Physical Chemistry
30. Canonical Ensemble

Ruifeng ZHOU

Faculty of Science &Institute for the Advancement of Higher Education
Hokkaido University2

1. Canonical distribution
2. Thermodynamic quantities
3. Adsorption

1. Canonical distribution

3• If a system is allowed to exchange heat with its
surroundings (closed system) the microcanonicalensemble method cannot be used.
• If the particles in the system are not independent,
the particle partition function cannot be defined
because there are no defined particle energies.
• Such system can only be treated as an entity.
• Suppose there are 𝑘𝑘 (𝑘𝑘 ≫ 1) systems with identicalproperties (e.g., volume, shape, number and typeof particles and potential energy etc.).

41. Canonical distribution• The systems form a canonical ensemble.
Canonical ensemble
𝑁𝑁, 𝑉𝑉𝑈𝑈1𝑁𝑁, 𝑉𝑉
𝑈𝑈2𝑁𝑁, 𝑉𝑉
𝑈𝑈3𝑁𝑁, 𝑉𝑉
𝑈𝑈4𝑁𝑁, 𝑉𝑉
𝑈𝑈5𝑁𝑁, 𝑉𝑉
𝑈𝑈6𝑁𝑁, 𝑉𝑉
𝑈𝑈7𝑁𝑁, 𝑉𝑉
𝑈𝑈8𝑁𝑁, 𝑉𝑉
𝑈𝑈9𝑁𝑁, 𝑉𝑉
𝑈𝑈10𝑁𝑁, 𝑉𝑉
𝑈𝑈11𝑁𝑁, 𝑉𝑉
𝑈𝑈12𝑁𝑁, 𝑉𝑉
𝑈𝑈13𝑁𝑁, 𝑉𝑉
𝑈𝑈14𝑁𝑁, 𝑉𝑉𝑈𝑈⋯

51. Canonical distribution• The systems only exchange heat with each other.
 A canonical ensemble is isolated. It is equivalent to
one system in the microcanonical ensemble.
Microcanonical ensemble
Canonicalensemble
1Canonicalensemble
2Canonicalensemble
3Canonicalensemble
4Canonicalensemble
5Canonicalensemble
6Canonicalensemble
7Canonicalensemble86

1. Canonical distribution• A system in the canonical ensemble is equivalent
to a “particle” in the microcanonical ensemble.
Particle
𝑈𝑈1Particle
𝑈𝑈2

Particle
𝑈𝑈3

Particle
𝑈𝑈4Particle
𝑈𝑈5Particle
𝑈𝑈6Particle
𝑈𝑈7

Particle
𝑈𝑈8Particle
𝑈𝑈9Particle
𝑈𝑈10Particle
𝑈𝑈11Particle
𝑈𝑈12Particle
𝑈𝑈13

Particle𝑈𝑈14Particle𝑈𝑈⋯

7

1. Canonical distribution• The systems behave like distinguishable particles.
• The theory of microcanonical ensemble can beused to study canonical ensemble.
• Suppose there are 𝑘𝑘𝑖𝑖 systems at the 𝑖𝑖 th system
state, which has internal energy 𝑈𝑈𝑖𝑖 .
𝑘𝑘 = � 𝑘𝑘𝑖𝑖𝑖𝑖

1𝑈𝑈 = � 𝑘𝑘𝑖𝑖 𝑈𝑈𝑖𝑖
𝑘𝑘𝑖𝑖

• 𝑈𝑈 is the mean internal energy, which a fixed value.

8

1. Canonical distribution• The number of accessible states is𝑘𝑘!
Γ 𝑘𝑘𝑖𝑖 =∏𝑖𝑖 𝑘𝑘𝑖𝑖 !
• According to the principle of indifference, the mostprobable distribution is the one with maximized Γ.
ln Γ = ln 𝑘𝑘! − � ln 𝑘𝑘𝑖𝑖 ! ≈ ln 𝑘𝑘! − � 𝑘𝑘𝑖𝑖 ln 𝑘𝑘𝑖𝑖 − 1𝑖𝑖

𝑖𝑖• Introduce the Lagrange multipliers, 𝛼𝛼 and 𝛽𝛽.

