father of statistical thermodynamics
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The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 2
Father of statistical thermodynamics
Ludwig Boltzmann (1844-1906)โข Academical legacy- Maxwell-Boltzmann distribution for molecular speed in a gas.- Opinion and belief in the reality of atom and molecule.- To quote Plank, "The logarithmic connection between
entropy and probability was first stated by L. Boltzmannin his kinetic theory of gases".
๐ = ๐พ โ ๐๐๐๐
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 3
Various statistical distribution
Maxwell-Boltzmann Statistics have to be distinguishable each other and one energy state can be occupied by two or more particles. Ex) gas molecules
Bose-Einstein Statistics have to be indistinguishable each other and one energy state can be occupied by two or more particles. Ex) phonon, photon
Fermi-Dirac Statistics have to be indistinguishable each other and one energy state can be occupied by only one particle. Ex) electron, hole
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 4
Interpretation of entropy as a mixed-up-ness of the system
Solid, ๐๐ ๐๐๐๐ liquid, ๐๐๐๐๐ข๐๐ Vapor, ๐๐๐๐๐๐
- the more โ mixed upโ the constituent particles of a system, the larger is the value of its entropy.
๐๐ ๐๐๐๐ < ๐๐๐๐๐ข๐๐ < ๐๐๐๐
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 5
The concept of microstate
Terminology1. Isolated systems, where S โฒ = S โฒ(Uโฒ,Vโฒ,N ) or U โฒ = U โฒ(Sโฒ,Vโฒ,N) - microcanonical2. Closed systems, where S โฒ=Sโฒ(T,Vโฒ,N) - canonical3. Open systems, where S โฒ = S โฒ(T,Vโฒ,ฮผ) - grand canonical
Statistical thermodynamics postulates that the equilibrium state of a system is simplythe most probable of all of its possible (i.e., accessible) microstates. Therefore,statistical thermodynamics is concerned with
โข The determination of the most probable microstateโข The criteria governing the most probable microstateโข The properties of this most probable microstate
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 6
The concept of microstate
- To derive relationship between entropy and mixed-up-ness, the quantization of the mixed-up-ness is necessary.
- Both Boltzmann (Ludwig Eduard Boltzmann, 1844โ 1906) and Gibbs found it convenientto examine the distribution of energies among the particles of the system by placing the energy of the particles into discrete compartments .
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 7
The Microcanonical approach
- Considering a hypothetical system comprising a perfect crystal in which all of the distinguishable sitesare occupied by identical particles; the theoretical condition for Maxwell-Boltzmann distribution
- The crystal contains n particles and has the fixed energy U สน and fixed volume V สน, and whole system is
considered isolated one.
- Statistical thermodynamics asks the following questions:
โข In how many ways can the n particles be distributed over the available energy levelssuch that the total energy of the crystal (i.e., Uสน ) remains the same?
โข Of the possible distributions, which is the most probable?
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 8
The Microcanonical approacha. All three particles on level 1
b. One particle on level 3, and the other two particles on level 0
c. One particle on level 2, one particle on level 1, and one particle on level 0
โข Distribution a . There is only one microstate of this distribution, since the interchangeof the particles among the three sites does not produce a different microstate.
โข Distribution b . Any of the three distinguishable sites can be occupied by any ofthe three particles of energy 3u , and the remaining two sites are each occupied bya particle of zero energy. Since the interchange of the particles of zero energy doesnot produce a different arrangement, there are three microstates in distribution b .
โข Distribution c . Any of the three distinguishable sites can be occupied by the particleof energy 2u . Either of the two remaining sites can be occupied by the particleof energy 1u , and the single remaining site is occupied by the particle of zeroenergy. The number of distinguishable microstates in distribution c is thus 3 ร 2ร 1 = 3! = 6.
- Probability: 1/10
- Probability: 3/10
- Probability: 6/10
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 9
Configurational entropy of differing atoms in a crystal
- Consider two crystals, one containing white atoms and the other containing gray atoms.- There is no difference in the energy of white/white, white/gray, and gray/gray bonds.(no mixing enthalpy)- When the two crystals are placed in physical contact with one another, a spontaneous process occurs in which the white atoms diffuse into the crystal of the gray atoms and the gray atoms diffuse into the crystal of the white atoms. (Configurational entropy)
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 10
Configurational entropy of differing atoms in a crystal
The mixing process can be expressed as
If ๐๐ด atoms of A are mixed with ๐๐ต atoms of B,
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 11
Configurational entropy of magnetic spins on an array of atoms
- The spin may take values of ยฑ1
2, which called up and down spins for convenient.
