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11.1 Energy and Change
We want to relate the measured equilibrium constants of spontaneous
reactions to the thermodynamic features of those reactions.
11.2 The Criterion for Spontaneity
- DG < 0 Spontaneous, exergonic.
- DG = 0 Equilibrium.
- DG > 0 Nonspontaneous, endergonic.
(typo in Campbell, here)
11.3 Standard States and the Standard Free Energy Change
Standard states are arbitrary, but are agreed upon to allow easy comparisons
between experimental results. Chemists and biochemists have agreed to
use these standard states:
- Concentration: 1.0 M
- Water: 1.0
- Pressure: 1.0 atm
Free energy changes under these standard state conditions are designated
as DG°.
Relationship between free energy (DG) and
the equilibrium constant:
DG = DG°
+ RT ln [Products]/[Reactants]
At equilibrium, DG = 0, and
0 = DG° + RT ln [Products]/[Reactants];
therefore,
DG° = -RT ln [Products]/[Reactants]
Since these are equilibrium concentrations of products and reactants,
we can write,
DG = DG° + RT
ln Keq. Rewriting this equation, we have:
DG° = -RT ln Keq
Thus, Keq and DG° are both
measures of how favorable a reaction will be.
- Some frequently used numbers at 300K (27°C):
- RT = 0.60 kcal/mol = 2.50 kJ/mol
- RTln10 = 2.303log10 = 1.38 kcal/mol = 5.76 kJ/mol
11.4 A Modified Standard State for Biochemical Applications
- To the list of standard conditions above, biochemists add another,
namely:
- [H+] = 10-7 M, i.e. pH 7.0.
The use of this standard state is indicated by DG°'.
If H+ is neither a reactant nor a product, DG°'
= DG°.
See the example of ATP hydrolysis in Campbell, p. 398.
Consider another example:
| |
pKa |
Ka (M-1) |
DG° (kJ/mol) |
DG°' (kJ/mol) |
| Acetic acid |
4.76 |
10-4.76 |
27.4 |
-12.9 |
| Alanine |
2.34 |
10-2.34 |
13.5 |
-26.8 |
Difference
{or ratio} |
2.42
|
{102.42}
|
13.9
|
13.9
|
For the comparison between the acid dissociation constants (Ka)
of acetic acid and alanine, we see that both DG°
and DG°' give identical values (as they
must). This comparison also leads to the question, Why does alanine
lose its H+ at a much lower pH than does acetic acid?
- Qualitatively, the positively charged -NH3+
group, on the Ca of alanine
facilitates proton dissociation from the adjacent carboxyl group.
- We can add to the above (correct) description by calculating how
much the -NH3+ group facilitates carboxyl dissociation:
- The answer shown in the table is that dissociation from alanine
is favored by 13.9 kJ/mol.
- You should convince yourself that the following simple expression
applies to comparisons of pKa's:
DDG =
2.303*RTDpKa
where,
DDG is the
difference in DG between the two ionizations
(acetic acid - alanine, above), and
DpKa is the difference between
the respective pKa values (second column, above).
11.5 Thermodynamics and Life
The free energy is also related to changes in the enthalpy (heat) and
entropy (disorder) of the system:
DG = DH
- TDS
- DG, DH, DS
refer to differences between the two states.
- DG, DH, DS
are thermodynamic values which depend only on the initial and
final state
- DH is the amount of heat generated/consumed
by the reaction
- DS is the change in entropy or disorder
of the system
- For a spontaneous reaction: DG<0
- For a non-spontaneous reaction: DG>0
If DS = 0, then DG
= DH, and all reactions that give off heat
are spontaneous (DH<0 therefore DG<0).
If DH = 0, then DG
= -TDS, and all reactions that increase entropy
(DS>0) are spontaneous.
The entropy (DS) is related to the number
of possible configurations (W) of the system:
S = R ln W
Consider the following simple example of vaporization of a gas. Two
molecules of methane which are restricted to three of the nine possible
compartments of a flask. The number of ways that the molecules can arrange
themselves is (3*2)/2! =3. If the two molecules, in the vapor phase
can occupy any of the nine cells then the number of possible arrangements
are (9*8/2!)=36. The entropy difference between these two states is:
DS=Sg-Sl=R (ln 36 ñ
ln 3) = 4.92 cal/mol-deg. The total free energy available for this reaction
is: DG=-TDS=1.5
kcal/mol at 300K.
The thermodynamics of methane transfer from an inert liquid to either
the vapor phase or to liquid water has been measured:
- First consider the process of going from the inert liquid (benzene)
to the gas:
- DG=-3.5 kcal/mol
- DH= 0.5 kcal/mol
- DS=+14 kcal/mol-deg
- The reaction is favorable because of the large change in
entropy. There is actually a loss of enthalpy because the methane
molecules no longer interact in the gas phase.
- Next consider the transfer of methane from an inert liquid to liquid
water:
- DG= +2.8 kcal/mol
- DH= -3.2 kcal/mol
- DS= -32.0 kcal/mol-deg
This reaction is unfavorable (hydrocarbons are hydrophobic).
The large unfavorable entropy observed here is due to the ordering of
water around the dissolved methane molecules. Since this restricts the
possible positions of the water molecules the entropy of the system
decreases. Note that the loss of free energy due to the entropic term
is partially offset by a gain in enthalpy due to the newly created water-water
hydrogen bonds. Thus the burial of hydrophobic amino acids during folding
is favorable.
Temperature Dependence of the Equilibrium Constant:
The temperature dependence of the equilibrium constant can be obtained
by equating our two equations for DG:
- Exothermic Reaction (DH<0, reaction
produces heat)
- Endothermic Reaction (DH>0, reaction
consumes heat)
The above is another example of Le Chatelier's principle. Note that
the enthalpy of a reaction can be obtained from a plot of lnK versus
1/T.
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