The Ideal Diatomic and Polyatomic Gases Canonical partition function for ideal diatomic gas Consider a system of N non-interacting identical molecules: A result from quantum mechanics: Atom of ideal gas in cubic box of volume V 2 Canonical partition function for ideal diatomic gas Compared to the monoatomic ideal gas, rotation and vibration need to be considered: As in the previous chapter: 3 Canonical partition function for ideal diatomic gas De Broglie wavelength: based on dual wave-particle nature of matter Atoms 1 and 2 do not have to be the same 4 Canonical partition function for ideal diatomic gas For the electronic partition function: 5 Canonical partition function for ideal diatomic gas For the electronic partition function: Additional information and approximations: -The differences in electronic levels are generally high and the corresponding terms are neglected. -In the previous chapter, the ground energy level of monoatomic gases was taken as reference for the calculations and assigned to zero; -The ground energy level for diatomic gases is not zero the ground state energy is related to the chemical bond formation 6 Canonical partition function for ideal diatomic gas For the electronic partition function: energy of the lowest vibrational mode of the molecule energy of chemical bond formation 7 8 https://www.youtube.com/watch?v=T7fRGXc9SBI 9 Canonical partition function for ideal diatomic gas For the rotational partition function: Computed based on the energy levels of a linear rigid rotor. From quantum mechanics, the energy level j is characterized by: Moment of inertia Distances from the atoms to the center of mass 10 Importance of rotational motion 11 12 Canonical partition function for ideal diatomic gas For the rotational partition function: For heteronuclear diatomic molecules (e.g., CO), j can take any integer value, 0 or above. For homonuclear diatomic molecules (e.g., H 2, O 2, N 2 ), j can be only even or odd. Let us begin with heteronuclear diatomic molecules: 13 Canonical partition function for ideal diatomic gas Note that the following group has units of temperature and is called rotational temperature: Then: 14 Canonical partition function for ideal diatomic gas We will make the assumption that the rotational energy levels are tightly spaced so that they can be treated as a continuous variable. This leads to: With: Replacing the summation with an integral is satisfactory if: 15 physical meaning of the rotational temperature 16 It gives us an estimate of the temperature at which the thermal energy (kT) equals the separation between rotational levels. At this T, the population of excited rotational states is significant. H 2 : 88 K HD: 65.8 K HCl: 15.2 K CO: 2.78 K NO: 2.45 K Sample values of : 17 most molecules are in excited rotational levels at ordinary Ts Canonical partition function for ideal diatomic gas What to do if the rotational partition function cannot be approximated by an integral? Expand it using a Taylor series: 18 Canonical partition function for ideal diatomic gas For the rotational partition function: Let us now consider homonuclear diatomic molecules: j can be only even or odd*. At high temperatures, when many terms need to be considered, each of these summations will have half of the terms used in a heteronuclear diatomic molecule: 19 due to symmetry properties of the wave function Canonical partition function for ideal diatomic gas For the rotational partition function: Symmetry number (=1, for heteronuclear diatomic molecule; =2, for homonuclear diatomic molecule) 20 Canonical partition function for ideal diatomic gas For the vibrational partition function: Computed based on the energy levels of a harmonic oscillator. From quantum mechanics, the energy level j is characterized by: Force constant between the atoms 21 22 Canonical partition function for ideal diatomic gas For the vibrational partition function: Note that the following group has units of temperature and is called vibrational temperature: 23 Canonical partition function for ideal diatomic gas For the vibrational partition function: 24 sum over states can be done exactly!!! Thermodynamic properties of diatomic ideal gases 25 Thermodynamic properties of diatomic ideal gases 26 Thermodynamic properties of diatomic ideal gases See expressions for several properties in pages 49 and 50 of the textbook; also problem Canonical partition function for ideal polyatomic gas As before, consider a system of N non-interacting identical molecules: A result from quantum mechanics: Atom of ideal gas in cubic box of volume V 28 Canonical partition function for ideal polyatomic gas As with diatomic ideal gases, rotation and vibration need to be considered: As in the previous chapter: 29 Canonical partition function for ideal polyatomic gas De Broglie wavelength: based on dual wave-particle nature of matter Atomic masses do not have to be the same 30 Canonical partition function for ideal polyatomic gas For the electronic partition function, as before: 31 32 For linear molecules there are two axes for rotation and two degrees of freedom associated with rotation. For non-linear molecules there are 3 axes and 3 moments of inertia. In a polyatomic molecule there is more than one vibrational mode. TypeTranslationRotationVibration Linear323n-5 Non- linear 333n-6 3n degrees of freedom total Canonical partition function for ideal polyatomic gas For the rotational partition function: Linear molecules such as CO 2 Rotational temperatures computed from the three principal moments of inertia of the nonlinear molecule Nonlinear molecule such as H 2 O 33 Canonical partition function for ideal polyatomic gas For the vibrational partition function: It accounts for chemical bond stretching motion (as with diatomic molecules) and for bond bending motion Linear molecules:Nonlinear molecules: 34number of vibrational modes: 35 36 Non linear rigid polyatomic 37 rigid non-linear polyatomic 38 3 moments of inertia 3 rotational constants if the three are equal, spherical top only two equal, symmetric top three different, asymmetric top 39 examples 40 spherical topsymmetrical top J=A, B, C 3 rotational temperatures Spherical top molecules 41 Allowed energies: J = 0, 1, 2, Degeneracy: Spherical top molecules 42 Can be solved analytically Rotational partition functions 43 Spherical top Asymmetric top Symmetric top Comparison to experiments 44 45 Thermodynamic properties of polyatomic ideal gases Formulas for Q, A, U, S, etc are rather long. Please refer to Section 4.4 in the textbook 46 A practical note on the heat capacities of ideal gases Practical empirical expressions for the heat capacities of ideal gases are of the form (or similar): 47 A practical note on the heat capacities of ideal gases Practical empirical expressions for the heat capacities of ideal gases are of the form (or similar): But the expressions derived in this and in the previous chapter are not power series in temperature 48 A practical note on the heat capacities of ideal gases Practical empirical expressions for the heat capacities of ideal gases are of the form (or similar): But the expressions derived in this and in the previous chapter are not power series in temperature There is one advantage of the simple expressions, though: the rigorous formulas are increasingly complicated as the number of atoms increases. Therefore, it may be better to use the simple power series for molecules with a large number of atoms. 49 A practical note on the heat capacities of ideal gases Inspired by the fact that rigorous formulas involve hyperbolic functions, the Design Institute of Physical Properties Research (DIPPR) now adopts the following empirical expressions for the heat capacities of ideal gases are of the form: Where k 1 -k 5 are fitted to experimental data 50