Lecture 7' - Thermodynamics and the 1st Law

8/14/2019 Lecture 7' - Thermodynamics and the 1st Law

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Introduction to StatisticalThermodynamics


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Introduction

Thermodynamics is often regarded as too abstract to berelevant to biochemists: emphasis on heat engines. abstract concepts of energy and entropy.

However, this abstractness makes Thermodynamics anideal tool for addressing biological systems. such systems are often not well-defined.

Example: Energetics of Protein Denaturation Temp-dependence of K eq:

Yields the enthalpy change, Ho during denaturation. so-called vant Hoff analysis.

knowledge of protein structure or composition unnecessary.


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Outline

Here, we briefly review some basic ideas from thermodynamics: Lecture 6:

Heat, Work, and Energy The 1 st Law of Thermodynamics.

Lecture 7: Entropy, Free Energy, and Equilibrium. The 2 nd Law of Thermodynamics.

Useful for understanding macromolecular structure. structure discussed in terms of distributions over accessible states. Free energies will relate to equilibrium probabilities of occupation.

Prerequisite for Statistical Thermodynamic modeling (Unit 4).


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Overview

Biochemical systems typically contain huge numbers of molecules:

Size

Avagadros Number (6.023 x 1023

molecules). These systems are extremely complex. Explicit modeling often beyond the capability of physical chemistry.

Howevera Thermodynamic analysis: independent of the specific details of molecular structure.

nevertheless, provides detailed predictions. essentially exact, for many-particle systems.

represents a statistical analysis Statistical thermodynamic predictions represent ensemble averages.


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Points to Remember

Thermodynamic predictions yield average quantities: conceptually, the most likely behavior.

Predictions exact only for systems with a very largenumber of particles. systems with a small number of particles may deviate.

e.g.: protein copy number is often small.

Predictions apply to systems at equilibrium. need to (separately) investigate times to equilibrium.

Thermo. provides no information about rates. thermodynamics means equilibrium thermodynamics.


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Fundamental Quantities

We first define and discuss some fundamental quantities:1. The System.2. The State of a System.3. Heat, q.4. Work, w.5. Internal Energy, E.6. Enthalpy, H = E + PV.

These quantities will allow us to state the 1 st Law of Thermodynamics.


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The System

Def. (System): A part of the universe chosen for study. Essentially, we divide the universe into 2 parts:

system of interest; the rest of the universe = the surroundings. our system will have a definite spatial boundary.


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System Characteristics

Very abstract notionbut with definite characteristics: Definite boundaries in space.

May be open or closed with respect to transfer of matter. constraint on exchange of particles?

May, or may not be thermally isolated from thesurroundings.

constraint on exchange of heat? If insulated, system is called adiabatic.

note the absence of a notion of time result: cannot discuss rates.


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Thermodynamic State of a System

At equilibrium, system assumed to occupy one of a set of abstract, thermodynamic states. no precise meaning away from equilibrium.

Each state unique; defined by the values of state variables: e.g.: two out of three of {temperature, pressure, volume}

plus the masses and identities of component chemical substances.

System state determines system properties:

provides a complete (macroscopic) specification of the system. Such properties will be of two types:

extensive properties require specification of amount of substance; e.g., volume, total energy.

intensive properties independent of system size. e.g., density, viscosityrequire less information.


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Changes of System State

Thermodynamics is concerned with: equilibrium distributions over accessible states. mean values of the properties of this distribution.

Also concerned with changes b/w equilibrium states. i.e. , system traversal from an initial to a final state. focus: resulting change in some macroscopic system property.

A change of state may be either reversible or irreversible. Reversible changes take a path near to equilibrium.

resulting system changes can be accurately predicted.

Irreversible changes deviate from equilibrium. resulting system changes cannot be accurately predicted.

however, generally bounded by equilibrium values.


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The Transfer of Energy:Heat vs. Work

Heat, q energy transferred into a system due to a temperature difference

between the system and its surroundings. positive heat = heat absorbed by a system (+ energy flows in).

Work, w any other exchange of energy between a system and surroundings.

e.g.: volume change vs. external pressure, electrical work, etc.

positive work = work done by a system (+ energy flows out ). Neither q or w is a system property or state variable:

have meaning only with respect to some energy transfer.

Why differentiate between the two?

transferring heat transfers entropywhile doing work does not.


