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Chapter 8 Student Medley

Biological PhysicsNelson

Updated 1st Edition

Slide 1-1

Chemical Forces & Self Assembly

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Sections

• 8.1 Chemical Potential

• 8.2 Chemical Reactions

• 8.3 Dissociation

• 8.4 Self-assembly of amphiphiles

• 8.5 and 8.6 will be extra reading

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CHEMICAL POTENTIAL

8.1

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Introduction

• Today’s topic is important not just in biology, but

also in saving the planet

• Think fuel cells (the opposite of electrolysis) the

Gibbs Free Energy, ΔG can lead to an efficiency

of 83% (far better than any motor engine)

Ref: D. Schroeder, Thermal Physics

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8.1 Chemical Potential

• Biological question: How can a molecular

machine, sitting in the middle of a well-mixed

solution, extract useful work? Doesn’t it need to

sit at the boundary between chambers of

different temperature, pressure or concentration,

like a heat engine, turbine, or osmotic cell?

• Physical idea: Even a well-mixed solution can

contain many different molecular species, at far

from-equilibrium concentrations. The deviation

from equilibrium generates a chemical force.

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8.1 Chemical potential ...

• How do we include far from equilibrium particle

concentrations?

– Generalize the free energy F to include a

chemical potential (8.1)

– more on this definition, but assuming so (8.2)

• Just like TA=TB for 2 systems at equilibrium

temperature

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Your Turn 8A

• First step: show that dS/dN for fixed 𝐸𝑘𝑖𝑛 is:-

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The chemical potential (HOMEWORK)

• Your Turn 8A gives

but we want if for dS/dN|E, where E = Ekin + εN

• Now show that dS/dN|E=dS/dN|Ekin-εdS/dEkin|Nand use this to derive

or

where we define

and N/Ekin=3/2kBT with c=N/V

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Your Turn 8B

• Show that in equilibrium:

𝑃𝑗 = 𝑍−1𝑒(−𝐸𝑗+𝜇𝑁𝑗)/𝑘𝐵𝑇 ,

𝑍 = 𝑗 𝑒(−𝐸𝑗+𝜇𝑁𝑗)/𝑘𝐵𝑇 is the grand partition function

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CHEMICAL REACTIONS

8.2

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8.2.1 Chemical equilibrium occurs when

chemical forces balance

• Spontaneous transitions between the two states

are rare.

• Energy conservation

• Equilibrium

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8.2.1• Gilbert says: Of course this would never happen in real

life. Energy doesn’t spontaneously organize itself from

thermal motion to any sort of potential energy. Rocks

don’t fly out of the mud.

• Sullivan: But transformations of individual molecules

can go in either direction. If a reaction can go forward, it

can also go backward, at least once in a while. Don’t

forget our buffalo.

• Gilbert: Yes, of course. I meant the net number

converting to the low-energy state per second must be

positive.

• Sullivan: But wait! We’ve seen before how even that

isn’t necessarily true, as long as somebody pays the

disorder bill. Remember our osmotic machine; it can

draw thermal energy out of the environment to lift a

weight.

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8.2.1

• Allowing conversions between the isomers is like

connecting two tanks of equal volume but with

different numbers of gas molecules.

• The energy to do that work came from the

thermal energy of the environment, but the

conversion from thermal to mechanical energy

was paid for by the increase of disorder as the

system equilibrated.

• Chemical equilibrium is the point where the

chemical forces balance.

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8.2.1 Chemical equilibrium occurs when


• Chemical equilibrium is the point where


• In general for mechanical/electrical/chemical

forces acting on a system then equilibrium

occurs when all are ZERO

• This is similar to the discussion in Ch. 6 Eq.

(6.24)

where in this case c2/c1=e-ΔE/kBT and ΔE=0

implies that two dilute solutions have μ1=μ2

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Your Turn 8C: Chemical equilibrium is

where chemical forces balance (Sec. 6.5.1)

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Recall:- 6.5.1 Fixed-volume case

• ‘a’ is not thermally isolated, ‘B’ is a block of steel

at temperature T

• ‘B’ is unaffected by ‘a’

• ‘a’+’B’ is isolated from rest of the world

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8.2.2 ΔG gives a universal criterion for the

direction of a chemical reaction

• 1. The equilibrium condition μ2 = μ1 is just a

restatement of the Second Law

• 2. Interconversions between two isomers are

interesting, but there’s a lot more to chemistry

than that.

• 3. It doesn't matter at all what happens inside

the “phone booth”.

• Our result for equilibrium holds even for

spontaneous reactions in solution, as long as

they are slow enough that we have well-defined

initial concentrations 𝑐1 and 𝑐2.

