Ludwig Boltzmann BZ-1 - Gustavus Adolphus...

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BZ-1 Ludwig Boltzmann Boltzmann is well-known as a chemical theorist – he worked on topics ranging from the kinetic theory of gases to the development of statistical thermodynamics… at a time when the atomic nature of matter was not understood or accepted! His most famous equation, S = k B ln W, is printed on his tombstone and is a key expression to understanding entropy in terms of statistical thermodynamics. However, the equation and Boltzmann’s work in general was so highly criticized that he committed suicide without ever knowing the impact he had on modern chemistry. k B = 1.38 × 10 -23 J·K -1 or 0.695 cm -1 ·K -1 k B N A = R

Transcript of Ludwig Boltzmann BZ-1 - Gustavus Adolphus...

Page 1: Ludwig Boltzmann BZ-1 - Gustavus Adolphus Collegehomepages.gac.edu/~anienow/CHE-371a/Lectures/Boltzmann...Ludwig Boltzmann BZ-1 Boltzmann is well-known as a chemical theorist – he

BZ-1Ludwig Boltzmann

Boltzmann is well-known as a chemical theorist – he worked on topics ranging from the kinetic theory of gases to the development of statistical thermodynamics… at a time when the atomic nature of matter was not understood or accepted!

His most famous equation, S = kB ln W, is printed on his tombstone and is a key expression to understanding entropy in terms of statistical thermodynamics. However, the equation and Boltzmann’s work in general was so highly criticized that he committed suicide without ever knowing the impact he had on modern chemistry.

kB = 1.38 × 10-23 J·K-1 or 0.695 cm-1·K-1 kB NA = R

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BZ-2What is statistical thermodynamics?

• A way to connect the microscopic and macroscopic

• In principle, if enough information about the molecules in a system are known, the behavior of the system could be calculated from QM

• Statistical mechanics overcomes this impossible problem by predicting the most probable behavior of a large collection (ensemble) of molecules

• Summary: In statistical thermodynamics, the averages of molecular properties are related to thermodynamic properties (such as pressure and temperature) of macroscopic systems.

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BZ-3Review from QM Slides

1) Energy states for molecules are quantized.2) Solve Schrödinger equation to find allowed states. 3) Given some T, how are these states populated?

How does temperature affect the population of states?

What is pj

if Ej

is big?

∑−

=

i

jTBk

iE

TBkjE

e

ep

Probability that a randomly chosen system will be in state j with Ej

Partition function

Boltzmann Distribution

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BZ-4Consider a 330 mL hot chocolate

There are about 1024 molecules (N) in this volume (V) and we know that they all interact with each other.

But there must be a set of allowed macroscopic energies that come from the allowed electronic, vibrational, rotational, and translational energies of the individual molecules (recall QM?!). This set of energies will be a function of N and V:

{Ej

(N,V)}

What is the probability that your

hot chocolate will be in state j

with energy Ej

?

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BZ-5Consider a tray of 106 hot chocolates!

jEj Cea β−=

The tray of hot chocolates is kept in thermal equilibrium with a heat reservoir. Each of the hot chocolates has the same N, V, and T. However, they can be in different quantum states. This collection of hot chocolates is an ensemble.

Total # of hot chocolates = A

# of hot chocolates in state j

(with Ej

) = aj

We want to know # of hot chocolates in each state! So what is the form of aj

?

Now… what are C

and β?

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BZ-6C, β, and Q …

?==∑ ∑ −

j j

Ej

jeCa β ?=CTo find C, sum both sides of equation:

∑ −

==

j

E

E

jj

j

j

eep

Aa

β

β pj

is the probability that a randomly chosen hot chocolate will be in quantum state j

TkB

1=β∑ −=

j

E jeVNQ ββ ),,(We will show later that:

Let the denominator = Q

What is Q

called?Ex-Boltz1

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BZ-7What is the partition function?1. The denominator in the Boltzmann probability (i.e., the

normalization constant).2. A measure of the extent to which the particles are able

to escape from the ground state (i.e., a measure of population in excited states).

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BZ-8One more slide on Q!

∑ ∑−

==j

VNEj

jj VNQeVNE

EpEj

),,(),( ),(

β

β

VN

QE,

ln⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−=

β

One can express all macroscopic properties of a system in terms of Q!

