The Ideal Monatomic Gas

Canonical ensemble: N, V, T 2

Canonical partition function for ideal monatomic gas A system of N non-interacting identical atoms: From quantum mechanics: Atom of ideal gas in cubic box of volume V 3

Canonical partition function for ideal monatomic gas Translational quantum numbers 4

With suitable assumptions: Approximate each sum and solve as an integral Then compute q Then compute Q 5

The calculated Q is the canonical partition function for ideal monatomic gas We have considered the contribution of atomic translation to the partition function However, we have not included other effects, such as contribution of the electronic and nuclear energy states. We will get back to this matter later in this slide set. 6

Canonical partition function for ideal monatomic gas Then: 7

Canonical partition function for ideal monatomic gas The internal energy of a N-particle monatomic ideal gas is: But U of a monatomic ideal gas is known to be: 8

Canonical partition function for ideal monatomic gas Then: Note that is only a function of the thermal reservoir, regardless of the system details. Therefore, this identification is valid for any system. 9

Relationship of Q to thermodynamic properties Then: We showed that: 10

Relationship of Q to thermodynamic properties Multiplying by kT 2 : For a system with fixed number of particles N: 11

Relationship of Q to thermodynamic properties Combining: But: 12

Therefore: 13

Relationship of Q to thermodynamic properties We identify that: Comparing this result with: 14

Relationship of Q to thermodynamic properties The entropy: The Helmholtz free energy: 15

Relationship of Q to thermodynamic properties The Helmholtz energy: This is a very important equation – knowing the canonical partition function is equivalent to having an expression for the Helmholtz energy as function of temperature, volume and number of molecules (or moles). From such expression, it is possible to derive any thermodynamic property 16

Relationship of Q to thermodynamic properties The chemical potential of a pure substance: The enthalpy: 17

Relationship of Q to thermodynamic properties Heat capacity at constant volume: 18

Relationship of Q to thermodynamic properties The previous expression can also be written as: This form will be useful when discussing fluctuations, later in this slide set 19

Thermodynamic properties of monatomic ideal gases The previous slides showed how to evaluate thermodynamic properties given Q It is time to discuss the effect of the electronic and nuclear energy states to the single atom partition function before proceeding with additional derivations We will assume the Born-Oppenheimer approximation: translational energy states are independent of the electronic and nuclear states Besides, we will assume the electronic and nuclear states are independent of each other 20

Thermodynamic properties of monatomic ideal gases With these assumptions : The single atom partition function is: As discussed in the previous class for independent energy modes: 21

Thermodynamic properties of monoatomic ideal gases In which: 22

Thermodynamic properties of monatomic ideal gases De Broglie wavelength: based on dual wave-particle nature of matter 23

Thermodynamic properties of monatomic ideal gases The electronic partition function : 24

Thermodynamic properties of monatomic ideal gases The electronic partition function : Additional information and approximations: -The degeneracy of the ground energy level is equal to 1 in noble gases, 2 in alkali metals, 3 in Oxygen; -The ground energy level is the reference for the calculations – it is conventional to set it to zero; -The differences in electronic levels are high. For example, argon: At room T: 25

Thermodynamic properties of monatomic ideal gases Then, the electronic partition function is approximated as: 26

Thermodynamic properties of monatomic ideal gases The nuclear partition function: The analysis is similar to that of the electronic partition function, only that the energy levels are even farther apart. It results: Also, in situations of common interest to chemical engineers, the atomic nucleus remains largely undisturbed. The nuclear partition function becomes only a multiplicative factor that will cancel out in calculations 27

Thermodynamic properties of monatomic ideal gases Compiling all these intermediate results: 28

Thermodynamic properties of monatomic ideal gases For the monoatomic ideal gas, the logarithm of the canonical partition function is: 29

Thermodynamic properties of monatomic ideal gases Let us now use this expression to compute several properties, beginning with the pressure: where N is the number of molecules and N av is Avogadro’s number. 30

Thermodynamic properties of monatomic ideal gases 31

Thermodynamic properties of monatomic ideal gases We derived this very famous equation from very fundamental principles – an amazing result 32

Thermodynamic properties of monatomic ideal gases Internal energy and heat capacity at constant volume: These expressions are more complicated if excited energy levels are taken into account – see eq and Problem

Thermodynamic properties of monatomic ideal gases Helmholtz energy: Before obtaining its expression, let us introduce Stirling’s approximation: This approximation is increasingly accurate the larger N is. Since N here represents the number of atoms, it is typically a very large number and this approximation is excellent. 34

Thermodynamic properties of monatomic ideal gases Helmholtz energy: Ignoring the nuclear partition function by setting it equal to 1: 35

Thermodynamic properties of monatomic ideal gases Entropy: known as Sackur-Tetrode equation 36

Thermodynamic properties of monatomic ideal gases Chemical potential: 37

Thermodynamic properties of monatomic ideal gases 38

Thermodynamic properties of monatomic ideal gases Now compare this formula and the formula well- known to chemical engineers of the chemical potential of a pure ideal gas: 39

Energy fluctuations in the canonical ensemble In the canonical ensemble, the temperature, volume, and number of molecules are fixed. The energy may fluctuate. Assume its fluctuations follow a Gaussian distribution : Mean of the distributionStandard deviation Variable Probability density 40

Energy fluctuations in the canonical ensemble Given this distribution, the average value of any function G(x) is calculated as follows: The variance (standard deviation to power 2) is: 41

Energy fluctuations in the canonical ensemble Let us apply this formalism to the average energy and its fluctuation: 42

Energy fluctuations in the canonical ensemble 43

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Energy fluctuations in the canonical ensemble We previously found that: 45

Energy fluctuations in the canonical ensemble The 2/3 factor is a particularity of using monoatomic ideal gases as example. However, the factor is common and shows that relative fluctuations decrease as the number of molecules increases. Comparing these two expressions: 46

Energy fluctuations in the canonical ensemble 47  = [E/(3NkT/2)]-1

Gibbs entropy equation But, using relationships developed in previous slides: 48

Gibbs entropy equation 49

Gibbs entropy equation Combining the expressions developed in the two previous slides (algebra omitted here): 50

Gibbs entropy equation If there is only one possible state: If there are only two possible states, assumed to have equal probability: If there are only three possible states, assumed to have equal probability: 51