Chemical Reactions in Ideal Gases

Non-reacting ideal gas mixture Consider a binary mixture of molecules of types A and B. The canonical partition function of this system is: 2

Non-reacting ideal gas mixture Consider a binary mixture of molecules of types A and B. The canonical partition function of this system is: Occupation numbers 3 # of molecules A in the jth energy state # of molecules B in the jth energy state

Non-reacting ideal gas mixture Molecules of type A are undistinguishable from each other and the same happens to the molecules of type B. Also, we take them as non-interacting species: For the general case of S non-interacting species: 4

Non-reacting ideal gas mixture The thermodynamic properties of ideal gas mixtures can be derived from the above equation (as done at the end of Chapter 3). Among other expressions, we have that: 5

Partition function for reacting ideal gas mixture The general case will be discussed later, but we will begin with a simple situation that exists in isomerization reactions: In terms of partition function, how is this situation different from a non-reacting mixture of two species A and B? 6

Partition function for reacting ideal gas mixture In terms of partition function, how is this situation different from a non-reacting mixture of two species A and B? 7

Partition function for reacting ideal gas mixture In terms of partition function, how is this situation different from a non-reacting mixture of two species A and B? A given molecule of substance A can be in any of the energy states accessible to a single molecule of substance A. 8

Partition function for reacting ideal gas mixture In terms of partition function, how is this situation different from a non-reacting mixture of two species A and B? A given molecule of substance A can be in any of the energy states accessible to a single molecule of substance A. But also, if it transforms into B, it also has access to any of the energy states accessible to a single molecule of substance B. 9

Partition function for reacting ideal gas mixture In terms of partition function, how is this situation different from a non-reacting mixture of two species A and B? A given molecule of substance A can be in any of the energy states accessible to a single molecule of substance A. But also, if it transforms into B, it also has access to any of the energy states accessible to a single molecule of substance B. A single molecule in the system then has access to all energy states of both substances A and B. 10

Partition function for reacting ideal gas mixture A single molecule in the system then has access to all energy states of both substances A and B. 11

Partition function for reacting ideal gas mixture For a single molecule: 12

Partition function for reacting ideal gas mixture Then: 13

Partition function for reacting ideal gas mixture Another approach to derive the same expression. For this stoichiometry, the total number of moles is conserved: Initial amounts 14

Partition function for reacting ideal gas mixture As long as the total number of molecules is conserved, any combination of numbers of molecules of A and B is possible. The partition function is: 15

Partition function for reacting ideal gas mixture The last equality comes from identifying the coefficients of the binomial expansion (refer to the Appendix of Chapter 5, page 83). 16

Chemical equilibrium constant in reacting ideal gas mixtures From classical thermodynamics: “The state of equilibrium has minimum value of the Helmholtz function compatible with given specifications of system temperature, volume, and initial amounts of each substance present”. 17

Chemical equilibrium constant in reacting ideal gas mixtures If these are the specifications in a reacting ideal gas mixture, the number of molecules of each substance will change so as to minimize the Helmholtz energy. The macroscopic state that minimizes the Helmholtz energy maximizes the canonical partition function. Let us then maximize it subject to the constraint that the total number of molecules remains constant. 18 For a particular choice of N A and N B

Chemical equilibrium constant in reacting ideal gas mixtures Then: Assuming we can treat the number of molecules as a continuous variable: 19

Chemical equilibrium constant in reacting ideal gas mixtures At the maximum of the canonical partition function: 20

Chemical equilibrium constant in reacting ideal gas mixture It is also possible to write a similar relationship in terms of molecular concentrations: 21 equilibrium constant of the reaction

Chemical equilibrium constant in reacting ideal gas mixture A second way of getting the same relationships The problem can be restated as finding the average number of molecules of A in the system (the average for B then follows immediately from the mass balance). The probability of having a certain number of molecules is: 22

Chemical equilibrium constant in reacting ideal gas mixture The average number of molecules of A is: 23

Chemical equilibrium constant in reacting ideal gas mixture Using properties of the binomial expansion (refer to the Appendix of Chapter 5 in the textbook): Analogous development for B gives: 24

Chemical equilibrium constant in reacting ideal gas mixture Then: 25 molecular concentration-based equilibrium constant

Chemical equilibrium constant in reacting ideal gas mixture A third way of getting the same relationships This method uses the “Maximum Term Method”. As in the second method, the starting point is the probability of having a certain number of molecules of type A: We will now look for the condition that maximizes this probability, i.e., the value of N A for which p(N A ) is maximum (and so is ln p(N A )) 26

Chemical equilibrium constant in reacting ideal gas mixture Note that the canonical partition function depends on the total number of molecules but not on the number of molecules of type A: 27

Chemical equilibrium constant in reacting ideal gas mixture 28

Chemical equilibrium constant in reacting ideal gas mixture At the maximum: Values in the most probable state 29

Chemical equilibrium constant in reacting ideal gas mixture Note: in the second method, we derived the chemical equilibrium constant based on the average number of molecules; In the third method, we derived the chemical equilibrium constant based on the most probable number of molecules. To explain why we obtained the same expression, it is necessary to examine the role of fluctuations in reacting ideal gas systems 30

