Entropy and Free Energy in Biology
|
|
- Hope Bishop
- 6 years ago
- Views:
Transcription
1 Entropy and Free Energy in Biology
2 Energy vs. length from Phillips, Quake. Physics Today. 59:38-43, kt = 0.6 kcal/mol = 2.5 kj/mol = 25 mev typical protein typical cell Thermal effects = deterministic ones!
3 Energy minimization principles mechanical equilibrium demands potential energy be minimized (net force is zero) optical trap works on this principle Assumes T = 0, i.e., no thermal fluctuations!!!
4 Statistics of N body systems what does equilibrium mean when there are thermal fluctuations? *in equilibrium, the properties of the system should not depend on time (which properties???) the properties that describe the system as a whole!
5 Statistics of N body systems D C more precisely, the relevant properties for equilibrium are the accessible measurables of a system, e.g., total energy, volume B A What would be a likely measurable of the system at left? { A = 5, B = 2, C = 1, D= 1 } microstate macrostate
6 Statistics of N body systems D C B A Flip of particles in different levels = change in microstate
7 { 5, 2, 1, 1 } { 5, 2, 1, 1 } Same macrostate Statistics of N body systems D C B A
8 Statistics of N body systems D C B A jump of one particle into different level
9 { 5, 2, 1, 1 } { 4, 2, 1, 2 } Change in macrostate Statistics of N body systems D C B A
10 Principle of equal a priori probabilities all accessible microstates are equally probable -NOTE: this is an assumption!!! but one with much evidence. *So for a given system, which is the most probable macrostate?
11 Principle of equal a priori probabilities If all microstates have equal probability, the macrostate with the largest number of microstates W is the most likely to be observed statistical mechanics becomes all about counting Example: how many ways to arrange N particles in M levels with a set number per level (a given macrostate)? W = N! n 1!n 2!...n M! N! sequences of particles, but we don t care about their order in each level (n! rearrangements per level) *This is how Boltzmann originally derived the connection between entropy and probability for an ideal gas
12 Large N approximations Factorials are not fun to work with (non-analytic) For large values of N, we can use Stirling s approximation x! (x/e) x W = 1 p n 1 1 pn pn M M 1 N ln W = M X i=1 p i ln p i S defined as Shannon entropy (useful in information theory, for example) thermodynamic entropy is the same, just with proper units through Boltzmann s constant k S = k ln W
13 Large N approximations Why do we define entropy as S = k ln W? Entropy needs to be an extensive quantity, i.e., it should scale with system size (intensive variables do not) W A S A W B W A+B =? S B S A+B =? W A+B = W A W B S A+B = S A + S B S(W A W B )=S(W A )+S(W B ) The only way to satisfy this is to define entropy using a logarithm
14 Large N approximations S = k ln W What is the probability distribution that maximizes S? *Need to use Lagrange multipliers and constraint on total probability MX p i =1 i=1 p i = 1 N Distribution is flat most disordered 1/N pi i N
15 Calculating entropy N p proteins bind non-specifically to N binding sites on DNA What is the entropy of this system? Will use Stirling s approximation: ln N! N ln N N PBoC 5.5 S = kn[c ln c +(1 c)ln(1 c)]
16 Calculating entropy Lac repressor - canonical example of gene regulation ~10 copies/cell, about 5*10 6 binding sites on E. coli DNA what is the entropy, assuming non-specific binding? (terrible assumption!) Entropy peaks when the relative concentration is 0.5 largest number of possible arrangements PBoC 5.5 S/Nk = 2.8 x 10-5 N
17 First Law of Thermodynamics U = Q + W conservation of energy in differential du ds dq S = W = T dv Z Vf V i dn S,V P dv the fundamental thermodynamic relation the partial derivatives define familiar properties ds = Q T! V,N du = T ds P dv + i µ i dn i *note that T, P, μ are intensive variables
18 First Law of Thermodynamics why ds but δq? P exact differential inexact differential P1,V1 WA entropy, volume, pressure, energy are state variables, i.e., they can be described by a differentiable function P2,V2 work/heat are path variables WB V work is area under the curve, so it depends on how we get from point 1 to point 2 W = Z Vf V i P dv
19 Consequences of entropy maximization exchange heat exchange volume exchange particles S = S(U,V,N) (all extensive variables of an ds 1 du 1 2 du 2 =0 T 1 = T 2
20 Real systems (not isolated) canonical (NVT) microcanonical (NVE) or isothermalisobaric (NpT) -typical biological system in contact with its environment -environment can be treated as a reservoir of heat and/or volume (T, p constant) -TOTAL entropy max. leads to SYSTEM free energy min.