ℒ = ln 𝑘𝑘! − � 𝑘𝑘𝑖𝑖 ln 𝑘𝑘𝑖𝑖 − 1 − 𝛼𝛼 � 𝑘𝑘𝑖𝑖 − 𝛽𝛽 � 𝑘𝑘𝑖𝑖 𝑈𝑈𝑖𝑖
𝑖𝑖𝑖𝑖𝑖𝑖1. Canonical distribution• To get maximum ln Γ,𝜕𝜕ℒ= − ln 𝑘𝑘𝑖𝑖 − 𝛼𝛼 − 𝛽𝛽𝑈𝑈𝑖𝑖 = 0𝜕𝜕𝑘𝑘𝑖𝑖𝑘𝑘𝑖𝑖 = 𝑒𝑒 −𝛼𝛼−𝛽𝛽𝑈𝑈𝑖𝑖 = 𝐶𝐶𝑒𝑒 −𝛽𝛽𝑈𝑈𝑖𝑖
𝐶𝐶 = 𝑒𝑒 −𝛼𝛼• This is the canonical (Boltzmann) distribution.
• 𝛼𝛼 is related to the chemical potential.
1𝛽𝛽 =𝑘𝑘B 𝑇𝑇

9102. Thermodynamic quantities• Define the canonical partition function.
𝒵𝒵 ≡ � 𝑒𝑒 −𝛽𝛽𝑈𝑈𝑖𝑖
𝑖𝑖• The probability of a system at the 𝑖𝑖th state is𝑘𝑘𝑖𝑖
𝐶𝐶𝑒𝑒 −𝛽𝛽𝑈𝑈𝑖𝑖
𝑒𝑒 −𝛽𝛽𝑈𝑈𝑖𝑖
𝑓𝑓𝑖𝑖 ≡ ==−𝛽𝛽𝑈𝑈𝑖𝑖
𝑘𝑘 ∑𝑖𝑖 𝐶𝐶𝑒𝑒𝒵𝒵• The observed internal energy is the mean internalenergy.
𝜕𝜕 ln 𝒵𝒵𝑈𝑈 = � 𝑓𝑓𝑖𝑖 𝑈𝑈𝑖𝑖 = −𝜕𝜕𝛽𝛽 𝑉𝑉𝑖𝑖112. Thermodynamic quantities

𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿 + 𝛿𝛿𝛿𝛿 = � 𝑈𝑈𝑖𝑖 𝑑𝑑𝑓𝑓𝑖𝑖 + � 𝑓𝑓𝑖𝑖 𝑑𝑑𝑈𝑈𝑖𝑖
𝑖𝑖𝑖𝑖• The change of state energies is related to work.
• The change of distribution is related to heat.

• Use𝛿𝛿𝛿𝛿 = 𝑇𝑇𝑇𝑇𝑇𝑇 = � 𝑈𝑈𝑖𝑖 𝑑𝑑𝑓𝑓𝑖𝑖
𝑖𝑖

� 𝑑𝑑𝑓𝑓𝑖𝑖 = 0𝑖𝑖

122. Thermodynamic quantities

ln 𝒵𝒵
1𝑇𝑇𝑇𝑇𝑇𝑇 = � 𝑈𝑈𝑖𝑖 𝑑𝑑𝑓𝑓𝑖𝑖 +� 𝑑𝑑𝑓𝑓𝑖𝑖 = � 𝛽𝛽𝑈𝑈𝑖𝑖 + ln 𝒵𝒵 𝑑𝑑𝑓𝑓𝑖𝑖
𝛽𝛽𝛽𝛽𝑖𝑖𝑖𝑖

𝑖𝑖

1
1= − � ln 𝑓𝑓𝑖𝑖 𝑑𝑑𝑓𝑓𝑖𝑖 = −� ln 𝑓𝑓𝑖𝑖 𝑑𝑑𝑓𝑓𝑖𝑖 + � 𝑑𝑑𝑓𝑓𝑖𝑖
𝛽𝛽𝛽𝛽𝑖𝑖𝑖𝑖

1
1= − � 𝑑𝑑 𝑓𝑓𝑖𝑖 ln 𝑓𝑓𝑖𝑖 = − 𝑑𝑑 � 𝑓𝑓𝑖𝑖 ln 𝑓𝑓𝑖𝑖
𝛽𝛽𝛽𝛽𝑖𝑖𝑖𝑖• Thus, we get the Gibbs entropy.

