Introductory biological thermodynamics. Entropy, temperature and Gibbs free energy Lecture #2 September 1 st
Main Questions and Ideas introduced in this lecture <ul><li>How can living organisms be so highly ordered ? </li></ul><ul><li>Equilibrium versus non-equilibrium systems. Living systems </li></ul><ul><li>are not at equilibrium, and they are open. Quasi equilibrium. </li></ul><ul><li>Interactions can lead to a spontaneous ordering even </li></ul><ul><li>at equilibrium. </li></ul><ul><li>Entropy can lead to a spontaneous ordering at equilibrium ! </li></ul><ul><li>Flow of information characterizes living organisms. </li></ul><ul><li>Develop our understanding of the utility of G (application to open, aqueous systems) </li></ul><ul><li>Introduce thermodynamic equilibrium (help us to determine G°´) </li></ul><ul><li>Effect of T on thermodynamic equilibrium (van’t Hoff plot; helped us to determine H°´ and S°´) </li></ul>
Biology is living soft matter Self-assembly High specificity Multi-component Information
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. ( H = E + PV ) The internal energy of a system plus the product of its volume and the external pressure exertedon the system.
Thermal energy and molecular length-scale - Boltzmann constant - characteristic energy scale DNA 2 nm 25 nm microtubule E. coli 1 m = 1000 nm Bacteriophage virus 170.000 bp DNA 100 nm
Statistical description of random World The collective activity of many randomly moving objects can be effectively predictable, even if the individual motions are not. If everything is so random in the nano-world of cells, how can we say anything predictive about what’s going there ?
Entropy. The 2 nd 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.
Entropy as measure of disorder Number of allowed states in A : Number of allowed states in B : Number of allowed states in joint system A + B : Entropy: Entropy is additive:
How to count states Total number of states: Using Stirling formula: Probability of each state: Molecules A Molecules B Entropy of system:
Entropy… Molecules A Molecules B Probability of each state:
Sequence Analysis Shannon definition of INFORMATION ENTROPY
Entropy of ideal gas Indistinguishablility For N molecules: For one molecule: - “cell” volume (quantum uncertainty ) V – total volume Free energy of ideal gas: density:
Hard-sphere crystal Hard-sphere liquid Hard-sphere freezing is driven by entropy ! Higher Entropy… Lower Entropy…
Entropy and Temperature Isolated (closed) system: System A System B Total energy: Number of allowed states in A Total number of allowed states Total entropy
Entropy Maximization System A System B A and B in thermal contact. Total system A+B is isolated. Total entropy: Total energy: Define Temperature:
Ordering and 2 nd law of thermodynamics <ul><li>Condensation into liquid (more ordered). </li></ul><ul><li>Entropy of subsystem decreased… </li></ul><ul><li>Total entropy increased! Gives off heat to room. </li></ul>System in thermal contact with environment Initially high Cools to room Equilibration
Summary – Gibbs free energy Reservoir, T Small system a G a = H a - TS a For closed system: For open (small) system Free energy is minimized: - Helmholtz free energy - Gibbs free energy
Biological systems are open <ul><li>Develop our understanding of the utility of G </li></ul><ul><li>Introduce thermodynamic equilibrium (helps us to determine G°´) </li></ul><ul><li>Effect of T on thermodynamic equilibrium </li></ul>
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?
Thermodynamics of open systems (reaction mixes) For one component (A): G A = G A °´ + n A RT .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 (µ A ) and we write: µ A = µ A °´ + RT .ln[ A ] Previously we had for the general case: dG = Vdp - SdT For open, multicomponent systems, we write: dG = Vdp - SdT + i µ i dn i
Thermodynamics of open systems (reaction mixes) In biological systems (constant p and T): dG = i µ i dn i or G = i µ i n i 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 = G products - G reactants = ( cµ c + dµ d ) - ( aµ a + bµ b )
Thermodynamics of open systems (reaction mixes) But µ A = µ A °´ + RT. ln[A] etc . so: G = [(cµ c °´ + dµ d °´) - (aµ a °´ + bµ b °´) + RT ln ( [C] c [D] d /[A] a [B] b ) which we express as: G = G°´ + RT ln ( [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
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.
Thermodynamic equilibrium <ul><li>We calculate G so that we can determine the spontaneous direction of a reaction (favourable or unfavourable). To do that we must determine: </li></ul><ul><ul><ul><li> G°´ </li></ul></ul></ul><ul><ul><ul><li>the concentration of each component </li></ul></ul></ul><ul><ul><ul><li>the stoichiometry of the reaction. </li></ul></ul></ul><ul><li>We need G < 0 for a favourable forward reaction. We can drive the reaction forward either by: </li></ul><ul><ul><ul><li>increasing [A] and/or [B] </li></ul></ul></ul><ul><ul><ul><li>decreasing [C] and/or [D] </li></ul></ul></ul><ul><li>Living cells can (sometimes) control reactant/product concentrations to ensure that G < 0 for desired reactions. </li></ul>
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. 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.
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.
Thermodynamic equilibrium 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°´ = -RT ln K = H°´ - T S°´ Thus, We can therefore plot ln K 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 <ul><ul><ul><li>The sign of G tells us the spontaneous direction of a reaction: G < 0, forward G > 0, reverse G = 0, equilibrium </li></ul></ul></ul><ul><ul><ul><li> G°´ , the standard free energy change for a reaction determines the relative concentrations of reactants and products that will be found at thermodynamic equilibrium . </li></ul></ul></ul><ul><ul><ul><li>(3) Neither quantity tells us about the rate of the reaction. </li></ul></ul></ul>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, 1 992) . 3. The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet F. Evans and H. Wennerstrom, (Wiley, 1 994). 4. Biological thermodynamics Donald T. Haynie (Cambridge University Press, 2001)