21 Real systems (not isolated) -cellular systems are NOT in static equilibrium, but often a dynamic one G = U - TS + pv A = U - TS Gibbs free energy Helmholtz free energy (volume not changing) Free energy is that available to do useful work accounts for thermal energy pushing system away from static equilibrium Biological systems minimize free energy
22 Real systems (not isolated) a consequence of contact with a thermal reservoir is that the average energy of the system is constrained - how does this affect the probability distribution? MX p i =1 i=1 total probability MX p i E i = hei i=1 average energy p i = e E i P e E i Boltzmann distribution pi partition function Z X e E i i
23 classic derivation of Boltzmann what is the probability of a particular microstate of our system in contact with a reservoir? Recall: P (E i ) W total (E i ) S = k ln(w ) S/ E = 1 T P (E i ) e E i/kt P (E i )= e E i/kt Z
24 calculating average quantities hai = X p i A i = P Ai e Z E i many quantities can be calculated directly from the partition function Z hei = P Ei e Z E i ln Z PBoC 6.1
25 Example: Hydrophobicity -one of the most important driving forces in biology! -hydrophobic effect ensures: -proteins fold -membranes form and membrane proteins insert -substrates bind -etc. -placing a non-polar substance in water disrupts the hydrogen bonding network, limiting the orientations (i.e., microstates) of a water molecule in contact -to maximize entropy (and minimize free energy) contact surface area is minimized, leading to aggregation of non-polar substance
26 Second Law of Thermodynamics S 0 -empirically known to be true, but dependent on a low-entropy state in the early universe (inflation?) -in equilibrium, equality holds ( ) S =0 -actually only statistically true -violations can briefly occur, particularly in microscopic systems
27 Maxwell s Demon - can he violate the 2nd law? NO! -the demon is part of the system, although he can decrease the entropy of the molecules in the box, his own entropy must go up
28 Brownian ratchet (credit to Feynman) -uses thermal motion of molecules randomly hitting propeller to drive it -ratchet and pawl assures that motion is only in one direction, even though molecules hit propeller from all directions -can this be used to do useful work and violate the 2nd law? NO! the ratchet also undergoes thermal fluctuations back and forth!
29 Modern-day statistical mechanics The free energy change between two states is related to the work done to move the system from one state to the other by ΔF <W> (Second Law) Inequality can be converted to an equality by accounting for fluctuations that transiently violate the Second Law e F/kT = e W/kT Example application: unfolding a small protein helix (see HW) Jarzynski equality (1997!!!) Work extension
30 Hill function Example: ligand binding to a protein PBoC 6.1 this system is highly degenerate, meaning most states have the same energy first count the number of microstates for bound and unbound states then determine their energies and, thus, the weight of each state P bound = c c 0 e β E 1+ c c 0 e β E
Entropy and Free Energy in Biology
Entropy and Free Energy in Biology Energy vs. length from Phillips, Quake. Physics Today. 59:38-43, 2006. kt = 0.6 kcal/mol = 2.5 kj/mol = 25 mev typical protein typical cell Thermal effects = deterministic
More informationChapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density
More informationA Brief Introduction to Statistical Mechanics
A Brief Introduction to Statistical Mechanics E. J. Maginn, J. K. Shah Department of Chemical and Biomolecular Engineering University of Notre Dame Notre Dame, IN 46556 USA Monte Carlo Workshop Universidade
More informationMCB100A/Chem130 MidTerm Exam 2 April 4, 2013
