Partition Function

7 Key excerpts on "Partition Function"

• 

  eBook - ePub

  Elements of Statistical Thermodynamics

  Second Edition

  • Leonard K. Nash(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  However it is computed, the magnitude of the Partition Function z reflects the net accessibility of the various quantum states at the given temperature. And precisely because it does so measure the “spread,” the Partition Function constitutes the essential link between the molecular parameters of the unit and the macroscopic parameters of an assembly of such units. That is, the Partition Function offers us the possibility of calculating all the thermodynamic parameters of a macroscopic assembly from spectroscopic measurements that establish the energies of the quantum states characteristic of the units constituting that assembly. We can capitalize on this important possibility only after we have mastered in detail the proper formulation and evaluation of Partition Functions. But before embarking on the somewhat tedious analysis required, we may do well to look at an example that highlights the importance of the Partition Function. This example demonstrates, for one peculiarly simple case, that knowledge of Partition Functions represents power to compute the equilibrium constant of an otherwise uncharacterized chemical reaction.

   

  Chemical equilibrium and the Partition Function. Gas-phase isomerization reactions (e.g., the interconversion of butane and isobutane, or of cis and trans butene-2) are important molecular equilibria of the type:

  A ( ) = B ( ).

  Figure 12 offers a highly schematic display of the possible quantum levels of A and B—each set comprising every possible combination of the translational, rotational, vibrational, and electronic quantum states of the respective pure species in its standard state. Each energy ∈Ai or ∈Bi is measured in the usual way from the respective ground-state energy ∈AO or ∈BO . As indicated, the two ground states will ordinarily differ in energy by some quantity we symbolize as . As usual, the superscript 0 refers to the standard-state condition, while the subscript 0 denote the temperature of 0°K at which all units must fall in their respective ground states. Representing the quantity of energy required to convert one molecule of A in its ground state into one molecule of B in its ground state, thus expresses the energy change when one molecule of A undergoes the above reaction at 0°K. Concerned as we now are with an equilibrium involving both A and B, we must of course take care to reduce all our energy measurements to a single self-consistent basis. This we can easily do. Setting our reference zero of energy to coincide with the ground state of A, i.e., taking ∈A0 ≡ 0, we need only add the quantity to each of the B-group energies. That is, for the energy of the ith quantum state of B we will write (∈Bi +

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  eBook - ePub

  Statistical Mechanics for Chemistry and Materials Science

  • Biman Bagchi(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  PV . Thus, the isothermal-isobaric Partition Function can be expressed in terms of the canonical Partition Function by the Laplace transform:

  Ξ ( N , P , T ) =

  1

  V 0

  ∫ 0 ∞

  d V exp ⁡ ( − β P V ) Q ( N , V , T )

  5.79

  5.79

  Where V 0 is a constant that has units of volume. Thus,

  Ξ ( N , P , T ) =

  1

  V 0

  N !

  h

  3 N

  ∫ 0 ∞

  d V

  ∫

  d X exp ⁡ ( − β ( ∏ ( X ) + P V ) )

  5.80

  5.80

  The Gibbs free energy is related to the Partition Function by

  G = −

  k B

  T   ln ⁡ Ξ ( N , P , T )

  5.81

  5.81

  Thus, Gibbs free energy is the thermodynamic potential of the isothermal-isobaric ensemble. This is, of course, anticipated from thermodynamics, and has been mentioned earlier.

  5.9 Physical Interpretation of Partition Function (PF) in Different Ensembles

  We have already discussed some aspects, but let us reiterate some points. First, a Partition Function provides the weight of the system, under the chosen conditions. This statement translates into certain extremely useful extremum conditions that we employ almost every day in our study of complex systems. Note that the thermodynamic potential of each Partition Function is given by the logarithm of the same. For example,

  S =

  k B

  ln ⁡ Ω

  5.82

  5.82

  A = −

  k B

  T ln ⁡ Q

  5.83

  5.83

  P V

  k B

  T

  = ln ⁡ Ξ

  5.84

  5.84

  For a given thermodynamic state the Partition Function is maximum at equilibrium. Let us elaborate on this often quoted statement, to gain some physical insight.

