Ch 9 pages Lecture 18 – Quantization of energy

 We have found that the spectral distribution of radiation emitted by a heated black body, modeled as a large number of atomic oscillators is predicted to be Summary of lecture 17 based on classical mechanical considerations. This result would predict that short wavelength radiation should be emitted with high intensity, contradicting experimental observations that short wavelength radiation is emitted with low intensities (bodies do not glow at low temperature).

Quantization of Energy In 1901, Max Planck published a quantum theory of radiation to explain the known spectral distribution of black body radiators. Unlike Raleigh, he only allowed the oscillators to adopt certain energy values, not all. Planck’s quantum hypothesis can be constructed as follows. He assumed that the black body is composed of a large number of oscillators whose energies obey the harmonic oscillator equation:

Quantization of Energy The frequency of the harmonic oscillation is given in terms of the constants m (mass) and  (spring force constant) as: Rearrange the equation for the harmonic oscillator as follows

Quantization of Energy The expression: Represents an ellipse with semi- axes a and b. Therefore, the trajectory of a harmonic oscillator can be represented in momentum- coordinate space as an ellipse:

Quantization of Energy Classically, when an oscillating atom emits radiation, its trajectory is modified as the momentum and the amplitude of displacement change. The energy emitted by an oscillator has no restricted values. But Planck assumed that in the black body, oscillator trajectories are restricted in such a way that only certain trajectories are possible.

Quantization of Energy E=nh where n=0,1,2,3… Stating that only certain trajectories are allowed means that only ellipses with certain values of a and b may exist. The area of an ellipse is  ab and we can express this quantum restriction on the motion and energy of an oscillator: Where is the oscillator frequency, h is a constant (Planck’s constant), and n is an integer. It follows that, under Plank’s quantum hypothesis, the energy of an oscillator is restricted by the quantization rule:

Quantization of Energy E=nh where n=0,1,2,3… We shall see next week that the correct expression for the quantized energy levels of a harmonic oscillator is actually

Quantization of Energy The second crucial hypothesis introduced by Planck was that, if an oscillator emits energy, it must pass from E=(n+1)h to say E=nh The quantum hypothesis restricts energy changes to  E=h. This means that energy is emitted into the cavity of the black body in discrete amounts or quanta. These energy particles are called photons and this hypothesis is crucial to explain spectroscopy, as we shall see later.

Quantization of Energy To formalize Plank’s equations, we can write his hypothesis to explain the black body phenomenon as follows. The intensity of radiation is still governed by the equation: And we can still calculate the energy using the relationship

Quantization of Energy But q now has the ‘quantized’ form: Therefore:

Quantization of Energy If we introduce the general expression: By expanding in terms of We find:

Quantization of Energy Where c is the speed of light so that Planck’s Quantum Theory of Black Body Radiation is summarized by the following expression for the light emitted as a function of temperature and frequency: By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x J-sec. The constant h is now called Planck’s constant. At high temperatures (kT>>h , Planck’s Radiation Law converges to the classical Jean’s Law.

Quantization of Energy By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x J-sec. The constant h is now called Planck’s constant. At high temperatures (kT>>h , Planck’s Radiation Law converges to the classical Jean’s Law.

Heat Capacities Revisited Heat capacities of diatomic gas molecules and crystalline solids are predicted to be C V =7R/2 and C V =3R, respectively, at room temperature These predictions are based on the assumption that vibrational motions contribute a factor of RT (per dimension) to the energy in accordance with the equipartition principle However, C V is closer to 5R/2 per mole of gas per diatomic molecules and C V is almost zero at room temperature for many solids.

Molecular Partition Function of a Diatomic Molecule From the discussion of the last class, the classical energy of a diatomic molecule is: This expression can be used to calculate the molecular partition function. First remember once again that, in Lecture 2, we mentioned the following fundamental property of the partition function To a high degree of approximation, the energy of a molecule in a particular state is the simple sums of various types of energy (translational, rotational, vibrational, electronic, etc.).

Molecular Partition Function of a Diatomic Molecule If Using this fact we can rearrange the form for the molecular partition function: then We have partitioned q according to:

Molecular Partition Function of a Diatomic Molecule We can calculate the energy from the relationship Notice that the translational, rotational, and vibrational partition functions all involve integrals of the form: Therefore:

Molecular Partition Function of a Diatomic Molecule From which it immediately follows that: The energy per mole E and the heat capacity C V are then:

Molecular Partition Function of a Diatomic Molecule or approximately 29 J/K-mole for a diatomic gas. This expression reflects the equipartition principle, each degree of freedom contributes 1/2R to the heat capacity or 1/2RT to the total energy of a system (per mole).

Molecular Partition Function of a Diatomic Molecule However, almost no diatomic gas obeys this expression. For example, for H 2, C V is approximately 20J/K-mole or approximately 5/2R An even more serious situation arises when we attempt to calculate the heat capacity of solids. If we regard the solid as a three-dimensional array of atoms, the motions executed by these atoms are vibrations Therefore, the motions of the atoms may be regarded as harmonic oscillations in three dimensions.

Molecular Partition Function of a Diatomic Molecule From the equipartition principle, we would expect the vibrational energy to be E=3RT and the contribution to the heat capacity from vibrational motions should be C V =3R In fact, at room temperature the vibrational heat capacity for crystalline solids is almost 0 and only approaches 3R at high temperatures These observations indicate serious failures of classical mechanics to accurately account for the behavior of polyatomic gases and solids These failures contributed to the birth of quantum mechanics.

Molecular Partition Function of a Diatomic Molecule Plank’s hypothesis can also be used to reexamine the heat capacities and their deviation from classical behavior as well Let us focus on diatomic gases by defining the average energy as: If we assume translational and rotational motions obey the equipartition principle, but that the vibrational motions obey quantum mechanical behavior, then we can write:

Molecular Partition Function of a Diatomic Molecule To be correct, as we shall see next week, the energy levels for a quantum mechanical 1-dimensional oscillator with characteristic frequency v are given by: Using Planck’s quantization hypothesis for harmonic oscillations and applying it to bond vibrations (homework), we can calculate the partition function to be:

Molecular Partition Function of a Diatomic Molecule If we limit ourselves to Planck’s description at this stage, then the partition function provided in the homework allows you to calculate properties such as the vibrational energy and specific heat:

Molecular Partition Function of a Diatomic Molecule In the low temperature limit (h >>kT): In the high temperature limit (h <<kT, using

Molecular Partition Function of a Diatomic Molecule The high temperature value of C V for diatomic molecules agrees with the equipartition principle, which is the result obtained using classical statistical mechanics. In this limit: kT>>h (notice that h is the separation between the vibrational energy levels). Classical statistical mechanics correctly predicts the vibrational heat capacity if the separation between the vibrational energy levels (i.e. energy quantization) is negligible compared to kT. But at low temperature, where quantization of energy levels is important, classical statistical mechanics fails and quantum effects become significant.

Molecular Partition Function of a Diatomic Molecule Later, when we develop a theory of quantum wave mechanics, we will show why quantization for vibrational motions is much more important at low temperatures than for translational and rotational motions Before we do that we will consider another problem that classical physics fails to explain: the stability of the hydrogen atom.