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Burning Hydrogen

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• Equilibrium is situation where world’s entropy:

Δ𝑆𝑡𝑜𝑡 = −Δ𝐺

𝑇= (2𝜇𝐻20 − 2𝜇𝐻2 − 𝜇𝑂2)/𝑇 = 0

• Must be no change in Δ𝑆𝑡𝑜𝑡 if reaction takes one

step to the left (or right) & using Eq. 8.3 implies

• And lumping 𝐾𝑒𝑞 ≡ 𝑒−(2𝜇𝐻20−2𝜇𝐻2−𝜇𝑂2)/𝑘𝐵𝑇 we

then find

𝐶𝐻2𝑂2

𝐶𝐻22(𝐶𝑂)

=𝐾𝑒𝑞

𝑐𝑜

Burning Hydrogen?

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Burning more hydrogen

• The above is just another way of phrasing the

2nd Law of thermodynamics:

• The condition for equilibrium is that a certain

combination of the concentrations (the reaction

quotient) must equal a concentration-

independent constant:

– the equilibrium constant divided by the

reference concentration

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8.2.2 Your Turn 8D:- 𝑲𝒆𝒒

• Using result from Your Turn 8A. Show that

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General reactions?

• A chemical reaction will run forward if the quantity ΔG is

a negative number, or backward if it’s positive.

• ΔG the net chemical force driving the reaction.

• Equilibrium is the situation where a reaction makes no

net progress in either direction, or ΔG = 0.

• Rephrase this condition by separating ΔG into its

concentration-independent part

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8.2.2 Your Turn 8E/F

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Biochemical conventions

• Biochemists make some special exceptions to

the convention that c0=1 M.

• In a dilute aqueous solution of any solute, the

concentration of water is always about 55M.

• Then instead of [H2O], Equation 8.16 has the

factor cH2O/c0,H2O = cH2O/(55M) ≈ 1.

• With this convention we can just omit this factor

altogether from the Mass Action Rule, even if

water participates in the reaction.

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Continued ...

• Similarly, when a reaction involves hydrogen

ions (protons, or H+), we choose the

concentration to be 10−7 M.

• In any case, the notation [Z] will always refer to

cZ/1M

• When we use the special conventions above, we

denote the corresponding quantities as ∆G′0,

and K’eq

• (“standard transformed free energy change” and

“transformed equilibrium constant”)

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More Biochem*

• When any foreign molecule is introduced into a

solvent like water, it disturbs the structure of the

neighboring solvent molecules, becoming

effectively a larger, somewhat blurry object,

loosely called a hydrated ion.

• When we speak of an ion like Na+, we mean this

whole complex:-

– Hydronium ion, H3O+

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Disturbing the Peace

• Another form of the second law:

• Le Chatelier’s Principle:

When one or more chemical reactions can occur

at rates fast enough to equilibrate on the time

scale of the experiment, equilibrium also implies

relations between the various 𝜇𝛼, namely, one

Mass Action rule (Equation 8.17) for each

relevant reaction

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8.2.3 Kinect Interpretation of Complex

Equilibria?

• X2+Y2 = 2XY

• It seems that at low concentration, the rate r+ of

the forward reaction (reactions per time) should

also be proportional to cX2 cY2

• Setting k+=k- , implies

• It looks good, but to good to be true…

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• These rates are often totally wrong!

• Sometimes there is no effect when you double

the concentration of Y2, sometimes doubling

concentration of X2 quadruples the rate.

• This is called:

zeroth order in Y2, and second order in X2.

Which means the rate is proportional to:

(CY2)0(CX2)

2

We were too naïve.

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• What’s going on?

• The slow step is called a bottleneck, or rate-

limiting process.

• The key point is that in equilibrium, each

elementary reaction in Eqn. 8.21 must

separately be in equilibrium.

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We don’t care or even need a phone booth

• Multiplying these three equations together

reproduces the usual Mass Action Rule for the

overall reaction, with Keq = Keq1 Keq2 Keq3:

• The details of the intermediate steps in a

reaction are immaterial for its overall equilibrium.

• Equilibrium doesn’t care what happens inside

the phone booth.(8.2.2)

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8.2.4 The primordial soup was not in

chemical equilibrium*

• Long, long ago…

• The early Earth was barren. There was plenty of

carbon, hydrogen, nitrogen, oxygen (though not

free in the atmosphere as it is today),

phosphorus, and sulfur.

• in a mixture of atoms at atmospheric pressure,

with overall proportions C:H:N:O=2:10:1:8

similar to our bodies’. To help form high-energy

molecules, let’s optimistically assume a

temperature of 500◦C. Mostly we get familiar

low-energy, low-complexity molecules H2O,

CO2, N2, and CH4.