  Consider a physical quantity, ξ . This can be a function other than the state functions used to specify the thermodynamic state of a system. For example, this can be a reaction coordinate in a chemical reaction, order parameter in a phase transition. We consider the extremum principle in the context of this parameter ξ . The PF is maximum at that value of ξ where the system spends maximum amount of time. This is obvious in a microcanonical ensemble, where the PF measures the number of microscopic states. In the canonical ensemble the time spent in each energy level is weighted by the Boltzmann factor. In the grand canonical ensemble a further weightage is coming from fugacity to take into account the weightage of the state with given N

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  eBook - ePub

  Phase Modeling Tools

  Applications to Gases

  • Michel Soustelle(Author)
  • 2015(Publication Date)
  • Wiley-ISTE

    (Publisher)

  6 Molecular Partition Functions

  6.1. Definition of the molecular Partition Function

  The molecular Partition Function is defined, for an ensemble composed of many particles of molecular dimensions each with the energy εi , by the equation:

  [6.1]

  6.2. Decomposition of the molecular Partition Function into partial Partition Functions

  The molecular Partition Function for a system includes terms that relate to different forms of energy: nuclear, electronic, vibrational energy of molecules, their rotational energy, their translational energy and interaction energies between different molecules.

  To simplify this, we know that different forms of energy, for a molecule, are independent (however, there are exceptions, e.g. the interdependence of vibrational and rotational energies of a molecule). Under these conditions, we can write the total energy of a molecule as the sum of different contributions by different forms of energy: nuclear ε

  n

  , electronic ε

  e

  , vibrational ε

  v

  , rotational εr , translational εt and interaction ε

  I

  , i.e.:

  [6.2]

  The Partition Function becomes:

  [6.3]

  This shows the partial Partition Functions related to the different forms of energy:

  [6.4]

  The overall Partition Function takes the form of the product of the partial Partition Functions:

  [6.5]

  Calculating partial Partition Functions requires that the number of degrees of freedom for each type of molecule movement is known.

  If a molecule is composed of N atoms, it has a total of 3N

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  eBook - PDF

  Introduction to Cell Mechanics and Mechanobiology

  • Christopher R. Jacobs, Hayden Huang, Ronald Y. Kwon(Authors)
  • 2012(Publication Date)
  • Garland Science

    (Publisher)

  The Partition Function is related to the free energy and can be used to calculate the free energy at equilibrium without having to enumerate the density of states. Consider the case in which we have a system of interest as defined in the canoni- cal ensemble. We know that the Boltzmann distribution gives the probability p(m s ) of a microstate with microscopic energy Q(m s ). To show that this distribu- tion is related to the free energy, consider the probability of the system having 140 CHAPTER 5: Statistical Mechanics Primer internal energy W at thermal equilibrium. We shall denote this probability p(W). We know that for a macroscopic value of W, the system may have multiple micro- states. The probability of the system having a macroscopic internal energy W can be calculated as the total number of microstates with energy Q = W, multiplied by the associated Boltzmann probability, p W Z e W W k T ( ) ( ). = - 1 B Ω (5.40) But because Ω( ) , W e e k T k T = = S S B B (5.41) we can see that Equation 5.40 is related to the free energy as p W Z e Z e W TS k T ( ) , ( ) = = - - - 1 1 B β Ψ (5.42) where β = 1/k B T and Ψ is the free energy. Equation 5.42 implies that analogous to how the Boltzmann distribution implies that the probability of a microstate goes as the inverse exponential of its microscopic energy the probability of a macrostate (which can be associated with a different number of microstates) goes as the inverse exponential of its free energy. With this relation in hand, we are now able to show that Z is related to the free energy. We previously showed that Z can be obtained by integrating the Boltzmann distri- bution over all microstates, as in Equations 5.38 and 5.39. But we also know that for Equation 5.42 to yield the correct probability for p(W), Z e dW W = - ∫ β Ψ( ) (5.43) or in the discrete case, Equation 5.43, Z e W W = - ∑ βΨ( ) .