MCB1A/Chem13 MidTerm Exam 2 April 4, 213 Name Student ID True/False (2 points each). 1. The Boltzmann constant, k b T sets the energy scale for observing energy microstates 2. Atoms with favorable electronic
More information(# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble
Recall from before: Internal energy (or Entropy): &, *, - (# = %(& )(* +,(- Closed system, well-defined energy (or e.g. E± E/2): Microcanonical ensemble & = /01Ω maximized Ω: fundamental statistical quantity
More informationLecture 13. Multiplicity and statistical definition of entropy
Lecture 13 Multiplicity and statistical definition of entropy Readings: Lecture 13, today: Chapter 7: 7.1 7.19 Lecture 14, Monday: Chapter 7: 7.20 - end 2/26/16 1 Today s Goals Concept of entropy from
More informationStatistical mechanics of biological processes
Statistical mechanics of biological processes 1 Modeling biological processes Describing biological processes requires models. If reaction occurs on timescales much faster than that of connected processes
More informationMCB100A/Chem130 MidTerm Exam 2 April 4, 2013
MCBA/Chem Miderm Exam 2 April 4, 2 Name Student ID rue/false (2 points each).. he Boltzmann constant, k b sets the energy scale for observing energy microstates 2. Atoms with favorable electronic configurations
More informationMolecular Interactions F14NMI. Lecture 4: worked answers to practice questions
Molecular Interactions F14NMI Lecture 4: worked answers to practice questions http://comp.chem.nottingham.ac.uk/teaching/f14nmi jonathan.hirst@nottingham.ac.uk (1) (a) Describe the Monte Carlo algorithm
More informationChapter 4: Going from microcanonical to canonical ensemble, from energy to temperature.
Chapter 4: Going from microcanonical to canonical ensemble, from energy to temperature. All calculations in statistical mechanics can be done in the microcanonical ensemble, where all copies of the system
More informationLecture 14. Entropy relationship to heat
Lecture 14 Entropy relationship to heat Reading: Lecture 14, today: Chapter 7: 7.20 end Lecture 15, Wednesday: Ref. (2) 2/29/16 1 Hemoglobin and probability Oxygen binding molecule. Its quaternary structure
More informationStatistical Mechanics. Atomistic view of Materials
Statistical Mechanics Atomistic view of Materials What is statistical mechanics? Microscopic (atoms, electrons, etc.) Statistical mechanics Macroscopic (Thermodynamics) Sample with constrains Fixed thermodynamics
More informationSCORING. The exam consists of 5 questions totaling 100 points as broken down in this table:
UNIVERSITY OF CALIFORNIA, BERKELEY CHEM C130/MCB C100A MIDTERM EXAMINATION #2 OCTOBER 20, 2016 INSTRUCTORS: John Kuriyan and David Savage THE TIME LIMIT FOR THIS EXAMINATION: 1 HOUR 50 MINUTES SIGNATURE:
More informationPhysics 172H Modern Mechanics
Physics 172H Modern Mechanics Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TAs: Alex Kryzwda John Lorenz akryzwda@purdue.edu jdlorenz@purdue.edu Lecture 22: Matter & Interactions, Ch.
More informationStatistical thermodynamics (mechanics)
Statistical thermodynamics mechanics) 1/15 Macroskopic quantities are a consequence of averaged behavior of many particles [tchem/simplyn.sh] 2/15 Pressure of ideal gas from kinetic theory I Molecule =
More informationIntroduction Statistical Thermodynamics. Monday, January 6, 14
Introduction Statistical Thermodynamics 1 Molecular Simulations Molecular dynamics: solve equations of motion Monte Carlo: importance sampling r 1 r 2 r n MD MC r 1 r 2 2 r n 2 3 3 4 4 Questions How can
More informationStatistical Mechanics Primer
Statistical Mechanics Primer David an alen January 7, 2007 As the data produced by experimental biologists becomes more quantitative, there becomes a need for more quantitative models. There are many ways
More informationLecture 4: Mechanical and Chemical Equilibrium In the Living Cell (Contd.)