VNB T

QTkE,

2 ln⎟⎠⎞

⎜⎝⎛

∂∂

−=

Postulate – the averaged ensemble energy is the observed energy of a given system. That is:

See Math Ch B

In terms only of Q

or

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BZ-9Ideal gas Law

!)],([),,(

NVqVNQ

Nββ =

Vh

mVq2/3

2

2),( ⎟⎟⎠

⎞⎜⎜⎝

⎛=

βπβ

The partition function, Q, for a monoatomic ideal gas can be written as:

q partition function for 1 atomm

mass of the atom

h

Planck’s constantN

number of atoms

where

From this information about the partition function, we can calculate the macroscopic properties (e.g., energy) for an ensemble (e.g., a large collection of monoatomic gas molecules). The derivation of Q will be discussed soon…

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BZ-10Find <E> for an ideal gas…

VN

QE,

ln⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−=

β !2),,(

2/3

2 NV

hmVNQ

NN

⎟⎟⎠

⎞⎜⎜⎝

⎛=

βπβ

Find ln(Q) and separate out the terms involving β

Differentiate with respect to β From the kinetic theory of gases, we know the internal energy of a monoatomic ideal gas is:

U=3/2 nRT

N=nNA and R=kB NA

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BZ-11Other Macroscopic PropertiesThe result of U = <E> on the previous slide demonstrates the power of statistical thermodynamics. From microscopic

properties, we can calculate any macroscopic

quantity we want!

VVV T

ETUC

⎟⎟

⎜⎜

∂=⎟⎟

⎞⎜⎜⎝

⎛∂∂

=

Other macroscopic quantities:

β,

ln

NB V

QTkP ⎟⎠⎞

⎜⎝⎛∂∂

=

Heat capacity: the energy required to raise the temperature of a given amount of substance by 1K.

Pressure

What is for a monoatomic ideal gas?VC

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BZ-12Heat capacity

2

2

125

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

Tkh

Tkh

BV

B

B

e

eTk

hRRCυ

υ

υ

From the partition function for a diatomic gas, you can derive the constant volume heat capacity…

Experiment vs. TheoryO2

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BZ-13<P> for Monoatomic Ideal Gas

β,

ln

NB V

QTkP ⎟⎠⎞

⎜⎝⎛∂∂

=

?ln

,

=⎟⎠⎞

⎜⎝⎛∂∂

=βN

B VQTkP

!2),,(

2/3

2 NV

hmVNQ

NN

⎟⎟⎠

⎞⎜⎜⎝

⎛=

βπβ

Find ln(Q) and separate out the terms involving V

Differentiate with respect to V

EX-BZ2

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BZ-14Obviously Q

is important!!

The partition function plays a significant role in statistical thermodynamics. Q comes from an understanding of allowed energies of a given system and this comes from quantum mechanics (i.e., what states are allowed by the laws of QM).

Q

as we have been using it, with N, V, and T

fixed, refers to the partition function for the system and can be referred to as the canonical partition function and the ensemble (of hot chocolates) that we constructed is the canonical ensemble. There are other ensembles!!

Q

is all we need to get macroscopic properties, but to do so for an arbitrary system, we need the eigenvalues for the N- body Schrodinger equation… no can do. But, we can get Q

from “individual” molecular information!

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BZ-15Q

in terms of q

...)()()(),( +++= VVVVNE ck

bj

ail εεε

∑ ∑ +++−− ==l kji

E ck

bj

ail eeTVNQ

...,,

...)(),,( εεεββ

)...,(),(),(),,( TVqTVqTVqTVNQ cba=

Consider a system of distinguishable, independent

particles…

The total energy of the system is:

∑ −=i

ieTVq βε),(

NTVqTVNQ )],([),,( =

The partition function becomes:

If all particles are identical:

Depends only on individual molecules/atoms!

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BZ-16Distinguishable? Get real…NTVqTVNQ )],([),,( = is a nice result, but not very useful.

Atoms/molecules are usually indistinguishable.

jLet be an atom with energy εj and only ε1 , ε2 , and ε3 are allowed. Consider the 3 situations below...

