Introductory biological thermodynamics' Entropy, temperature and Gibbs free energy

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Introductory biological thermodynamics. Entropy,
temperature and Gibbs free energy

• Lecture 2
• September 1st

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Main Questions and Ideas introduced in this
lecture

• How can living organisms be so highly ordered ?

• Equilibrium versus non-equilibrium systems.
  Living systems
• are not at equilibrium, and they are open.
  Quasi equilibrium.

• Interactions can lead to a spontaneous ordering
  even
• at equilibrium.

• Entropy can lead to a spontaneous ordering at
  equilibrium !

• Flow of information characterizes living
  organisms.

• Develop our understanding of the utility of
  ?G(application to open, aqueous systems)
• Introduce thermodynamic equilibrium (help us to
  determine ?G)
• Effect of T on thermodynamic equilibrium(vant
  Hoff plot helped us to determine ?H and ?S)

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Biology is living soft matter
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A couple of definitions
Internal energy, E. The energy within the system
(translational energy of the molecules,
vibrational energy of the molecules, rotational
energy of the molecules, the energy involved in
chemical bonding, the energy involved in
nonbonding interactions between molecules or
parts of the same molecule) Enthalpy, H.
(HEPV) The internal energy of a system plus the
product of its volume and the external pressure
exertedon the system.
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Thermal energy and molecular length-scale
- Boltzmann constant
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Statistical description of random World
The collective activity of many randomly moving
objects can be effectively predictable, even if
the individual motions are not.
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Entropy. The 2nd law of thermodynamics
Isolated system always evolve to
thermodynamic equilibrium. In equilibrium
isolated system has the greatest possible
ENTROPY (disorder) allowed by the physical
constraints on the system.
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Entropy as measure of disorder
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How to count states
Total number of states
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Entropy
Molecules A
Molecules B
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Sequence Analysis Shannon definition of
INFORMATION ENTROPY
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Entropy of ideal gas
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Disordered Liquid
Ordered Solid
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Hard-sphere liquid
Hard-sphere freezing is driven by entropy !
Hard-sphere crystal
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Entropy and Temperature
Isolated (closed) system
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Entropy Maximization
A and B in thermal contact. Total system AB is
isolated.
System B
System A
Total entropy
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Ordering and 2nd law of thermodynamics
System in thermal contact with environment
Cools to room
Initially high

• Condensation into liquid (more ordered).
• Entropy of subsystem decreased
• Total entropy increased! Gives off heat to room.

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Summary Gibbs free energy
Small system
a
Reservoir, T
Ga Ha -TSa
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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

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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 ltgt 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?
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Thermodynamics of open systems (reaction mixes)
For one component (A) GA GA
nART.lnA 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.lnA Previously we had for the general
case dG Vdp - SdT For open,
multicomponent systems, we write dG Vdp
- SdT ?i µidni
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Thermodynamics of open systems (reaction mixes)
In biological systems (constant p and T)
dG ?i µidni or ?G ?i µi?ni
We can now calculate ?G for a specific
reaction aA bB ltgt 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)
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Thermodynamics of open systems (reaction mixes)
But µA µA RT.lnA etc.
so ?G (cµc dµd) - (aµa
bµb) RTln(CcDd/AaBb) which we
express as ?G ?G RTln(CcDd/Aa
Bb) (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 lt----gt 1C
2D which is the same as A A B lt----gt
C D D
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Thermodynamics of open systems (reaction mixes)
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.
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.
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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 lt 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 lt 0 for desired reactions.

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Thermodynamic equilibrium
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. Lets see what
happens We have If ?G lt 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.
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Thermodynamic equilibrium
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.
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Thermodynamic equilibrium
Note if ?G lt 0, then (CD)eq gt
(AB)eq (for abcd1) if ?G gt 0,
then (CD)eq lt (AB)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).
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The effect of temperature
We can write ?G -RTlnK ?H -
T?S Thus,
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 Vant Hoff plot. It will give a
straight-line if ?H is independent of T
(usually true for narrow ranges of T).
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Summary

• The sign of ?G tells us the spontaneous
  direction of a reaction ?G lt 0,
  forward ?G gt 0, reverse ?G 0,
  equilibrium
• ?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.

See a textbook for more details on ?G and ?G -
we will see this later on at transition state
theory
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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). 3.
The Colloidal Domain Where Physics,
Chemistry, Biology and Technology Meet F.
Evans and H. Wennerstrom, (Wiley, 1994). 4.
Biological thermodynamics Donald T. Haynie
(Cambridge University Press, 2001)