Lecture 4: Mechanical and Chemical Equilibrium In the Living Cell (Contd.) Lecturer: Brigita Urbanc Office: 12-909 (E-mail: brigita@drexel.edu) Course website: www.physics.drexel.edu/~brigita/courses/biophys_2011-2012/
More informationPart II: Statistical Physics
Chapter 6: Boltzmann Statistics SDSMT, Physics Fall Semester: Oct. - Dec., 2014 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function
More information4.1 Constant (T, V, n) Experiments: The Helmholtz Free Energy
Chapter 4 Free Energies The second law allows us to determine the spontaneous direction of of a process with constant (E, V, n). Of course, there are many processes for which we cannot control (E, V, n)
More informationPart II: Statistical Physics
Chapter 6: Boltzmann Statistics SDSMT, Physics Fall Semester: Oct. - Dec., 2013 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function
More informationBoltzmann Distribution Law (adapted from Nash)
Introduction Statistical mechanics provides a bridge between the macroscopic realm of classical thermodynamics and the microscopic realm of atoms and molecules. We are able to use computational methods
More informationLecture 6 Free Energy
Lecture 6 Free Energy James Chou BCMP21 Spring 28 A quick review of the last lecture I. Principle of Maximum Entropy Equilibrium = A system reaching a state of maximum entropy. Equilibrium = All microstates
More informationChE 210B: Advanced Topics in Equilibrium Statistical Mechanics
ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 1 Reading: 3.1-3.5 Chandler, Chapters 1 and 2 McQuarrie This course builds on the elementary concepts of statistical
More informationStatistical. mechanics
CHAPTER 15 Statistical Thermodynamics 1: The Concepts I. Introduction. A. Statistical mechanics is the bridge between microscopic and macroscopic world descriptions of nature. Statistical mechanics macroscopic
More informationLecture Notes 2014March 13 on Thermodynamics A. First Law: based upon conservation of energy
Dr. W. Pezzaglia Physics 8C, Spring 2014 Page 1 Lecture Notes 2014March 13 on Thermodynamics A. First Law: based upon conservation of energy 1. Work 1 Dr. W. Pezzaglia Physics 8C, Spring 2014 Page 2 (c)
More informationto satisfy the large number approximations, W W sys can be small.
Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath
More informationStatistical thermodynamics L1-L3. Lectures 11, 12, 13 of CY101
Statistical thermodynamics L1-L3 Lectures 11, 12, 13 of CY101 Need for statistical thermodynamics Microscopic and macroscopic world Distribution of energy - population Principle of equal a priori probabilities
More information1. Thermodynamics 1.1. A macroscopic view of matter
1. Thermodynamics 1.1. A macroscopic view of matter Intensive: independent of the amount of substance, e.g. temperature,pressure. Extensive: depends on the amount of substance, e.g. internal energy, enthalpy.
More informationPhysics is time symmetric Nature is not
Fundamental theories of physics don t depend on the direction of time Newtonian Physics Electromagnetism Relativity Quantum Mechanics Physics is time symmetric Nature is not II law of thermodynamics -
More informationSolid Thermodynamics (1)
Solid Thermodynamics (1) Class notes based on MIT OCW by KAN K.A.Nelson and MB M.Bawendi Statistical Mechanics 2 1. Mathematics 1.1. Permutation: - Distinguishable balls (numbers on the surface of the
More informationStatistical Physics. The Second Law. Most macroscopic processes are irreversible in everyday life.
Statistical Physics he Second Law ime s Arrow Most macroscopic processes are irreversible in everyday life. Glass breaks but does not reform. Coffee cools to room temperature but does not spontaneously
More informationStatistical thermodynamics Lectures 7, 8
Statistical thermodynamics Lectures 7, 8 Quantum classical Energy levels Bulk properties Various forms of energies. Everything turns out to be controlled by temperature CY1001 T. Pradeep Ref. Atkins 7
More informationChapter 17. Free Energy and Thermodynamics. Chapter 17 Lecture Lecture Presentation. Sherril Soman Grand Valley State University
Chapter 17 Lecture Lecture Presentation Chapter 17 Free Energy and Thermodynamics Sherril Soman Grand Valley State University First Law of Thermodynamics You can t win! The first law of thermodynamics
More informationAdvanced Thermodynamics. Jussi Eloranta (Updated: January 22, 2018)
Advanced Thermodynamics Jussi Eloranta (jmeloranta@gmail.com) (Updated: January 22, 2018) Chapter 1: The machinery of statistical thermodynamics A statistical model that can be derived exactly from the
More informationChapter 3. Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc.