𝑆𝑆 = −𝑘𝑘B � 𝑓𝑓𝑖𝑖 ln 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖

132. Thermodynamic quantities

𝑒𝑒 −𝛽𝛽𝑈𝑈𝑖𝑖= 𝑘𝑘B � 𝑓𝑓𝑖𝑖 ln 𝒵𝒵 + 𝛽𝛽𝑈𝑈𝑖𝑖
𝑆𝑆 = −𝑘𝑘B � 𝑓𝑓𝑖𝑖 ln𝒵𝒵𝑖𝑖
𝑖𝑖
� 𝑓𝑓𝑖𝑖 = 11𝑈𝑈
𝑖𝑖
= 𝑘𝑘B ln 𝒵𝒵 + � 𝑓𝑓𝑖𝑖 𝑈𝑈𝑖𝑖 = 𝑘𝑘B ln 𝒵𝒵 +𝑇𝑇𝑇𝑇
𝑖𝑖

𝐴𝐴 = 𝑈𝑈 − 𝑆𝑆𝑆𝑆 = −𝑘𝑘B 𝑇𝑇 ln 𝒵𝒵• Since,𝛿𝛿𝛿𝛿 = −𝑃𝑃𝑃𝑃𝑃𝑃 = � 𝑓𝑓𝑖𝑖 𝑑𝑑𝑈𝑈𝑖𝑖
𝑖𝑖

𝑑𝑑𝑈𝑈𝑖𝑖
1𝑑𝑑𝑈𝑈𝑖𝑖 1 𝜕𝜕 ln 𝒵𝒵−𝛽𝛽𝑈𝑈𝑖𝑖= − � 𝑒𝑒
=𝑃𝑃 = − � 𝑓𝑓𝑖𝑖𝑑𝑑𝑑𝑑𝒵𝒵
𝑑𝑑𝑑𝑑 𝛽𝛽 𝜕𝜕𝜕𝜕 𝑇𝑇𝑖𝑖𝑖𝑖

142. Thermodynamic quantities• Suppose the system is composed of 𝑁𝑁
distinguishable independent particles. The 𝑗𝑗th𝑗𝑗
𝑗𝑗thparticle is on the 𝑖𝑖
state with energy 𝐸𝐸𝑖𝑖 .
𝑈𝑈 𝑖𝑖 1 , ⋯ , 𝑖𝑖 𝑁𝑁

𝒵𝒵 = � ⋯ � 𝑒𝑒
𝑖𝑖 1= � 𝑒𝑒𝑖𝑖 1𝑖𝑖 𝑁𝑁−𝛽𝛽𝐸𝐸𝑖𝑖

1−𝛽𝛽 ∑𝑁𝑁
𝑗𝑗=1 𝐸𝐸𝑖𝑖⋯ � 𝑒𝑒𝑖𝑖 𝑁𝑁

𝑗𝑗−𝛽𝛽𝐸𝐸𝑖𝑖

𝑁𝑁𝑁𝑁= � 𝐸𝐸𝑖𝑖
𝑗𝑗=1𝑗𝑗𝑁𝑁= � ⋯ � � 𝑒𝑒𝑖𝑖 1𝑁𝑁𝑖𝑖 𝑁𝑁 𝑗𝑗=1= � 𝓏𝓏𝑗𝑗
𝑗𝑗=1−𝛽𝛽𝐸𝐸𝑖𝑖

𝑗𝑗152. Thermodynamic quantities• 𝓏𝓏𝑗𝑗 is the particle partition function of the 𝑗𝑗th particle.
𝓏𝓏𝑗𝑗 = � 𝑒𝑒𝐸𝐸𝑖𝑖 𝑗𝑗−𝛽𝛽𝐸𝐸𝑖𝑖

𝑗𝑗⋯𝑖𝑖 1 = 2𝑖𝑖 2 = 1𝑖𝑖 3 = 6𝑖𝑖 𝑗𝑗 = 4

2. Thermodynamic quantities

16• For a single type of particles (indistinguishable), thepermutations of particles are overcalculated.
𝑁𝑁𝓏𝓏𝐸𝐸𝒵𝒵 =𝑁𝑁!

