215.1 IntroductionStatistical thermodynamics provides deep insight into the classical description of a MONATOMIC ideal gas.In classical thermodynamics, the principle of equipartition of energy fails to give the observed value of the specific heat capacity for diatomic gases.The explanation of the above discrepancy was considered to be the most important challenge in statistical theory.
315.1 The quantized linear oscillator A linear oscillator is a particle constrained to move along a straight line & acted on by a restoring forceF= -Kx = ma =If displaced from its equilibrium position and released, the particle oscillates with simple harmonic motion of frequency , given byNote that the frequency depends on K and m, and is independent of the amplitude X.
4Consider an assembly of N one-dimensional harmonic oscillators, in which the oscillators are loosely coupled so that the energy exchange among them is small.In classical mechanics, a particle can oscillate with any amplitude and energy.From quantum mechanics, the single particle energy levels are given byEJ = (J + ½)h , J = 0, 1, 2, …..The energies are equally spaced and the ground state has non-zero energy.
5The states are nondegenerate, i.e. gj = 1 The internal degrees of freedom include vibrations, rotations, and electronic excitations.For internal degrees of freedom, Boltzmann Statistics applies. The distinguishable property arises from the fact that those diatomic molecules have different translational energy.The states are nondegenerate, i.e. gj = 1The partition function of an oscillator
6Introducing the characteristic temperature θ = h/k The solution for the above eq. is (in class derivation)
11At low temperature limit On has So approaching zero faster than the growth of (θ/T)2 as T → 0
12Therefore Cv 0 as T 0The total energy of a diatomic molecule is made up of four contributions that can be separately treated:
131. The kinetic energy associated with the translational motion The vibrational motionRotation motion (To be discussed later)Example: 15.1 a) Calculate the fractional number of oscillators in the three lowest quantum states (j=0, 1, 2,) forSol:
1715.4 Rotational modes of diatomic molecules The moment of inertia, where μ is the reduced mass r0 is the equilibrium value of the distance between the nuclei.
18From quantum mechanics, the allowed angular momentum states are where l = 0, 1, 2, 3… … From classical mechanics, the rotational energy equals with w is angular velocity. The angular momentum therefore, the energy
19Define a characteristic temperature for rotation θrot can be found from infrared spectroscopy experiments, in which the energies required to excite the molecules to higher rotational states are measured. Different from vibrational motion, the energy levels of the above equation are degenerate. for level
20Now, one can get the partition function For , virtually all the molecules are in the few lowest rotational states. As a result, the series of can be truncated with negligible errors after the first two or three terms!
21For all diatomic gases, except hydrogen, the rotational characteristics temperature is of the order of 10 k (Kelvin degree). At ordinary temperature, Therefore, many closely spaced energy states are excited. The sum of may be replaced by an integral. Define:
22Note that the above result is too large for homonuclear molecules such as H2, O2 and N2 by a factor of 2… why? The slight modification has no effect on the thermodynamics properties of the system such as the internal energy and the heat capacity!
23Using (Note: ) again At low temperature Keeping the first two terms
25Characteristic Temperatures Characteristic temperature of vibration of diatomic moleculesSubstance θvib(K)HONHClCONOClCharacteristic temperature of rotation of diatomic moleculesSubstance θrot(K)HONHClCONOCl
2615.5 Electronic Excitation The electronic partition function is where g0 and g1, are, respectively, the degeneracies of the ground state and the first excited state. E1 is the energy separation of the two lowest states. Introducing
27For most gases, the higher electronic states are not excited (θe ~ 120, 000k for hydrogen). therefore, At practical temperature, electronic excitation makes no contribution to the external energy or heat capacity!
2815.6 The total heat capacity For a diatomic molecule systemSinceDiscussing the relationship of T and Cv (p )
30Example I (problem 15.7) Consider a diatomic gas near room temperature. Show that the entropy is Solution: For diatomic molecules
31For translational motion, the molecules are treated as non-distinguishable assemblies
32For rotational motion (they are distinguishable in terms of kinetic energy)
33Example II (problem 15-8) For a kilomole of nitrogen (N2) at standard temperature and pressure, compute (a) the internal energy U; (b) the Helmholtz function F; and (c) the entropy S.Solution:calculate the characteristic temperature first!