Chapter 3 Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. Concepts Energy functions F and G Chemical potential, µ Partial Molar properties
More informationUC Berkeley. Chem 130A. Spring nd Exam. March 10, 2004 Instructor: John Kuriyan
UC Berkeley. Chem 130A. Spring 2004 2nd Exam. March 10, 2004 Instructor: John Kuriyan (kuriyan@uclink.berkeley.edu) Enter your name & student ID number above the line, in ink. Sign your name above the
More informationPhysics 4230 Final Examination 10 May 2007
Physics 43 Final Examination May 7 In each problem, be sure to give the reasoning for your answer and define any variables you create. If you use a general formula, state that formula clearly before manipulating
More informationPhysics 132- Fundamentals of Physics for Biologists II
Physics 132- Fundamentals of Physics for Biologists II Statistical Physics and Thermodynamics It s all about energy Classifying Energy Kinetic Energy Potential Energy Macroscopic Energy Moving baseball
More informationStatistical thermodynamics Lectures 7, 8
Statistical thermodynamics Lectures 7, 8 Quantum Classical Energy levels Bulk properties Various forms of energies. Everything turns out to be controlled by temperature CY1001 T. Pradeep Ref. Atkins 9
More information[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B
Canonical ensemble (Two derivations) Determine the probability that a system S in contact with a reservoir 1 R to be in one particular microstate s with energy ɛ s. (If there is degeneracy we are picking
More information6.730 Physics for Solid State Applications
6.730 Physics for Solid State Applications Lecture 25: Chemical Potential and Equilibrium Outline Microstates and Counting System and Reservoir Microstates Constants in Equilibrium Temperature & Chemical
More informationRemoving the mystery of entropy and thermodynamics. Part 3
Removing the mystery of entropy and thermodynamics. Part 3 arvey S. Leff a,b Physics Department Reed College, Portland, Oregon USA August 3, 20 Introduction In Part 3 of this five-part article, [, 2] simple
More informationBefore the Quiz. Make sure you have SIX pennies
Before the Quiz Make sure you have SIX pennies If you have more than 6, please share with your neighbors There are some additional pennies in the baskets up front please be sure to return them after class!!!
More informationLecture 2 and 3: Review of forces (ctd.) and elementary statistical mechanics. Contributions to protein stability
Lecture 2 and 3: Review of forces (ctd.) and elementary statistical mechanics. Contributions to protein stability Part I. Review of forces Covalent bonds Non-covalent Interactions: Van der Waals Interactions
More informationLecture 2-3: Review of forces (ctd.) and elementary statistical mechanics. Contributions to protein stability
Lecture 2-3: Review of forces (ctd.) and elementary statistical mechanics. Contributions to protein stability Part I. Review of forces Covalent bonds Non-covalent Interactions Van der Waals Interactions
More informationFirst Law Limitations
First Law Limitations First Law: During any process, the energy of the universe is constant. du du du ZERO!!! universe sys surroundings Any energy transfer between system and surroundings is accomplished
More information2. Thermodynamics. Introduction. Understanding Molecular Simulation
2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular
More informationLecture 27: Entropy and Information Prof. WAN, Xin
General Physics I Lecture 27: Entropy and Information Prof. WAN, Xin xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ 1st & 2nd Laws of Thermodynamics The 1st law specifies that we cannot get more energy
More informationChapter 20 Entropy and the 2nd Law of Thermodynamics
Chapter 20 Entropy and the 2nd Law of Thermodynamics A one-way processes are processes that can occur only in a certain sequence and never in the reverse sequence, like time. these one-way processes are
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability
More informationExam Thermodynamics 12 April 2018
1 Exam Thermodynamics 12 April 2018 Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets. Put your name and student number on each sheet. The examination time is 12:30 until 15:30.
