Chapter 8 Student Medley

Biological PhysicsNelson

Updated 1st Edition

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Chemical Forces & Self Assembly

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Proposed sections

• 8.1 Chemical Potential:- Wade

• 8.2 Chemical Reactions:- Siyu & Zhijin

• 8.3 Dissociation:- Daniel & May

• 8.4 Self-assembly of amphiphiles (what is an amphiphile please as well):- Dennis & Taku

• 8.5 and 6 will be homework (extra reading).

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CHEMICAL POTENTIAL8.1

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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

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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|N

and use this to derive

or

where we defineand N/Ekin=3/2kBT with c=N/V

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Your Turn 8B

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CHEMICAL REACTIONS8.2

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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 forces balance• Chemical equilibrium is the point where

chemical forces balance• 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 the point where chemical forces balance

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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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Burning Hydrogen

• Equilibrium is the situation where the world’s entropy Stot is a maximum.

• There must be no change in Stot if the reaction takes one step to the left (or right).

•• Reaction quotient• 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

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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 tothe 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 their standard concentration to be 10−7 M.

• In any case, the notation [Z] will always refer tocZ/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 introduced into a solvent like water 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 second law:

• Le Chatelier’s Principle:• The chemical reaction will run in the direction

that partially undoes the change we made.

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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-,

• It looks good!

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• No! those are often totally wrong!• Sometimes there is no effect when you double

the concentration of Y2, sometimes doublingconcentration 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 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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• 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 insidethe phone booth.(8.2.2)

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8.2.4 The primordial soup was not inchemical 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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Next time

• 8.3 Dissociation

• 8.4 Self-Assembly