- Assume that there is eight sight in the system, So there will be nine distinct groups.
- The most probable microstate has a total magnetization of zero, and this is considered to be the equilibrium state. It can also be seen that the average value of M over all microstates is also zero. A material in this state is called a paramagnet .
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 12
The Boltzmann distribution
- If n particles are distributed among the energy levels such that ๐0 are on levelฮต0 , ๐1 on level ฮต1 , ๐2 on level ฮต2 ,โฆ , and ๐๐ on ฮต๐ , the highest level of occupancy, thenthe number of arrangements, ฮฉ , is given by
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 13
The Boltzmann distribution
or
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 14
The Boltzmann distribution
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 15
The influence of temperature
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 16
The influence of temperature
Helmholtz free energy
- Consider now a system of particles in thermal equilibrium with a constant-temperatureheat bath .U โฒ = ๐โฒ๐๐๐๐ก๐๐๐๐๐ ๐ ๐ฆ๐ ๐ก๐๐ + ๐โฒโ๐๐๐ก ๐๐๐กโVโฒ = ๐โฒ๐๐๐๐ก๐๐๐๐๐ ๐ ๐ฆ๐ ๐ก๐๐ + ๐โฒโ๐๐๐ก ๐๐๐กโn = the number of particles in the system + the heat bath of fixed size
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 17
Heat flow and the production of entropy
- Consider two closed systems, A and B . Let the energy of A be ๐โฒ๐ดand the numberof complexions of A be ฮฉ๐ด. Similarly, let the energy of B be ๐โฒ๐ต and its number ofcomplexions be ฮฉ๐ต.
- When thermal contact is made between A and B , the product ฮฉ๐ด ฮฉ๐ต will, generally, not have its maximumpossible value, and thermal energy will be transferred either from A to B or from B to A.
- The flow of thermal energy ceases when ฮฉ๐ด ฮฉ๐ต reaches its maximum value
The Third Law .Heat capacity, Enthalpy, Entropy .Fundamental equations .Statistical interpretion of entropy . 18
Heat flow and the production of entropy
19The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Application and practicality of thermodynamic method
- The main power of the thermodynamic method stems from its provision of criteria for equilibrium.
- The practical usefulness of this power is determined by the practicality of the equations of state for the system.
- It is important to establish variables that is easy to measure and easy to control.
๐๐ = ๐ฟ๐ โ ๐ฟ๐ค = ๐ โ๐ฟ๐
๐โ ๐ โ ๐๐ = ๐๐๐ โ ๐๐๐
- From a practical point of view, the choice of S and V as the independent variables is inconvenient.
- Entropy can be neither simply measured nor simply controlled.
20The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Practical thermodynamic variables
- From a practical point of view, a convenient pair of independent variables would be T and P, since these variables are easily measured and controlled.
- From the theoreticianโs point of view, a convenient choice of independent variables would be V and T,since they are easily examined by the methods of statistical mechanics.
- These equations are the fundamental equations that can be obtained by using the first law of thermodynamic,then we will define more practical thermodynamic variables H, A, G, ๐๐.
H: the enthalpy A: the Helmholtz free energyG: the Gibbs free energy ๐๐: the chemical potential
21The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The enthalpy, H
At the condition of constant pressure,
This equation shows that the change of state of a simple closed system at constant pressure, during which only P -V work is done, is the change in the enthalpy of the system and equals the thermal energy entering or leaving the system, ๐๐ . For this reason, it was called the heat function at constant pressure by Gibbs
22The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Helmholtz free energy, A
- If process is done at constant volume,โ๐ = ๐โ๐๐ ๐ฆ๐
๐ฟ๐๐ ๐ฆ๐ = โ๐ฟ๐๐ ๐ข๐
- For a system undergoing a change of state from state 1 to state 2
If system is closed,
And if the process is also isothermal,
23The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Helmholtz free energy, A
- During a reversible isothermal process, for which โ๐๐๐๐ is zero, the amountof work done by the system is a maximum.