More informationThe physics of information: from Maxwell s demon to Landauer. Eric Lutz University of Erlangen-Nürnberg
The physics of information: from Maxwell s demon to Landauer Eric Lutz University of Erlangen-Nürnberg Outline 1 Information and physics Information gain: Maxwell and Szilard Information erasure: Landauer
More informationPHY 5524: Statistical Mechanics, Spring February 11 th, 2013 Midterm Exam # 1
PHY 554: Statistical Mechanics, Spring 013 February 11 th, 013 Midterm Exam # 1 Always remember to write full work for what you do. This will help your grade in case of incomplete or wrong answers. Also,
More informationEntropy. Physics 1425 Lecture 36. Michael Fowler, UVa
Entropy Physics 1425 Lecture 36 Michael Fowler, UVa First and Second Laws of A quick review. Thermodynamics First Law: total energy conserved in any process: joules in = joules out Second Law: heat only
More information763620SS STATISTICAL PHYSICS Solutions 5 Autumn 2012
7660SS STATISTICAL PHYSICS Solutions 5 Autumn 01 1 Classical Flow in Phase Space Determine the trajectories of classical Hamiltonian flow in -dim corresponding to a particle in a constant gravitational
More informationIrreversible Processes
Irreversible Processes Examples: Block sliding on table comes to rest due to friction: KE converted to heat. Heat flows from hot object to cold object. Air flows into an evacuated chamber. Reverse process
More information4/19/2016. Chapter 17 Free Energy and Thermodynamics. First Law of Thermodynamics. First Law of Thermodynamics. The Energy Tax.
Chemistry: A Molecular Approach, 2nd Ed. Nivaldo Tro First Law of Thermodynamics Chapter 17 Free Energy and Thermodynamics You can t win! First Law of Thermodynamics: Energy cannot be created or destroyed
More informationSome properties of the Helmholtz free energy
Some properties of the Helmholtz free energy Energy slope is T U(S, ) From the properties of U vs S, it is clear that the Helmholtz free energy is always algebraically less than the internal energy U.
More informationProperties of Entropy
Properties of Entropy Due to its additivity, entropy is a homogeneous function of the extensive coordinates of the system: S(λU, λv, λn 1,, λn m ) = λ S (U, V, N 1,, N m ) This means we can write the entropy
More informationStatistical Mechanics
42 My God, He Plays Dice! Statistical Mechanics Statistical Mechanics 43 Statistical Mechanics Statistical mechanics and thermodynamics are nineteenthcentury classical physics, but they contain the seeds
More informationStatistical Mechanics Notes. Ryan D. Reece
Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................
More informationMathematical Structures of Statistical Mechanics: from equilibrium to nonequilibrium and beyond Hao Ge
Mathematical Structures of Statistical Mechanics: from equilibrium to nonequilibrium and beyond Hao Ge Beijing International Center for Mathematical Research and Biodynamic Optical Imaging Center Peking
More informationStatistical Thermodynamics. Lecture 8: Theory of Chemical Equilibria(I)
Statistical Thermodynamics Lecture 8: Theory of Chemical Equilibria(I) Chemical Equilibria A major goal in chemistry is to predict the equilibria of chemical reactions, including the relative amounts of
More informationAppendix 1: Normal Modes, Phase Space and Statistical Physics
Appendix : Normal Modes, Phase Space and Statistical Physics The last line of the introduction to the first edition states that it is the wide validity of relatively few principles which this book seeks
More informationLecture 6. Preliminary and simple applications of statistical mechanics
Lecture 6. Preliminary and simple applications of statistical mechanics Zhanchun Tu ( 涂展春 ) Department of Physics, BNU Email: tuzc@bnu.edu.cn Homepage: www.tuzc.org Main contents Fundamental concepts and
More informationChemistry 431. Lecture 27 The Ensemble Partition Function Statistical Thermodynamics. NC State University
Chemistry 431 Lecture 27 The Ensemble Partition Function Statistical Thermodynamics NC State University Representation of an Ensemble N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T N,V,T
More informationLecture 27 Thermodynamics: Enthalpy, Gibbs Free Energy and Equilibrium Constants
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
More informationThe Second Law of Thermodynamics
he Second Law of hermodynamics So far We have studied the second law by looking at its results We don t have a thermodynamic property that can describe it In this chapter we will develop a mathematical
More informationShort Announcements. 1 st Quiz today: 15 minutes. Homework 3: Due next Wednesday.
