Physical Principles in Biology Biology 3550 Fall 2017 Lecture 27 Thermodynamics: Enthalpy, Gibbs Free Energy and Equilibrium Constants Wednesday, 1 November c David P. Goldenberg University of Utah goldenberg@biology.utah.edu
The Second Law of Thermodynamics For a spontaneous process: S univ = S sys + S surr Entropy change for the system (constant T): S sys = q rev T = nk ln Ω 2 Ω 1 Entropy change for the surroundings: S surr = q T q is heat absorbed for specified process. Flow of heat to surroundings represents an increase in entropy of the surroundings. Entropy of the system can decrease in a spontaneous process, if there is a flow of heat to the surroundings.
Thermodynamics and Chemical Reactions Changes in potential energy: A B Forming a chemical bond reduces potential energy of molecules and releases energy. Breaking a chemical bond increases potential energy of molecules and absorbs energy. If the potential energy is released, where can that energy go? Thermal energy (temperature) increases. Heat flows out of the system. (q < 0) Work is done by the system. (w < 0) Some combination of the above.
J. Willard Gibbs 1839 1903 Arguably the first great American theoretical scientist. Consolidated thermodynamics into a consistent theory. Applied thermodynamics to chemistry. Also made important contributions in math and optics.
A New State Function: Enthalpy (H) Enthalpy, H, is closely related to the internal energy, E. H = E + PV The product PV has the units of energy or work. For a process at constant pressure: H = E + P V P V is work due to a change in volume at constant pressure. Two new quantities: w p = P V = work done on system in constant pressure process q p = heat absorbed by system in constant pressure process From the first law: E = q p + w p = q p P V H = E + P V = q p
A New State Function: Enthalpy (H) Two consistent definitions for H at constant pressure: E minus work due to volume change H = E + P V = E w p If there is no volume change, then w p = 0, and H = E. The heat absorbed during the constant pressure process: H = q p Chemical and biochemical reactions are usually studied under conditions of constant (or nearly constant) pressure, and V is usually very small. H can be experimentally measured.
Measuring H with a Calorimeter Thermometer Reaction Water Insulation If H < 0, heat flows from the reaction vessel and warms the water. If H > 0, heat flows into the reaction vessel, and water temperature decreases. T water H = mass of water (g) (in calories)
Clicker Question #1 Which of the following is true at constant pressure? Choose two. 1 H = E + P V 2 H = E + V P 3 H = q rev 4 H = w rev 5 H = q p 6 H = w p
The Gibbs Free Energy G = H TS sys For a process at constant temperature: G = H T S sys For a process at constant temperature and constant pressure: G = q p T S sys Since S surr = q/t, q p = T S surr at constant pressure. G = T S surr T S sys = T ( S surr + S sys ) = T S univ Or, S univ = G/T
The Gibbs Free Energy is Used to Apply the Second Law From the previous slide: S univ = G/T If G < 0 S univ > 0, and the process will be spontaneous. If G > 0 S univ < 0, and the reverse process will be spontaneous. If G = 0 S univ = 0, and the process is reversible. (Can be pushed in either direction.) G = 0 defines a system at equilibrium. All based on state functions of the system! Don t have to worry about the surroundings.
Clicker Question #2 Which of the following is true at constant pressure and constant temperature? Choose two. 1 G = H T S univ 2 G = H T S sys 3 G = T S surr T S sys 4 G = T S surr T S sys 5 G = T S surr q p /T
What About Free Energy as Energy Available to do Work? The change in Helmholtz free energy, as previously defined: F = w rev The maximum work available from a process. From the first law: E = q rev + w rev w rev = E q rev F = E q rev For a constant-temperature process: S sys = q rev T q rev = T S sys F = E T S sys
The Helmholtz Free Energy For a constant temperature process: F = E T S sys The standard definition of the Helmholz free energy: F = E S sys Compare to: G = H T S Since H = E w p (a few slides back): G = F w p w p is work due to volume change at constant pressure.
The Helmholtz Free Energy From previous slide: G = F w p If volume change is small and w p can be ignored: G = F If G < 0: Process is favorable, and maximum work available from the process = F = G If G > 0: Process is unfavorable, and minimum work required to drive the process = F = G Commonly made assumptions for chemical reactions in dilute solutions: P V = w p 0 H E G F
The Equilibrium Constant for a Chemical Reaction A B The reaction can occur in either direction. (at least in principle) The probability that any molecule of A will be converted to a molecule of B is the same as for any other molecule of A. The total rate of conversion of A to B is proportional to the concentration of A. Similarly, the total rate of conversion of B to A is proportional to the concentration of B. A differential equation: d[a] = k dt f [A] + k r [B] The constants k f and k r are called rate constants.
The Equilibrium Constant for a Chemical Reaction A B From the previous slide: d[a] dt = k f [A] + k r [B] Similarly: d[b] dt = k f [A] k r [B] = d[a] dt Whatever the starting concentrations, [A] and [B] will eventually adjust so that they no longer change. At equilibrium: d[a] dt = d[b] dt = 0 k f [A] + k r [B] = 0
The Equilibrium Constant for a Chemical Reaction A B At equilibrium: d[a] dt = d[b] dt = k f [A] eq + k r [B] eq = 0 [A] eq and [B] eq are the concentrations of A and B at equilibrium. Rearranging: k r [B] eq = k f [A] eq [B] eq [A] eq = k f k r = K eq K eq is defined as the equilibrium constant.
The Equilibrium Constant for a Chemical Reaction A B K eq = [B] eq [A] eq At equilibrium, a little bit of work can push the reaction in either direction. This defines the condition where G = 0 Definition of G for a chemical reaction: The free energy change for converting one mole of reactants to one mole of products, at specified concentrations, without significantly changing concentrations(!). Implies a very, very large volume! Or, a mechanism that restores concentrations continuously.