- The amount of work done is equal to the decrease in the value of the Helmholtz free energy.
- ๐๐โฒ๐๐๐ > 0 means the process is spontaneous(forward), then ๐๐ดโฒ would be negativein the spontaneous process.
- ๐๐โฒ๐๐๐ = 0 means the process is reversible, then ๐๐ดโฒ would be zero in the reversible process.
- ๐๐โฒ๐๐๐ < 0 means the process is backward, then ๐๐ดโฒ would be positive in the backward reaction.
โปRemember that we are dealing with isochoric process.
24The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Helmholtz free energy, A
๐1 > 0K๐0 = 0K ๐2 > ๐1
25The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Gibbs free energy, G
- The Gibbs free energy is defined as
- For a system undergoing a change of state from state 1 to state 2
- If the process carried out isothermal and isobar,
26The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Gibbs free energy, G
- For an isothermal, isobaric process, and no form of work other than P-V work is done,
- ๐๐โฒ๐๐๐ > 0 means the process is spontaneous(forward), then ๐๐บ would be negativein the spontaneous process.
- ๐๐โฒ๐๐๐ = 0 means the process is reversible, then ๐๐บ would be zero in the reversible process.
- ๐๐โฒ๐๐๐ < 0 means the process is backward, then ๐๐บ would be positive in the backward reaction.
- Since โ๐๐๐๐ is a criterion for thermodynamic equilibrium, then an increment of an isothermalisobaric process occurring at equilibrium requires that
27The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Several forms of equilibrium
๐๐
๐๐
Gibbs free energy versus temperature
๐๐บ
๐๐=0, two phase can coexist
Gibbs free energy versusInteratomic distance
๐๐บ
๐๐=0,
๐2๐บ
๐๐2> 0, equilibrium
Gibbs free energy versusParticle radius during solidification
๐๐บ
๐๐=0,
๐2๐บ
๐๐2< 0, not equilibrium
28The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The fundamental equation
- Since ๐๐ = ๐ฟ๐ โ ๐ฟ๐ค,
๐ ๐ผ = ๐ป๐ ๐บ โ ๐ท๐ ๐ฝ โฆโ
- Since ๐๐ป = ๐๐ + ๐๐๐ + ๐๐๐,
๐ ๐ฏ = ๐ป๐ ๐บ + ๐ฝ๐ ๐ท โฆโก
- Since ๐๐ด = ๐๐ โ ๐๐๐ โ ๐๐๐,
๐ ๐จ = โ๐บ๐ ๐ป โ ๐ท๐ ๐ฝ โฆโข
- Since ๐๐บ = ๐๐ป โ ๐๐๐ โ ๐๐๐,
๐ ๐ฎ = โ๐บ๐ ๐ป + ๐ฝ๐ ๐ท โฆโฃ
29The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The fundamental equation
- If the system is also under the influence of an applied magnetic field,
30The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The chemical potential
- The chemical potential of the i th species in a homogeneous phase is theincremental change of the Gibbs free energy that accompanies an incrementalincrease of the species to the system at constant temperature, pressure, and numbersof moles of all of the other species.
= ๐๐
31The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The chemical potential
32The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The chemical potential
- The term ๐๐๐๐๐ is the chemical work done by the system, which was denoted as wสน
โ ๐บ๐ด๐ = ๐โ๐๐๐๐๐๐ ๐๐๐ก๐๐๐ก๐๐๐ ๐๐ ๐ด ๐๐ก ๐๐ด = ๐๐ต
โ ๐บ๐ต๐ = ๐โ๐๐๐๐๐๐ ๐๐๐ก๐๐๐ก๐๐๐ ๐๐ ๐ต ๐๐ก ๐๐ด = ๐๐ต
33The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Thermodynamic relations
34The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Maxwell's relations
- If Z is a state function and x and y are chosen as the independent thermodynamic variables in a closed system of fixed composition
35The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The first ๐ป๐ ๐บ equation- The term
๐๐
๐๐ ๐can be shown to equal ๐ผ/๐ฝ๐
- For an isothermal expansion,
- For an isentropic expansion
36The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The first ๐ป๐ ๐บ equation
- If the substance is 1 mole of an ideal gas
- the integration of which (assuming constant c v ) between the states 1 and 2,
- For an isothermal expansion of an ideal gas
- If the temperature of an ideal gas is raised at constant volume
- For an isentropic expansion of an ideal gas
37The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The second ๐ป๐ ๐บ equation
- For an isothermal reversible change of pressure
- For an isentropic process
38The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The second ๐ป๐ ๐บ equation
- For an isothermal reversible change of pressure of an ideal gas
- For an isentropic process of an ideal gas
- For an isobaric reversible change in temperature of an ideal gas
39The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
S and V as dependent variables and T and P as independent variables
40The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
๐๐ป = ๐๐๐ + ๐๐๐
๐(๐, ๐) H(๐, ๐)
An energy equation
41The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
Another important formula
42The Third Law .Heat capacity, Enthalpy,
Entropy .