211

2

3

1

23

233

1

2

1

12

123

1

3

2

21

Each of these situations represent the same state and should be “counted” only once (i.e., since these states are indistinguishable but are the same, there is an “over-counting”).

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BZ-17Now consider a (not so)

special case

!)],([

NTVqQ

N

=

Consider a system in which no two atoms have the same energy. In that case, you can divide by N! to account for over-counting.

How special is this case? … If the number of quantum states with energies less than ~kB

T is much greater than N, then odds are very good that no two particles will have the same energy.

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BZ-18Let me “translate” this conceptRemember translational energies vs kB T (EX-QM5)?

At room temperature, there are typically so many translational states accessible to a particle that the number of states is >> N. So Q = qN/N! holds.

Trans ~ 10-6 – 10-15 cm-1 kB T ~ 200 cm-1 at 300 K

388,496,4)!641(!6

!41=

Everybody randomly pick a lottery number. Anybody have the same?

possibilities

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BZ-19Boltzmann Statistics

18

23

2

<<⎟⎟⎠

⎞⎜⎜⎝

⎛Tmk

hVN

B

The criterion that the number of states exceeds the number of particles is:

A system where this is valid obeys Boltzmann statistics.

Favors large m, high T, and low N/V.

!)],([

NTVqQ

N

=

For a system obeying Boltzmann statistics, partition function is:

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BZ-20Boltzmann statistics

Which of these follow Boltzmann statistics? (17.1)

What characteristics cause some not to follow Boltzmann statistics?

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BZ-21Q vs q

∑ −

=

j

E

E

j j

j

eep β

β

∑ −

=

j

j j

j

ee

βε

βε

π

Remember the probability of a member of an ensemble is in quantum state j

is:

∑ −

=

j

j vibj

vibj

ee

βε

βε

π

Similarly, the probability (πj

) that a molecule is in its jth molecular energy state is:

We can further designate the probability as the probability of a molecule to be in a vibrational, rotational, translational or electronic state.

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BZ-22, Energy of a molecule

∑ ∑∑∑

===j j vib

vibj

j

j

vibj

vibj

vibj

vib

qe

ee

vibj

vibj

vibj βε

βε

βε

εεεπε

ε

TqTkq vib

Bvib

∂∂

=∂

∂−=

lnln 2

β

TqTk elec

Belec

∂∂

=ln2ε

TqTk vib

Bvib

∂∂

=ln2ε

TqTk rot

Brot

∂∂

=ln2ε

TqTk trans

Btrans

∂∂

=ln2ε

transrotvibelectot εεεεε +++=

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BZ-23Degeneracy

∑ −=statesj

jeTVq,

),( βε

Partition functions have been written so far as a summation over states. States with the same energy we call levels. The levels have a degeneracy, g.

Writing the partition function as a summation over levels can be more convenient.

∑ −=levelsj

jjegTVq

,),( βε

Terms representing a degenerate level are repeated gj

times

Terms representing a degenerate level are written once and multiplied by gj

EX-BZ3

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BZ-24Summary: Key ConceptsWe want to know the macroscopic thermodynamic properties of a system (e.g., P, Cv , U).

The ensemble average of one of these properties is equal to the observed (macroscopic) value (e.g., <E>=U).

We can get all kinds of ensemble averages from the ensemble partition function Q.

We can get Q

from molecular partition functions q

and these come directly from quantum mechanics.

...And we are going to derive these next!!

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BZ-25Summary: Key Equations

∑−

=

i

jTBk

iE

TBkjE

e

ep

TkB

1=β

∑ −=j

E jeVNQ ββ ),,(

∑ ∑−

==j

VNEj

jj VNQeVNE

EpEj

),,(),( ),(

β

β

VNB T

QTkE,

2 ln⎟⎠⎞

⎜⎝⎛

∂∂

−=

!)],([),,(

NVqVNQ

Nββ =

N=nNAR=kB NA

VVV T

ETUC

⎟⎟

⎜⎜

∂=⎟⎟

⎞⎜⎜⎝

⎛∂∂

=

β,

ln

NB V

QTkP ⎟⎠⎞

⎜⎝⎛∂∂

=

Boltzmann Distribution

Partition Function

For Boltzmann statistics…

Conversions…

Macroscopic Properties