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8.2.4

• Then molecular hydrogen comes in at a mole

fraction of about 1%, acetic acid at 10−10, and so

on. The first really interesting biomolecule on the

list is lactic acid, at an equilibrium mole fraction

of 10−24! Pyruvic acid is even farther down the

list, and so on.

• Eqn 8.17 is a very rapidly decreasing function.

The concentrations of biomolecules in the

biosphere today are nowhere near equilibrium.

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8.2.4

• Biomolecules must be produced by the

transduction of some abundant source of free

energy. Ultimately this source is the Sun .

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DISSOCIATION

8.3

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8.3.1 Ionic and partially ionic bonds

dissociate readily in water

• Electronegativity

– An attraction of a free electron to electrically

neutral atom to have negative charge

– Ion has lower internal energy than neutral

• Ionic bond:

– bond between two ions by transferring

electron

– Stays by electrostatic attraction energy

• E=qV, and

where r is the diameter of an ion

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8.3.1 Ionic and partially ionic bonds

dissociate readily in water

• But it can dissolve in water

– When an ion pair separates, entropy

increases > free energy cost

• Polar molecules: soluble in water too

– Hydrogen bonding in water

– Dipole-dipole interaction (electrostatic

interactions of permanent dipoles in

molecules), alcohol….etc

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8.3.2 The strengths of acids and bases

reflect their dissociation equilibrium

constants

• Water

• Dissociation of water does cost more free

energy than that of ionic bond (ex. NaCl)

• In pure water

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8.3.2 The strengths of acids and bases

reflect their dissociation equilibrium

constants

• Definition:

• If pH = 7, neutral

• If pH > 7, base

• If pH < 7, acid

• Equal amount of HCl and NaOH

• Mix, neutral, saltwater!!!!!!!!!!!!

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8.3.3 The charge on a proteins varies with

its environment

• Each protein subunit (amino acids [except

proline]) provides the same backbone structure

to the protein: -NH-CH-CO-

• The side chains of the amino acids are different.

• Side chains interact with each other and water to

form folded structure

• This structure define the protein function

• In short proteins are extremely complicated.

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8.3.3

• Some amino acids liberate H+, while others

attract H+

• The probability for α be protonated depends on

Keq,alpha and on the pH of the surrounding fluid,

this probability is denoted to Palpha

• Protonated and deprotonated subunits will work

on each other, but not individually

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8.3.4

• At a certain pH a proteins charge will be neutral,

this is called the isoelectric point

• This can be used to separate proteins from each

other in column chromatography or

electrophoresis

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Self assembly of amphiphiles

8.4

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The Invisible Hand

• Without any help from the “creation”,

appropriate molecules come together, following

chemical forces to make functioning structures.

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Recall cell membranes

• Organisms undergo metabolism,

maintain homeostasis, possess a

capacity to grow, respond to stimuli,

reproduce and, through natural

selection, adapt to their environment

in successive generations.

http://en.wikipedia.org/wiki/Homeostasis

http://en.wikipedia.org/wiki/Cell_growth

http://en.wikipedia.org/wiki/Stimulus_(physiology)

http://en.wikipedia.org/wiki/Reproduce

http://en.wikipedia.org/wiki/Natural_selection

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The building blocks of micelles

• Amphiphilic molecules reduce the oil-water

interface tension

• surfactant, emulsifier, detergents

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Micelles are entropically favored.

• A mixture of oil droplets in water is not a self-

assembly structure. It's merely a result of

hydrophobic interactions.

• Surfactant molecules however can go with

another option to save their entropic cost.

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Micelles

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The “construction” of micelles

• Suppose the soap is potassium oleate.

• K+ → osmotic pressure

oleate molecule → thermodynamic equilibrium

between

individual molecule ↔ aggregates of N ions

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Osmotic Pressure Contribution

; when half the monomers are free and half are

assembled into micelles.

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Now recall, osmotic pressure:

…

…

…

…

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• At the critical micelle concentration (CMC) – the

ratio of independently moving objects to all ions

droppes sharply

• See next slide

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Thermodynamic Contributions

• Not only the concentration (CMC), but also

temperatures (CMT) contributes the formation,

hence the proof of entropic contribution

dG = dH-TdS

• Concentration less than CMC:

monomers ↔ micelles [entropically disfavored]

• Concentration greater than CMC

concentrated micelles ↔ diluted micelles [entropically favored]

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Next time

• Chapter 11-:

– Subsection presentations by students (TBA)

• Extra reading

– 8.5 Curve fitting

– 8.6 Self-assembly in cells

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