Statistical interpretion of
entropy .Fundamental equations .
The Gibbs-Helmholtz equation
43The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Heat capacity
โ๐ = ๐ ๐1, ๐ โ ๐ ๐2, ๐ = ๐1
๐2
๐ถ๐ฃ ๐๐
44The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
History of heat capacity
Dulong-Petit DebyeEinstein
45The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Dulong-petit law
- An empirical rule which states that the molar heat capacities (๐ถ๐ฃ) of all solid elements have the value 3R.- Although the molar heat capacities of most elements at room temperature have values which are closeto 3R , subsequent experimental measurement showed that the heat capacity usually increases slightly with increasing temperature and can have values significantly lower values than 3R at low temperatures.
46The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Einsteinโs calculation
- Einstein considered the properties of a solid containing n atoms, each of which behaves as a quantumharmonic oscillator vibrating independently in three orthogonal directions about its position.
- the behavior of each of the 3n oscillators is not influenced by the behavior of its neighbors, and have a singlefrequency ๐ฃ to each of the oscillators.(Einstein solid)
- Equipartition theorem: the average energy of each quadratic contribution to the energy is ๐พ๐/2
โปReview
- Each atom has six degree of freedom.- It reveals 3R of specific heat for 1 mole.
47The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Einsteinโs calculation
- For a fixed frequency of vibration, the energy levels of a quantum harmonicoscillator take values of the ๐ th energy level as
- We defined the partition function ๐ฉ as
48The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Einsteinโs calculation
Where โ๐ข
๐๐ต= ๐๐ธ. ๐๐ธ is called Einstein characteristic temperature.
49The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Einsteinโs calculation
- Einsteinโs model fits well when T is sufficiently high and ๐ถ๐ฃ is close to 3R, and when ๐ โ 0, ๐ถ๐ฃ โ 0.
- But the theoretical values of the Einstein model approach zero more rapidly thando the actual values.- This discrepancy is caused by the fact that the quantum oscillators do not vibrate with a single frequency.
50The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Debye model
- Debye suggested the oscillation model that have the range of frequency of vibration.- The lower limit of the wavelength of these vibrations is determined by the interatomic distances in the solid.- Taking this minimum wavelength,๐๐๐๐ , to be in the order of 5 ร 10โ 10 m, and the wave velocity, ๐ , in the solidto be about 5 ร 103 m/sec,
51The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Debye model
- For very low temperature,
- This is called the ๐ท๐๐๐ฆ๐ ๐3 ๐๐๐ค for low temperature heat capacities.
๐ฆ โ 25.98
52The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Debye model
- Debyeโ s theory does not consider the contribution made to the heat capacityby the uptake of energy by free electrons at the Fermi level in a metal at lowtemperatures.- For a metal at low temperatures, the heat capacity varies as
53The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
The empirical representation of heat capacities
Monoclinic, Below to 1478K
Monoclinic, 1478~2670K
54The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Enthalpy as a function of temperature and composition
55The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Enthalpy as a function of temperature and composition
โ โก
โข
Since ๐ป is state function, ๐ป = 0,
โ
โก
โข
56The Third Law .Fundamental equations .Statistical interpretion of
entropy .
Heat capacity, Enthalpy,
Entropy .
Enthalpy as a function of temperature and composition
57Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
The dependence of entropy on temperature
and
58Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
The Third law of thermodynamics
- If the value of ๐0 for a reaction could be determined, โ๐บ would be known as a function of temperature as well,and hence, the reaction thermodynamics would be known.