17

2. Thermodynamic quantities• For an ideal mixture of different types of particles,
the overall partition function is𝓏𝓏𝑖𝑖 𝑁𝑁𝑖𝑖
𝒵𝒵 = � 𝒵𝒵𝑖𝑖 = �𝑁𝑁𝑖𝑖 !
𝑖𝑖𝑖𝑖• 𝒵𝒵𝑖𝑖 , is the canonical partition function of the 𝑖𝑖 thcomponent.
• 𝓏𝓏𝑖𝑖 is the particle partition function of the 𝑖𝑖thcomponent.
• 𝑁𝑁𝑖𝑖 is the number of particles of the 𝑖𝑖 th component.

18

2. Thermodynamic quantities

ln 𝒵𝒵 = � ln 𝒵𝒵𝑖𝑖 = � 𝑁𝑁𝑖𝑖 ln 𝓏𝓏𝑖𝑖 − 𝑁𝑁𝑖𝑖 ln 𝑁𝑁𝑖𝑖 + 𝑁𝑁𝑖𝑖
𝑖𝑖𝑖𝑖

• For each component,𝜕𝜕𝜕𝜕𝜕𝜕 ln 𝒵𝒵′= −𝑘𝑘B 𝑇𝑇
𝜇𝜇𝑖𝑖 =
𝜕𝜕𝑁𝑁𝑖𝑖 𝑇𝑇,𝑉𝑉
𝜕𝜕𝑁𝑁𝑖𝑖 𝑇𝑇,𝑉𝑉𝜕𝜕 𝑁𝑁𝑖𝑖 ln 𝓏𝓏𝑖𝑖 − 𝑁𝑁𝑖𝑖 ln 𝑁𝑁𝑖𝑖 + 𝑁𝑁𝑖𝑖
≈ −𝑘𝑘B 𝑇𝑇
𝜕𝜕𝑁𝑁𝑖𝑖𝓏𝓏𝑖𝑖= −𝑘𝑘B 𝑇𝑇 ln𝑁𝑁𝑖𝑖𝑇𝑇,𝑉𝑉• This result is the same as calculated usingmicrocanonical ensemble.

193. Adsorption• The adsorption of gas molecules on solid surface isan example of canonical ensemble.
• Suppose the gas is monoatomic ideal gas. Its
chemical potential is
2𝑃𝑃2𝜋𝜋ℏ
𝜇𝜇 g ′ = 𝑘𝑘B 𝑇𝑇 ln
𝑘𝑘B 𝑇𝑇 𝑘𝑘B 𝑇𝑇𝑚𝑚

32

• Suppose there are 𝓀𝓀 adsorption sites on thesurface, each accommodates at most one atom.
• There are 𝑁𝑁 atoms adsorbed on the surface.
• Adsorbed atoms vibrate on the solid surface.

203. Adsorption

𝑁𝑁𝓀𝓀

213. Adsorption

E• The energy difference between the vibration zero
point and the stationary particle in gas is 𝑜𝑜.

0𝑜𝑜𝛿𝛿

stationary particle in gas
R22

3. Adsorption• The particles have interaction!
• The canonical partition function of the adsorbed
phase is𝓀𝓀!
−𝛽𝛽𝑈𝑈
vib
𝒵𝒵vib
=𝒵𝒵 = � � 𝑒𝑒
𝑁𝑁! 𝓀𝓀 − 𝑁𝑁 !
ad vib• ∑ad is the sum over all adsorption states.
• 𝒵𝒵vib is the canonical partition function of vibration.
• ∑vib

𝒵𝒵vib = � 𝑒𝑒 −𝛽𝛽𝑈𝑈vib
vib

is the sum over all vibrational states.

