

This constant depends on the chemical natures of the reactants and products. We measure G°´ by measuring the concentrations of A, B, C and D once the reaction has reached equilibrium.

G o ' RT lnC c

D d

A aB b

eq

K C c

D d

A aB b

eq


Note: if G°´ < 0, then ([C][D])eq > ([A][B])eq (for a=b=c=d=1)

if G°´ > 0, then ([C][D])eq < ([A][B])eq

Thus, G°´ determines whether the reactants or products predominate at equilibrium. T

Thermodynamic equilibrium is not a static state (conversion of A and B to C and D and back again keeps going but there is no net change in their concentrations).

The effect of temperature

We can write: G°´ = -RTlnK = H°´ - TS°´

Thus, ln K Go'

RT

Ho'

R

1

T

So'

R

We can therefore plot lnK vs 1/T to determine H°´ and S°´, the standard enthalpy and entropy of the reaction respectively.

Such a plot is known as a Van’t Hoff plot. It will give a straight-line if H°´ is independent of T (usually true for narrow ranges of T).

Summary

(1) The sign of G tells us the spontaneous direction of a reaction:

G < 0, forwardG > 0, reverseG = 0, equilibrium

(2) G°´, the standard free energy change for a reaction determines the relative concentrations of reactants and products that will be found at thermodynamic equilibrium.

(3) Neither quantity tells us about the rate of the reaction.

K G ' RTe C c

D d

A aB b

eq

See a textbook for more details on G and G°´ - we will see this later on at transition state theory

References

1. Biological Physics. Energy, Information, Life Philip Nelson, (Freeman and Company, New York, 2004).

2. Principles of Physical Biochemistry, chapter 2, pp. 69-89 Kensal E. van Holde, W. Curtis Johnson and P. Shing Ho (Pretice Hall, Upper Saddle River, 1992).

Documents

Biological systems are open