Short Announcements 1 st Quiz today: 15 minutes Homework 3: Due next Wednesday. Next Lecture, on Visualizing Molecular Dynamics (VMD) by Klaus Schulten Today s Lecture: Protein Folding, Misfolding, Aggregation
More informationEntropy A measure of molecular disorder
Entropy A measure of molecular disorder Second Law uses Entropy, S, to identify spontaneous change. Restatement of Second Law: The entropy of the universe tends always towards a maximum (S universe > 0
More informationESCI 341 Atmospheric Thermodynamics Lesson 12 The Energy Minimum Principle
ESCI 341 Atmospheric Thermodynamics Lesson 12 The Energy Minimum Principle References: Thermodynamics and an Introduction to Thermostatistics, Callen Physical Chemistry, Levine THE ENTROPY MAXIMUM PRINCIPLE
More informationalthough Boltzmann used W instead of Ω for the number of available states.
Lecture #13 1 Lecture 13 Obectives: 1. Ensembles: Be able to list the characteristics of the following: (a) icrocanonical (b) Canonical (c) Grand Canonical 2. Be able to use Lagrange s method of undetermined
More informationLecture 2: Intro. Statistical Mechanics
Lecture 2: Intro. Statistical Mechanics Statistical mechanics: concepts Aims: A microscopic view of entropy: Joule expansion reviewed. Boltzmann s postulate. S k ln g. Methods: Calculating arrangements;
More information2m + U( q i), (IV.26) i=1
I.D The Ideal Gas As discussed in chapter II, micro-states of a gas of N particles correspond to points { p i, q i }, in the 6N-dimensional phase space. Ignoring the potential energy of interactions, the
More informationLecture 5: Entropy Rules!
Lecture 5: Entropy Rules! Lecturer: Brigita Urbanc Office: 12-909 (E-mail: brigita@drexel.edu) Course website: www.physics.drexel.edu/~brigita/courses/biophys_2011-2012/ 1 DEFINITION OF A MICROSTATE Example:
More information1 Foundations of statistical physics
1 Foundations of statistical physics 1.1 Density operators In quantum mechanics we assume that the state of a system is described by some vector Ψ belonging to a Hilbert space H. If we know the initial
More information213 Midterm coming up
213 Midterm coming up Monday April 8 @ 7 pm (conflict exam @ 5:15pm) Covers: Lectures 1-12 (not including thermal radiation) HW 1-4 Discussion 1-4 Labs 1-2 Review Session Sunday April 7, 3-5 PM, 141 Loomis
More informationAdiabatic Expansion (DQ = 0)
Adiabatic Expansion (DQ = 0) Occurs if: change is made sufficiently quickly and/or with good thermal isolation. Governing formula: PV g = constant where g = C P /C V Adiabat P Isotherms V Because PV/T
More informationComputer simulation methods (1) Dr. Vania Calandrini
Computer simulation methods (1) Dr. Vania Calandrini Why computational methods To understand and predict the properties of complex systems (many degrees of freedom): liquids, solids, adsorption of molecules
More informationRandom arrang ement (disorder) Ordered arrangement (order)
2 CHAPTER 3 In any spontaneous energy conversion, the entropy of the system increases. Regardless of the nature of energy conversion, the entropy of the universe tends toward a maximum. (more probable)
More informationPhysics 132- Fundamentals of Physics for Biologists II. Statistical Physics and Thermodynamics
Physics 132- Fundamentals of Physics for Biologists II Statistical Physics and Thermodynamics QUIZ 2 25 Quiz 2 20 Number of Students 15 10 5 AVG: STDEV: 5.15 2.17 0 0 2 4 6 8 10 Score 1. (4 pts) A 200
More informationAtkins / Paula Physical Chemistry, 8th Edition. Chapter 16. Statistical thermodynamics 1: the concepts
Atkins / Paula Physical Chemistry, 8th Edition Chapter 16. Statistical thermodynamics 1: the concepts The distribution of molecular states 16.1 Configurations and weights 16.2 The molecular partition function
More informationPhysics 4230 Final Exam, Spring 2004 M.Dubson This is a 2.5 hour exam. Budget your time appropriately. Good luck!