3. Adsorption• 𝑈𝑈vib is the total vibrational energy.
• The vibrations of atoms are independent anddistinguishable.
𝒵𝒵vib = 𝓏𝓏vib 𝑁𝑁• 𝓏𝓏vib is the particle partition function of vibration.
𝑒𝑒 −𝛽𝛽𝛽𝛽𝓏𝓏vib =1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽• Thus,𝓀𝓀!
𝑒𝑒 −𝑁𝑁𝛽𝛽𝑜𝑜
𝒵𝒵 =𝑁𝑁! 𝓀𝓀 − 𝑁𝑁 ! 1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽 𝑁𝑁

23

3. Adsorption

24• The Helmholtz free energy of the adsorbed phase
is𝐴𝐴 = −𝑘𝑘B 𝑇𝑇 ln 𝒵𝒵≈ −𝑘𝑘B 𝑇𝑇�𝓀𝓀 ln 𝓀𝓀 − 𝑁𝑁 ln 𝑁𝑁 − 𝓀𝓀 − 𝑁𝑁 ln 𝓀𝓀 − 𝑁𝑁 − 𝑁𝑁𝛽𝛽𝑜𝑜− 𝑁𝑁 ln 1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽 �• The chemical potential of the adsorbed phase is𝜕𝜕𝜕𝜕𝑁𝑁ad+ 𝑜𝑜 − 𝑘𝑘B 𝑇𝑇 ln 1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽= 𝑘𝑘B 𝑇𝑇 ln𝜇𝜇 ′ =𝜕𝜕𝑁𝑁 𝑇𝑇,𝑉𝑉𝓀𝓀 − 𝑁𝑁
• At equilibrium, the chemical potentials of the gas
phase and absorbed phase must be identical.

25

3. Adsorption
32𝑃𝑃 2𝜋𝜋ℏ2𝑘𝑘B 𝑇𝑇 ln
𝑘𝑘B 𝑇𝑇 𝑘𝑘B 𝑇𝑇𝑚𝑚
𝑁𝑁+ 𝑜𝑜 − 𝑘𝑘B 𝑇𝑇 ln 1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽= 𝑘𝑘B 𝑇𝑇 ln
𝓀𝓀 − 𝑁𝑁• Rearrange.
𝑁𝑁𝑃𝑃 2𝜋𝜋ℏ2=𝓀𝓀 − 𝑁𝑁 𝑘𝑘B 𝑇𝑇 𝑘𝑘B 𝑇𝑇𝑚𝑚

321 − 𝑒𝑒 −𝛽𝛽𝛽𝛽𝑘𝑘B 𝑇𝑇𝑘𝑘B 𝑇𝑇𝑚𝑚
𝑃𝑃0 ≡1 − 𝑒𝑒 −𝛽𝛽𝛽𝛽 2𝜋𝜋ℏ232𝑒𝑒−𝑜𝑜𝑜𝑜𝑘𝑘B 𝑇𝑇 =𝑒𝑒 𝑘𝑘B 𝑇𝑇

𝑃𝑃𝑃𝑃03. Adsorption• Define the coverage.

••••

26

𝑁𝑁𝑃𝑃𝜃𝜃 ≡ =𝓀𝓀 𝑃𝑃 + 𝑃𝑃0This is the Langmuir adsorption isotherm.
The Langmuir adsorption model can be derived
from kinetics as well.
However, Langmuir adsorption is only qualitatively
accurate because of the over simplified model.
Modification: adsorbate/adsorbate interactions,roughness, multilayer absorption, etc.

27

3. Adsorption

𝑃𝑃→∞

1.00.8Coverage

𝑃𝑃lim 𝜃𝜃 =𝑃𝑃→0𝑃𝑃0• This is
another
form ofHenry’slaw.
• The surface
is saturated
whenlim 𝜃𝜃 = 1

0.6

0.4

0.2

0.0

0123

P/P045