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Colligative propertiesAll about the properties of any solvent whosemole fraction (X) is less than = 1

i.e., diluted solvents

vapor pressure is lowermelting point is lowerboiling point is higher

osmotic pressure is greater thanatmospheric pressure

Use this for Problem 6 of HW # 4 Due next Friday

3

,

2 3

1 ......2 6 !

m AVyRT

ny

a e e

y y ya e yn

−Π

= =

= = + + +

Perhaps the most important power series on the Planet!

Constantly used in a process called LINEARIZATIONwhen y is small compared to 1, e.g., if y = - 0.02

yea y +≈= 1

The “nice” thing about osmotic pressure is that you get activity directlyfrom a Boltzmann-like (or equilibrium constant-like expression)

this is known as “linearization”

exponential power series

4

a =

y = -πVA/RT

Fact:

5

25 atm is huge (850 ft column of water)p=F/areaif area = 1 m2

1 m sq r= 1.5 ft 25 bar is 275 tonsWorld’s largest bull dozer only 145 tons

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9Printed in book, The Second Law byHenry Bent

Motions of Biological Molecules and Rates of Chemical Reactions Chapters 8 and 9

361-18 Lec 28Mon 22oct18

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http://www.musicofnature.org/home/american_toad_dream/

Henry Bent In human thermodynamics education, Henry Albert Bent (c.1927-) was is an American physical chemist noted for his 1965 book The Second Law, for his 1971 article “Haste Makes Waste: Pollution and Entropy”, in which he attempts at a connection between the maintenance of the environment and entropy, and for his thermodynamics workshops, where he attempted to educate students on how to live ethically according to the laws of thermodynamics. [1]

Specifically, instead of a national energy policy, Bent states:

"What we need is a personal entropy ethic."

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Chapters 2-4,6,7: Thermodynamics(equilibrium states)

All about what is possible or not(never whether or how how fast)

Chapters 8-10: Study rate at which physical and chemical processes that are NOT AT EQILIBRIUM proceed to equilibrium

Chapter 9: Kinetics: rates of chemical reactions

Chapter 8: About processes that depend on how fast molecules move:DiffusionSedimentationElectrophoresis first we examine velocities of molecules

Derive: kinetic energy of translation of an ideal gas = n(3/2) RT

Pressure =p = average force per unit areaof a billion billion violent collisionshttp://en.wikipedia.org/wiki/ Kinetic_theory_of_gases

Volume = Area x length = A x c

z Force = Fz = mass x acceleration

Fz = m dvz/dt = m∆vz/ ∆t

Velocity in z direction = vz

∆vz = change of vz during collision= 2 vz∆t = length/ velocity = 2c/ vz

<Fz >= m 2 <vz2 > / 2c = m <vz

2>/c (=2KE/c)

vz -vz

A

a b

c

z

Area = abVolume = abc

pressure = force/Area: p = <Fz >/ A = m <vz2>/ abc

= <vz2>/V (because abc = V)

pressure = force/Area: p = <Fz >/ A = m <vz2>/ abc

abc = V

½ m <vz2> = Uz (the kinetic energy due to m and vz )

therefore p = 2Uz/V multiplying by V gives: pV = 2Uz = (2/3) U (where U = total kinetic energy)(this is because U = Ux + Uy + Uz and that = Ux = Uy = Uz by symmetry)

UZ = 1/3 U (the total kinetic energy)2Uz =(2/3 )U

but pV = nRT (the ideal gas law established by experiments at low pressure)(n is the number of mols of ideal gas, T is temperature in Kelvin, and R is the universal gas constant)

giving pV = nRT = (2/3) U or U = (3/2) nRTDividing both sides by n gives the molar translational kinetic energy:

Um = (3/2) RT

Um = (3/2) RT = ½ M <v2> (where M= molecular wt.)

= Average Molar Translational Kinetic Energy

Now calculate the square root of the averageof the square of the speed.

(3/2 RT/ ½ M )1/2 = (<v2>)1/2 = vrms

What is another word for average?

The root mean square speed = vrms = (3RT/M)1/2

Reading in Chapter 5, p. 155-156

vrms is the only speed you will need in this class for estimating

Example: N2 (gas) at 300 K½ mass times velocity squared

The root mean square speed = vrms = (3RT/M)1/2

1/2 1/22 1/2

rms

3RT 3 8.3145 300v0.028

v 517m/s 1700 ft/sM

× × < > = =

= =

This very close to the speed of sound!

Sound travels only by the motion of molecules

2 31 v RT2 2M < >=

www.cs.toronto.edu/~gpenn/csc401/soundASR.pdf

Traveling waves of adiabatic compression and expansionso also a temperature wave

Depends on how fast themolecules are moving.

NO SOUND in VACUUM

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

We now turn to considering the distributution of speeds

The Maxwell-Boltzmann distribution

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Boltzmann Distribution

RTMgh

Tkmgh

TkU

eeegg

PP

BB

)()()(

1

2

1

2∆−∆−∆−

===

We have seen how the Boltzmann equation allowed us to calculate the ratio of the ratio of numbers of molecules at equilibrium at different elevations in the atmosphere using the potential energy, mgh

Recall Gibbs, Boltzmann said, independently in ~1880

where, g2/g1 is the ratio of number of microstates available in the two states.i.e., same as we have called W2/W1, where S = kBlnW also given by Boltzmann.

g = W = “degeneracy” of the state was nearly independent of elevation

Units ????

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Maxwell-Boltzmann Distribution

Tkm

BePP )v(

2

221

v)0()v( −

∝

is about the probability to find a molecule in different kinetic energy levels, ½ mv1

2 , and ½ mv22 , where we let v1 = 0

here the “degeneracy” of the state is proportional to the square of the speed, v2.More often written as a histogram, or frequency distribution:

RTM

e)v(

2

221

v−

∝

RTM

eN)v(

2

221

vv)(−

∝where N(v) is understood to be the number of molecules with speed lying between v and v + dv

i.e., a histogram

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21

=1200 K

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Next we are going to get some ideas about some things you might be curious about that lead to:estimates of rates of diffusion and rates of chemical reactions

1. how many collisions per second a molecule suffers = z (little z)2. how many total collisions per second happen in a liter of gas =Z (big Z)3. how FAR a molecule travels between collisions

The text has formulas for 3 kinds of “average speeds” used in the expressionsfor these 3 things, BUT THEY ARE ALL ESTIMATES.Why? Because they are all the collisions of marbles flying around andcolliding!

In this course, THESE ARE ALL the SAME! We will always use vrms

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