Fluctuations in a Chemically Reacting System Let us compare the partition function of the reacting system: and 31

Fluctuations in a Chemically Reacting System … and the partition function of a non-reacting system calculated with the most probable number of molecules of A and B: 32

Fluctuations in a Chemically Reacting System But: Then: 33

Fluctuations in a Chemically Reacting System Therefore, our comparison leads to: Which are identical! 34 complete partition function maximum term of the partition function (equilibrium state)

Fluctuations in a Chemically Reacting System Therefore, our comparison leads to: They would not have been identical if we had used a more refined version of Stirling’s approximation: But they would not differ by much anyway, even if we had done that 35

Fluctuations in a Chemically Reacting System Therefore, our comparison leads to: Interpretation of this result: The equilibrium state is much more likely than any other state of the system. The next slides show this 36

Fluctuations in a Chemically Reacting System The probability of a state of a state with N A and N B molecules of A and B, respectively, is: 37

Fluctuations in a Chemically Reacting System To obtain the probability of observing states with different distribution of molecules to that of the equilibrium state, we now expand the probability using a Taylor series expansion around the equilibrium state: 38

Fluctuations in a Chemically Reacting System Getting the first derivative: 39

Fluctuations in a Chemically Reacting System Getting the second derivative: 40

Fluctuations in a Chemically Reacting System The probability of observing states with distribution of molecules different from that of the equilibrium state is: 0 41

Fluctuations in a Chemically Reacting System The probability of observing states with distribution of molecules different from that of the equilibrium state is: 42

Fluctuations in a Chemically Reacting System Neglecting terms of higher order, the probability of observing states with distribution of molecules different from that of the equilibrium state is: 43

Fluctuations in a Chemically Reacting System Neglecting terms of higher order, the probability of observing states with distribution of molecules different from that of the equilibrium state is: 44

Fluctuations in a Chemically Reacting System Numerical example: one mol of reacting mixture, at equilibrium mole fractions of A and B are equal. 45

Fluctuations in a Chemically Reacting System Fluctuations such as these are beyond the sensitivity of the instruments commonly used in chemical laboratories and industries. Common measurements are accurate to 10^(-5) approximately. 46

Chemically Reacting Systems: General Case A chemical reaction can be written in the form: Stoichiometric coefficient of component i Chemical formula of component i The chemical equilibrium condition is such that: 47

Chemically Reacting Systems: General Case The chemical potential of ideal gases is: Using this formula in that of the chemical equilibrium condition: 48

Chemically Reacting Systems: General Case 49

Chemically Reacting Systems: General Case The partition function of an ideal gas reacting system can be computed based on the number of molecules of each substance in the most probable state. For a binary mixture, we used before that: For a general reacting mixture: 50

Chemically Reacting Systems: General Case For a general reacting mixture, using Stirling’s approximation: 51

Chemically Reacting Systems: General Case Because of the chemical reaction, changes in the number moles are interrelated: where: Extent of reaction 52

Chemically Reacting Systems: General Case Pressure: 53

Chemically Reacting Systems: General Case However, a few slides back, we showed that: Therefore: 54

Chemically Reacting Systems: General Case Internal energy: 55

Chemically Reacting Systems: General Case Heat capacity at constant volume: 56

Chemically Reacting Systems: General Case Heat capacity at constant volume: 57

Chemically Reacting Systems: General Case Heat capacity at constant volume: Internal energy change of reaction on a molecular basis 58

Chemically Reacting Systems: General Case Example: gas ionization A certain gas “A” ionizes originating its ion “i” and an electron “e”. 59

Chemically Reacting Systems: General Case Example: gas ionization Based on the results derived in previous slides: Assuming the gas is not initially ionized: 60

Chemically Reacting Systems: General Case Example: gas ionization Then: 61

Chemically Reacting Systems: General Case Example: gas ionization The partition function of each species is: State of zero energy taken as the atom at rest Electronic energy difference between atom and ions has been assigned to the ions 62

Chemically Reacting Systems: General Case Example: gas ionization The partition function of each species is: Electronic degeneracies: 1 for atom 2 for the ion 2 for the electron 63

Chemically Reacting Systems: General Case Example: gas ionization Then: 64

Chemically Reacting Systems: General Case Example: gas ionization Using the ideal gas pressure expression: 65

Chemically Reacting Systems: General Case Example: gas ionization Y is called degree of ionization 66

Chemically Reacting Systems: General Case Example: gas ionization 67

Chemically Reacting Systems: General Case Example: gas ionization Internal energy: 68

Chemically Reacting Systems: General Case Example: gas ionization Per mol of A initially present: 69

Chemically Reacting Systems: General Case Example: gas ionization In the previous slide, the following identification was made: No such identification is made in the textbook, what seems to be a mistake 70

Chemically Reacting Systems: General Case Example: gas ionization Enthalpy per mol of A initially present: 71

Chemically Reacting Systems: General Case Example: gas ionization Heat capacity at constant pressure per mol of A initially present: 72

73 Results for Argon the lines are different pressures; 0.01, 0.1, 1., and 33.6 bar respectively

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