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Internal Energy

Internal Energy, E: the total energy within the system.

includes energies which change during a chemical process: kinetic energy: translational, vibrational, and rotational; energy of chemical bonding (covalent bonds); energy of non-bonding interactions b/w molecules .

E is a function of system state (a state variable). specifying a systems state fixes its E.

how the system attained the state irrelevant.

E values defined relative to an arbitrary standard state: which has been defined to have zero energy.


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Enthalpy, H

System Enthalpy is the sum of: the internal energy (E).

the product of system volume, and the external pressureexerted on our system (PV).H = E + PV

H is also a function of system state (state variable).

each state characterized by a unique value of H. changes between states A and B yield a well-defined

enthalpy change, H = H B-HA. H independent of the intermediate set of conformations.

also referred to as the path.


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The Heat Capacity

The heat capacity, C of a given substance: the heat input required to raise the temperature of a given mass of

a substance by 1 :

C = mc = Q/ T. depends on m, the mass of the substance (an extrinsic property). c = specific heat (heat capacity/graman intrinsic property).

C refers to the molar heat capacity (an intrinsic property).

C depends on the manner in which heating occurs: constant volume: C = C v.

constant pressure: C = C p.

in general, C v


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First Law of Thermodynamics

The 1 st Law is a statement of Conservation of Energy in terms of heat, work, and internal energy. For a change in state, the 1 st Law states:

E = q-w For small changes, we use differential symbols for q,w: dE = dq dw

slashes indicate that q and w do depend on path. In this treatment, slashes will generally be omitted. E is a state variable, so that dE does not depend on path.

dE is an exact differential.

The 1 st Law completely general. does not rely on assumptions, such as reversibility (equilibrium).


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PdV Work

PdV work refers to: work involving only a change of system volume, dV in response to an external pressure, P.

If only PdV work is done, dw = PdV, and the 1 st Law becomes:

dE = dq - PdV Similarly we can write for dH:

dH = d(E + PV)= dE + PdV + VdP

= (dq - dw )+ PdV + VdP

= dq + VdP


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Measuring

E These two expressions help to relate system variable

changes to a measurable quantity: a change in temperature, T .

For state changes at constant volume:

dE = dq pdV = dq heat absorbed at constant volume, q V measures

dE.

for a finite change, E = q V . Since C = q/ T, we have:

E = C v T = mc v T.

A constant-volume calorimeter: E = q V accompanying a chemical reaction, measured by T.


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Measuring H

For state changes at constant pressure: dH = dq + VdP = dq

for a finite change,

H = q p. heat absorbed at constant pressure, q P

measures H.

Since C = q/ T, we have:

H = C p

T = mc p

T. facilitates measuring H b/w 2 states:

choose a path at constant pressure and measure T.

A constant-pressure calorimeter shown at right: H accompanying a chemical reaction measured by T.


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The Difference between E and H

In general, E and H will differ slightly for the samechange of state. in biological systems, we generally have ~ constant pressure. consider an infinitesimal change of state, at constant pressure

dH = dE + PdV + VdP = dE + PdV .

The difference is thus the work done to enforce the change in systemvolume, dV.

In biological systems , volume change generally small since most occur in liquid or solid (rather than gas).

Essentially incompressible.

thus, we often neglect the difference b/w E and H.


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Molecular Interpretation

Together, the 1 st Law and the concept of a state variable: allow some very general conclusions. we could now apply the 1 st Law to many different conditions

combinations of constant P, V, and T.

Our focus however, is a molecular interpretation: how do E, H relate to the behavior of atoms and molecules?

In particular, if we add energy to the system can we predict where has the energy has gone?

into kinetic energy.translation and rotation. for a complex biopolymer: vibrational energy, intermolecular

interactions.

How is the energy distributed ?


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Statistical Thermodynamics The Ensemble

Concern: behavior of a large number of molecules. or, the mean behavior of 1 particle over a long time.

We therefore take a statistical view: we consider an ensemble at equilibrium:

O(10 23) identically prepared copies of our biopolymer. we then investigate the ensemble distribution:

how members are distributed over particle states. each overall distribution = 1 system state.

Here, thermodynamic state: refers to the state of our system of N particles.

not equal to particle state. e.g., system energy is the sum over all particle energies .

E =

i E i.