1 Physics 4230 Final Exam, Spring 2004 M.Dubson This is a 2.5 hour exam. Budget your time appropriately. Good luck! For all problems, show your reasoning clearly. In general, there will be little or no
More informationThermodynamics. Energy is driving life. Energy of sun ultimately drives most of life on Earth
Sci 190E Lecture 09 Thermodynamics Thermodynamics is the only physical theory of universal content which, within the framework of the applicability of its basic concepts, I am convinced will never be overthrown.
More informationEnergy Barriers and Rates - Transition State Theory for Physicists
Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle
More informationChapter 19 Chemical Thermodynamics Entropy and free energy
Chapter 19 Chemical Thermodynamics Entropy and free energy Learning goals and key skills: Explain and apply the terms spontaneous process, reversible process, irreversible process, and isothermal process.
More informationStatistical thermodynamics for MD and MC simulations
Statistical thermodynamics for MD and MC simulations knowing 2 atoms and wishing to know 10 23 of them Marcus Elstner and Tomáš Kubař 22 June 2016 Introduction Thermodynamic properties of molecular systems
More informationThermodynamics & Statistical Mechanics SCQF Level 9, U03272, PHY-3-ThermStat. Thursday 24th April, a.m p.m.
College of Science and Engineering School of Physics H T O F E E U D N I I N V E B R U S I R T Y H G Thermodynamics & Statistical Mechanics SCQF Level 9, U03272, PHY-3-ThermStat Thursday 24th April, 2008
More informationGrand Canonical Formalism
Grand Canonical Formalism Grand Canonical Ensebmle For the gases of ideal Bosons and Fermions each single-particle mode behaves almost like an independent subsystem, with the only reservation that the
More informationThermochemical Properties
Thermochemical Properties Materials respond to Thermal stimuli (temperature) Chemical stimuli (composition or environment) Electromagnetic stimuli (electric or magnetic fields) Mechanical stimuli (mechanical
More informationPhysics 408 Final Exam
Physics 408 Final Exam Name You are graded on your work (with partial credit where it is deserved) so please do not just write down answers with no explanation (or skip important steps)! Please give clear,
More informationStatistical Physics. How to connect the microscopic properties -- lots of changes to the macroscopic properties -- not changing much.
Statistical Physics How to connect the microscopic properties -- lots of changes to the macroscopic properties -- not changing much. We will care about: N = # atoms T = temperature V = volume U = total
More informationStatistical Mechanics
Statistical Mechanics Newton's laws in principle tell us how anything works But in a system with many particles, the actual computations can become complicated. We will therefore be happy to get some 'average'
More informationComputational Biology 1
Computational Biology 1 Protein Function & nzyme inetics Guna Rajagopal, Bioinformatics Institute, guna@bii.a-star.edu.sg References : Molecular Biology of the Cell, 4 th d. Alberts et. al. Pg. 129 190
More informationPHY214 Thermal & Kinetic Physics Duration: 2 hours 30 minutes
BSc Examination by course unit. Friday 5th May 01 10:00 1:30 PHY14 Thermal & Kinetic Physics Duration: hours 30 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED
More informationClass 22 - Second Law of Thermodynamics and Entropy
Class 22 - Second Law of Thermodynamics and Entropy The second law of thermodynamics The first law relates heat energy, work and the internal thermal energy of a system, and is essentially a statement
More informationChapter 20 The Second Law of Thermodynamics
Chapter 20 The Second Law of Thermodynamics When we previously studied the first law of thermodynamics, we observed how conservation of energy provided us with a relationship between U, Q, and W, namely
More informationEntropy in Macroscopic Systems
Lecture 15 Heat Engines Review & Examples p p b b Hot reservoir at T h p a a c adiabats Heat leak Heat pump Q h Q c W d V 1 V 2 V Cold reservoir at T c Lecture 15, p 1 Review Entropy in Macroscopic Systems
More informationPhysics 9 Wednesday, February 29, 2012
Physics 9 Wednesday, February 29, 2012 learningcatalytics.com class session ID: 410176 Today: heat pumps, engines, etc. Aim to cover everything you need to know to do HW #8. Friday: start electricity (lots
More information