- Consideration of the value of ๐0 lead to the statement of the Third Law of Thermodynamics.- The values of โ๐บ and โ๐ป asymptotically approached each other at low temperatures with slopesthat approached zero.
and
- ๐ต๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐
59Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
The Third law of thermodynamics
- ๐๐๐๐๐๐ (Max Karl Ernst Ludwig Planck, 1858โ 1947) extended the Nernstโ s heattheorem by positing to the effect that the entropy of any homogeneous substancewhich is in complete internal equilibrium is zero at 0 K.
Point defect line defect planar defect
60Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
Apparent contradictions to the third law of thermodynamics
Internal equilibrium Non-internal equilibrium
๐(๐ด โ ๐ต) = 1
๐ ๐ด โ ๐ด = ๐ ๐ต โ ๐ต = 0
0 < ๐(๐ด โ ๐ต) < 1
๐ ๐ด โ ๐ด + ๐(๐ด โ ๐ต)
2> ๐(๐ด โ ๐ต)
0 < ๐(๐ด โ ๐ต) < 1
๐ ๐ด โ ๐ด + ๐(๐ด โ ๐ต)
2= ๐(๐ด โ ๐ต)
โ โก
61Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
Apparent contradictions to the third law of thermodynamics
โข Even chemically pure elements are mixtures of isotopes, and because of the chemicalsimilarity between isotopes, it is to be expected that completely random mixingof the isotopes occurs. For example, solid chlorine at 0 K is a solid solution of Cl35 โ Cl35 , Cl35 โ Cl37 ,and Cl37 โ Cl37 molecules.
โฃ At any finite temperature, a pure crystalline solid contains an equilibrium numberof vacant lattice sites, which, because of their random positioning in the crystal
โ๐บ = โ๐ป๐ฃ โ ๐โ๐= โ๐ป๐ฃ๐๐ฃ โ ๐(โ๐๐ฃ๐๐ฃ โ ๐ ๐๐ฃ๐๐๐๐ฃ โ ๐ 1 โ ๐๐ฃ ๐๐(1 โ ๐๐ฃ))
๐คโ๐๐๐ ๐โ๐๐ฃ๐๐ฃ = ๐กโ๐๐๐๐๐ ๐๐๐ก๐๐๐๐ฆ, ๐ ๐๐ฃ๐๐๐๐ฃ โ ๐ 1 โ ๐๐ฃ ๐๐(1 โ ๐๐ฃ) = ๐๐๐๐๐๐๐ข๐๐๐ก๐๐๐๐๐ ๐๐๐ก๐๐๐๐ฆ
๐๐บ
๐๐๐ฃ= โ๐ป๐ฃ โ ๐โ๐๐ฃ + ๐ ๐๐๐๐๐ฃ + ๐ ๐ โ ๐ ๐๐๐ 1 โ ๐๐ฃ โ ๐ ๐
= โ๐ป๐ฃ โ ๐โ๐๐ฃ + ๐ ๐๐๐๐๐ฃ โต 1 โ ๐๐ฃ โ 1
๐๐๐๐๐ ๐กโ๐๐๐๐๐ ๐๐๐ก๐๐๐๐ฆ ๐๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐,๐๐บ
๐๐๐ฃ= โ๐ป๐ฃ + ๐ ๐๐๐๐๐ฃ = 0 at equilibrium.
๐ฟ๐ = ๐๐๐(โโ๐ฏ๐๐น๐ป)
62Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
Apparent contradictions to the third law of thermodynamics
โค Random crystallographic orientation of molecules in the crystalline state can alsogive rise to a nonzero entropy at 0 K
For the Third Law to be obeyed, โ๐๐ผ๐ = 0
63Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
Experimental verification of the third law
- With the constant-pressure molar heat capacity of the solid expressed in the form
64Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
The influence of pressure on enthalpy and entropy
- For one mole of a closed system of fixed composition undergoing a change of pressure at constant temperature
65Heat capacity, Enthalpy,
Entropy .Fundamental equations .
Statistical interpretion of
entropy .The Third Law .
The influence of pressure on enthalpy and entropy
- For a closed system of fixed composition undergoing changes in both pressure and temperature
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