General Physical Chemistry I

Lecture 4

Aleksey Kocherzhenko

February 5, 2015"

Last time…"

Maxwell’s distribution"

f (v) = 4⇡

✓m

2⇡kBT

◆ 32

v2 exp

✓� mv2

2kBT

◆ Maxwell’s distribution of molecular speeds  

Ø  RMS speed:!

vrms =

r3kBT

m

Ø  Mean speed:!

v =

r8kBT

⇡m

vp =

r2kBT

mØ  Most probable speed:!

T =2Ek

3kBKinetic definition of temperature"

Same form for y- and z-components of the velocity: space is isotropic"

For the x-component of the velocity"

� (vx

) =

rm

2⇡kBTexp

✓� mv2

x

2kBT

◆

All molecular speeds are possible for all temperature values, but the average speed/kinetic energy is higher @ higher T

Diffusion and effusion"

Diffusion"

Two substances (solids, liquids, or gasses) are initially separate, but mix together due to molecular motion"

Diffusion for two solids is extremely slow,for a solid in a liquid or two liquids it is much faster,for two gasses it is very fast"

We will study the diffusion process quantitatively towards the end of the semester "

Effusion"

Example: compare effusion rates for H2 and CO2"

The escape of gas moleculesfrom a closed volume through a small opening "

Graham’s law: At a given pressure and temperature, the rate of effusion of a gasis inversely proportional to the"square root of its molar mass:"

Thomas Graham"

r / 1pM

rH2 / 1pMH2

rCO2 / 1pMCO2

Same proportionalityconstant" MH MC MO

(g/mol)

MCO2 = 44.009 MH2 = 2.016) rH2

rCO2

=

sMCO2

MH2

⇡ 4.67

Lighter molecules escape faster"

Effusion"Ø  The origin of Graham’s law is that lighter molecules move faster à

they are more likely to reach the opening and escape through it sooner"

Ø  Graham's Law is only valid for perfect gases at low pressures(so that gas molecules escape through the tiny pinhole slowly)and for very small openings through which the gas escapes(so that no collisions occur as the gas molecules pass through)"

Ø  Light gasses (H2, He) leak out of containers and through diaphragms readily, due to high effusion rates"

Ø Effusion through porous barriers important in separation of uranium isotopes (“uranium enrichment”)"

M�238UF6

�

M (235UF6)⇡ 1.008

Very small ratio:"multiple effusion steps needed"

Ø  Once used to determine molar mass by comparison of effusion rate of a gas or vapor with that of a gas of known molar mass(more precise methods available now, such as mass spectrometry)"

Molecular collisions"

Molecular collisions"Mean free path, : average distance a molecule travels between collisions"�

Time of flight, : average time a molecule spends between two collisions"z�1Collision frequency, : average # of collisions made by one molecule per s"z

Speed = (distance travelled) / (time it took)"

a measure of the average velocity"

vrms =

r3kBT

m

Notice that we know this quantity: it only depends on and"m T

) If we know or , we can find the other parameter"� z

vrms =�

z�1= �z

)

Ø  Basic kinetic model: gas atoms/molecules are considered point objects!

Ø  To include collisions, make them hard spheres!"Ø  If two molecules come within distance

of each other, a collision occurs"d

Math review: Vectors"Sum and difference of two vectors:"

~v1

~v2

✓~v1

~v2✓~v = ~v1 � ~v2

~v0 = ~v1 + ~v2

Scalar product of two vectors:"~v1 · ~v2 = |~v1| |~v2| cos ✓

|~v| =p~v · ~v

Absolute value of a vector:"

�~v2(same magnitude as , but opposite direction)"

~v2

Let’s calculate the average value of the relative speed of two molecules:"Ø  Relative velocity of two molecules:"~v = ~v1 � ~v2Ø  Absolute value of relative velocity (speed):" |~v| =

p~v · ~v =

=p

(~v1 � ~v2) · (~v1 � ~v2) =p~v1 · ~v1 + ~v2 · ~v2 � 2~v1 · ~v2

|~v|2 = |~v1|2 + |~v2|2 � 2~v1 · ~v2Ø  RMS:" and are random and uncorrelated; for every velocity there is an equal and opposite velocity à this term averages out to zero"

~v2~v1

= 0= v2rms

) |v|2 = 2v2rms

Molecular collisions"

Collision cross-section, :target area presented by one molecule to another"

�

� = ⇡d2 Rudolph Clausius"

l

A molecule flying through space collides with another molecule if the latter’s center of mass lies within a cylinder of radius around the axis along which the first molecule is moving "

d

Since the rms relative speed of a molecule with respect to other molecules is (see previous slide) "vrms

p2

Average time between collisions") Vcyl = ⇡d2|{z}

�

vrms

p2 z�1

| {z }l

A collision occurs if a molecule encounters another molecule within this volume:"n Vcyl = 1# of molecules per unit volume"

Molecular collisions"

) n�vrms

p2 z�1 = 1

n Vcyl = 1# of molecules per unit volume"

Vcyl = �vrms

p2 z�1 Average time

between collisions"

We established the following conditions that determine the average time between molecular collisions in a gas:"

Relate this to pressure and temperature of a gas"

Perfect gas law:" pV = ⌫RT Amount of gas:" ⌫ =pV

RT)

) # of molecules:"N = ⌫NA =pV NA

RT

) pNA�vrms

p2

RTz�1 = 1 or" z =

pNA�vrms

p2

RT

) # of molecules per unit volume:" n =N

V=

pNA

RT

✓=

p

kBT

◆

Molecular collisions"Mean free path, : average distance a molecule travels between collisions"�

, where we have used"

z =pNA�vrms

p2

RT

� = vrmsz�1 =

RT

pNA�p2Velocity" Time

between collisions"

Ø  Why is the rms relative speed of two molecules , not ? "vrms

p2 2vrms

(The molecular collision is not head-on, the second molecule is usually moving at an angle to the first)"

Ø  Typical mean free path @ standard ambient pressure and temperature?"(Of the order of 0.1 μm)"

Ø  – smaller mean free path for larger collision cross-section "� / ��1

Ø  ( ) – smaller mean free path/more collisions at higher p � / p�1 z / p

Ø  – since , heavy molecules collide less frequently than light molecules with the same collision cross-section"z / vrms vrms / M� 1

2

Real gasses"

Molecular interactions"

Real gas: intermolecular interactions are non-negligible…!

Perfect gas: (mean free path greatly exceeds interaction distance)no intermolecular interactions; total energy = kinetic energy of molecules!

� � d

@ distances ~ several molecular diameters, molecules attract each other à this is the energy that holds together liquids and solids"

@ distances < molecular diameter, molecules repel each other à liquids and solids do not collapse to infinite density, have finite volume"

Molecular interactions affect the equation of state for real gasses"

Perfect and real gas isotherms"

Perfect gas:! pV = ⌫RT

) p =⌫RT

V

= const

(when and are constant)"

⌫ T

) p / 1

V

Real gas:!Ø  At high , looks similar to perfect

gas isotherm "T

Ø  At lower , there is a regime where the real gas can be compressed without change in "

T

p

CO2"

(Boyle’s law)"