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  Molecular Physical Chemistry

  A Concise Introduction

  • Keith A McLauchlan(Author)
  • 2007(Publication Date)
  • Royal Society of Chemistry

    (Publisher)

  Notably, it is expressed here in terms of sums over quantum states, i . e . in terms of individual values. But each state may be degenerate and consist of many levels of equal energy; to account for this we have expressed the distribution throughout this book in terms of levels rather than states, i . e . we have included the degeneracies, g , in our equations. 35 Partition Functions We have assumed in the derivation that the molecules are independent and non-interacting. Now we generalise our treatment to the case where they might interact. 2.3 CANONICAL Partition Function In many real systems molecules interact, for example through inter-molecular forces, and/or change their positions in a crystal lattice to become distinguishable from one another. The interactions allow them collectively to explore all the possible energy states available to the molecules individually, with the total energy of the system conserved if it is isolated from its surroundings. Our focus changes from that of the Boltzmann distribution in that we are concerned now with the distribu-tion of energy between the whole series of possible energies for the complete system containing all the molecules, rather than how the mol-ecules are distributed between their individual quantum states. This is because at any one instant the collection can adopt any one of a range of energies, depending on the precise distribution of molecules between their quantum states. Sometimes the total energy will exceed the average and sometimes will be less than it. But over time the energy, which we wish to calculate, is constant. The situation is one of constant dynamic change with molecules jumping between states due to their interactions ( e . g . in collisions) and we must consider all the possible distributions that are explored, and take their average over time to obtain the mean properties of the system. There is no known way of doing this and so we resort to an elegant alternative.

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  Physical Chemistry

  • David Ball(Author)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  This value is called the equilibrium constant for the reaction. For an ideal gas, the Partition Function Q is a simple function of volume (again, from q trans ) times a more complicated function of temperature (from sev-eral other q ’s): Q 5 f ( T ) # V It is convenient to divide each molecular Q by volume to get a volume-independent Partition Function: Q V 5 f ( T ) By substituting this volume-independent Partition Function into the Partition Function expression for the equilibrium constant, we can get an equilibrium con-stant, labeled K ( T ), which is characteristic of the chemical species in the reaction and dependent solely on T : K ( T ) ; a Q C V b n C # a Q D V b n D a Q A V b n A # a Q B V b n B (18.60) Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 18.9 | Equilibria 657 Unless otherwise noted, all art on this page is © Cengage Learning 2014. Equation 18.60 shows that statistical thermodynamics can calculate temperature- dependent equilibrium constants from Partition Functions. Because the Partition Functions themselves are ultimately determined from the energy levels of the chemi-cal species, we see once again how a knowledge of energy levels—obtained from spectroscopy—helps us make thermodynamic predictions about chemical reactions. A word about units is necessary. It is important to keep track of all units in equation 18.60, and when comparing calculated results to experimental ones, units must be consistent. Equation 18.60 is the concentration-based equilibrium constant, K c .

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  Statistical and Thermal Physics

  Fundamentals and Applications

  • M.D. Sturge(Author)
  • 2018(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  When one applies statistical mechanics to a particular system, the h rst and most important task is usually to determine Z . 5.4 The Many-Particle Partition Function There is nothing in the derivation of the Partition Function or the Boltz-mann factor that restricts it to a microsystem. The only condition is that it be much smaller than the heat bath 0 , which we can make as large as we please. We shall often h nd it convenient to use the Partition Function for an entire macrosystem, which we call Z N to distinguish it from the Partition Function of the constituent microsystems. We call Z N the many-particle Partition Function to distinguish it from the single-particle Partition Function Z (or Z 1 when we need to emphasize the distinction). The mean energy of a macrosystem is simply U , so that Equation (5.9) becomes U = ln Z N . (5.11) To h nd Z N in terms of Z 1 , we limit ourselves here to the case where the microsystems are weakly interacting, in the sense that, while they can ex-change energy, the energy levels of any one microsystem are una ected by the state of any other. We furthermore assume for now that the microsys-tems are distinguishable . What this means is that it is possible in principle to determine which microsystem is in which state. For example, the spins in Chapter 3 are distinguishable, since each spin is a certain position in the solid, and one can imagine doing an experiment in which one determines whether a particular spin is up or down. 3 Identical particles in motion, on the other hand, are indistinguishable; their combined wave function is either symmetric or antisymmetric in exchange of particles, and it is impos-sible even in principle to determine, and meaningless to ask, which particle is in which state (see Footnote 4 of Chapter 9 ). We begin by considering two identical, but distinguishable, spins, la-belled A and B.

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