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The Most Probable Distribution

Generally, our system will be constrained constraints will vary, depending our system of interest. For the microcanonical ensemble , we have:

constant total energy (E). constant total number of particles (N).

Each distribution which satisfies our constraints is called accessible.

What is the most probable distribution , at equilibrium?

this system state will be useful: it will help characterize mean particle behavior. mean properties = ensemble averages.

averages taken over all particle states correspond to macroscopic observables.


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The Fundamental Assumption

Fundamental Assumption of Statistical Thermodynamics: For a particle in an isolated system, each particle state equally likely

isolated means constant E, N.

Implied: Each elementary distribution equally likely. such an elementary distribution referred to as a system microstate. in a system microstate, each particle assigned a distinct energy.

Each system state is a macroscopic state:

only the total number of particles in each particle-state is specified. each system state = a sum over a set of equivalent microstates.

The most probable system state: ensemble distribution with the largest number of distinct ways to arrange

the particles over the particle states.


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Example

Consider a simple system: 6 particles (N = 6);

distributed over 6 quantum states, i = {1,,6}. here, particle state i has energy, i = i*

where is the energy of the ground state.

Let the total (system) energy be constrained: E = 10 .

Many distributions consistent with this constraint. let n i denote the number of particles in particle state i;

system state then an ordered list: (n 1, n 2, n 3, n 4, n 5, n 6).

three such distributions are shown on the next slide


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Example (cont.)

Each can be achieved in a number of particlearrangements. State a (5, 0, 0, 0, 1, 0) can be achieved in 6 ways. State b (3, 2, 1, 0, 0, 0) can be achieved in 60 ways.

State c (2, 4, 0, 0, 0, 0) can be achieved in 15 ways.


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System State Degeneracy, W

The number of ways, W of arranging N particles n1 in one group;

n2

in a second group, etc.

is given by

The most probable distribution will maximize W. subject to our constraints:

N = constant; E = constant.


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The Boltzmann Distribution

For a large number of particles, W maximized when:

ni = n 1exp[ - ( i- 1 )];

ni = number of particles in particle state, i.

i = energy of particle state, i.

state i = 1 is the particle state of lowest energy (ground state).

Referred to as the Boltzmann distribution : the most probable distribution at constant system N, E.

for a system of particles at equilibrium.

constant given by, = 1/(k BT).

k B = R/N A is the Boltzmann constant. T = absolute tem erature.


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Particle State Degeneracy, g i

The Boltzmann distribution, as written describes: a distribution of the N particles of our system over particle states

Assumption: each particle state has a unique energy.

however, many particle states may have the same energy; these are energetically equivalentand thus, can be grouped.

more appropriate: distribution over particle energy levels.

Lets redefine i to refer to the i th energy level. So, for the new degenerate levels, we have n i = g ini,

Substituting into the Boltzmann distribution we get:

ni /g i= (n1 /g 1)exp[-( i- 1) /k BT ];

Now, n i = tot. number of particles in energy level i (we re-defined ni).

g i = degeneracy of the ith

energy level. For convenience we now dro the rimes on n from here on


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The Partition Function Summing over all energy levels, i yields the total particle

number:

N = i ni = (n 1/g1) i g i exp[-( i- 1)/k BT]

Then the probability that a randomly selected particle will befound in state i is:

Here the denominator is the partition function:

very useful for calculating system average quantities.


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Conclusion

In this Lecture, we have discussed: The concepts of system and equilibrium state

Identified several state variables ( e.g ., P, V, T, E, and H),

Distinguished between reversible and irreversible changes of state. The relationship between heat (q), work (w), and internal energy (E):

The 1 st Law of Thermodynamics.

The concept of an Ensemble, And identified the Boltzmann distribution as the most probable distribution of

ensemble members, at equilibrium.

In Lecture 7, we will: re-express system degeneracy in terms of a macroscopic quantity:

the Entropy, S.

This concept will enable us to formulate the 2 nd Law of Thermodynamics, to identify the conditions for equilibrium, and to convert the Boltzmann distribution to a more familiar form.


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Midterm Test Postponed

The Midterm is postponed until next class: Thursday, Dec. 20 th (next class)

Same material (does not include todays lecture material)

The test format has also changed: Mainly a short answer format

Lecture 7' - Thermodynamics and the 1st Law

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