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Lecture 18 – Quantization of energy Ch 9 pages 442-444 Summary of lecture 17 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 I ( , T ) 8 4 E 8 4 kT 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: p x2 x 2 E 2m 2 Quantization of Energy p x2 x 2 E 2m 2 The frequency of the harmonic oscillation is given in terms of the constants m (mass) and (spring force constant) as: 2 m Rearrange the equation for the harmonic oscillator as follows p x2 x 2 p x2 p x2 x 2 x 2 E 1 2 2 1 2m 2 2mE 2 E a b 2E a 2mE and b Quantization of Energy E p x 2 2m 2 2 x p x2 x 2 p x2 p x2 x 2 x 2 E 1 2 2 1 2m 2 2mE 2 E a b 2E a 2mE and b The expression: p x2 x 2 2 1 2 a b Represents an ellipse with semiaxes a and b. Therefore, the trajectory of a harmonic oscillator can be represented in momentumcoordinate 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 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: nh ab 2mE 1/ 2 2E / 1 / 2 2E k /m E 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: E=nh where n=0,1,2,3… 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 E ( n 1 / 2)hv Quantization of Energy E ( n 1 / 2)hv 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 DE=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 E ( n 1 / 2)hv 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: I ( , T ) 8 4 E And we can still calculate the energy using the relationship E kT 2 ln q T Quantization of Energy E ( n 1 / 2)hv I ( , T ) 8 4 E E kT 2 ln q T But q now has the ‘quantized’ form: n0 n0 q e E ( n )/ kT e nh / kT Therefore: nhe nh / kT ln q 1 q n 0 E kT 2 kT 2 T q T nh / kT e n 0 Quantization of Energy I ( , T ) 8 4 nhe nh / kT ln q 1 q n 0 E kT 2 kT 2 T q T nh / kT e n 0 E By expanding in terms of x e h / kT 1 2 x 3x 2 4 x 3 E hx 1 x x2 x3 If we introduce the general expression: 1 x n 1 nx n(n 1) x 2 .... (n 1 r )! x r 2! We find: (n 1)! r! (1 x ) 2 hx h E hx (1 x ) 1 1 x e h / kT 1 Quantization of Energy I ( , T ) 8 4 (1 x ) 2 hx h E hx (1 x ) 1 1 x e h / kT 1 E 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: I ( , T ) 8 4 E 8 e 4 h h / kT 1 8hc 5 e hc / kT 1 Where c is the speed of light so that c By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x10-34 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 I ( , T ) 8 4 E 8 e 4 h h / kT 1 8hc 5 e hc / kT 1 By fitting the equation for I(,T) to experimental data, Planck determined that the constant h=6.626x10-34 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 CV=7R/2 and CV=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, CV is closer to 5R/2 per mole of gas per diatomic molecules and CV 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: E Etrans Erotate Evibrate 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 E Etrans Erotate Evibrate If then q e Etr / kT e Erot / kT e Evib / kT ... q tr qrot qvib ... Using this fact we can rearrange the form for the molecular partition function: p x2 p 2y p z2 p2 p2 dp dp exp q V dp x dp y dp z exp 2lkT 2 kT p R2 / R 2 d Re xp 2kT We have partitioned q according to: q V qtrans qrotate qvibrate Molecular Partition Function of a Diatomic Molecule p x2 p 2y p z2 p2 p2 dp dp exp q V dp x dp y dp z exp 2lkT 2 kT p R2 / R 2 d Re xp 2kT We can calculate the energy from the relationship E ln Vxqtrans xqrot xqvib E ln q kT 2 kT 2 N T T Notice that the translational, rotational, and vibrational partition functions all involve integrals of the form: ax e dx 2 0 Therefore: 1 2 a qtrans kT 3 / 2 ; q rot kT ; q vib kT Molecular Partition Function of a Diatomic Molecule p x2 p 2y p z2 p2 p2 dp dp exp q V dp x dp y dp z exp 2lkT 2 kT 2 2 d Re xp p R / R 2kT qtrans kT 3 / 2 ; q rot kT ; q vib kT From which it immediately follows that: 7/2 E 7 2 ln q 2 ln Vxqtrans xqrot xqvib 2 ln T E kT kT kT kT N T T T 2 The energy per mole E and the heat capacity CV are then: E 7 RT 2 7R E CV 2 T V Molecular Partition Function of a Diatomic Molecule E 7 RT 2 7R E CV 2 T V 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 H2, CV 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 CV=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: E E trans E rotate E vibrate If we assume translational and rotational motions obey the equipartition principle, but that the vibrational motions obey quantum mechanical behavior, then we can write: E E trans E rotate E vibrate 5kT E 2 vibrate Molecular Partition Function of a Diatomic Molecule E E trans E rotate E vibrate 5kT E 2 vibrate Using Planck’s quantization hypothesis for harmonic oscillations and applying it to bond vibrations (homework), we can calculate the partition function to be: q e En / kT e nhv / kT (1 e hv / kT ) 1 n n 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: 1 E n ( n )hv 2 e hv / 2 kT q 1 e hv / kT Molecular Partition Function of a Diatomic Molecule E E trans E rotate E vibrate 5kT E 2 vibrate 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: E E CV T V T 5kT E 2 vibrate 5kT h h / kT 2 e 1 2 Nh 5Nk 5NkT h h / kT h / kT Nk e e 1 2 e h / kT 1 2 kT 2 Molecular Partition Function of a Diatomic Molecule E CV T V T 2 Nh 5Nk 5NkT h h / kT h / kT Nk e e 1 2 e h / kT 1 2 kT 2 In the low temperature limit (h>>kT): 2 5Nk 5Nk 5Nk h h / kT h / kT h CV Nk e 1 Nke h / kT e 2 2 2 kT kT 2 2 In the high temperature limit (h<<kT, using e hv / lT 1 hv / kT 2 5Nk 5Nk h h h h / kT h / kT h CV Nk e 1 Nk 1 e 1 1 2 2 kT kT kT kT 2 2 2 5Nk 7 Nk Nk 2 2 Molecular Partition Function of a Diatomic Molecule E CV T V T 2 Nh 5Nk 5NkT h h / kT h / kT Nk e e 1 2 e h / kT 1 2 kT 2 The high temperature value of CV 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 E CV T V T 2 Nh 5Nk 5NkT h h / kT h / kT Nk e e 1 2 e h / kT 1 2 kT 2 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.