1. Chem 125 Lectures 12/02-04/02 Projected material This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further. It is not readily understood without reference to notes from the lecture.

2. Josiah Willard Gibbs (1839-1904) ~1855

3. Gibbs/Nernst

4. Gibbs 1902

5. Mechanical Theory of Heat Rudolph Clausius (1822-1888) “Thermo dynamics” “heat consists in a motion of the ultimate particles of bodies and is a measure of the vis viva of this motion.” (1852) particles

6. J. Willard Gibbs (1889) “Clausius was concerned with the mean values of various quantities which vary enormously in the smallest time or space which we can appreciate.”

7. J. Willard Gibbs (1889) “Maxwell occupied himself with the relative frequency of the various values which these quantities have. In this he was followed by Boltzmann.”

8. J. Willard Gibbs (1889) “In reading Clausius, we seem to be reading mechanics; in reading Maxwell, and in much of Boltzmann’s most valuable work, we seem rather to be reading in the theory of probabilities.”

9. James Clerk Maxwell (1831-1879) Distribution of Velocities On the Motions and Collisions of perfectly elastic Spheres (1859) f (v x ) probability of x-velocity between v x and v x + d v x

10. vxvx vzvz vyvy v v 2 = v x 2 + v y 2 + v z 2 Assume v x, v y, v z are independent g(v 2 ) = g(v x 2 + v y 2 + v z 2 ) = f(v x ) f(v y ) f(v z ) Product Sum g(v x 2 + v y 2 + v z 2 ) = c 3 e -a (v x 2 + v x 2 + v x 2 )  f(v x ) = c e -a v x 2

11. f(v) = C v 2 e -a v 2 vxvx vzvz vyvy v

12. v f(v) 1D 2D 3D Maxwell Velocity Distribution

13. Got to here on December 2. Remaining slides for December 4.

14. On the Relationship between the Second Law of Thermodynamics and Probability Calculation regarding the laws of Thermal Equilibrium (1877) S = k ln W Ludwig Boltzmann 1844 - 1906 Confirmed Maxwell by considering the implications of random distribution of energy. Generalized to include non-translational energy: Vibration Rotation etc.

15. Random Distribution of 3 “Bits” of Energy among 4 “Containers” How many “complexions” have N bits in the first container? 3 N#N# 3 1 2

16. Random Distribution of 3 “Bits” of Energy among 4 “Containers” How many “complexions” have N bits in the first container? 6 N#N# 123 31

17. How many “complexions” have N bits in the first container? 0 6 N#N# 123 31 Random Distribution of 3 “Bits” of Energy among 4 “Containers” 10

18. 0 6 N#N# 123 31 30 bits of energy in 20 molecules 3 bits of energy in 4 “molecules” 30 in 20

19. (N)(N) E  (E) e -E/kT Boltzmann showed Exponential limit for lots of infinitesimal energy bits E ave = 1/2 kT If all “complexions” for a given E total are equally likely, shifting energy to any one degree of freedom of any one molecule is disfavored. By reducing the energy available elsewhere, this reduces the number of relevant complexions.

20. Boltzmann (1898) Lectures on Gas Theory “I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that when the theory of gases is again revived, not too much will have to be rediscovered.”

21. Max Planck 1858-1947 On the Theory of the Law of Energy Distribution in the Normal Spectrum (December 14, 1900) “I had always regarded the search for the absolute as the loftiest goal of all scientific activity” 2 nd Law - Irreversibility

22. Max Planck 1858-1947 According to Maxwell’s EM Theory light emission is proportional to: Energy 2 On the Theory of the Law of Energy Distribution in the Normal Spectrum (December 14, 1900)

23. Ferdinand Kurlbaum (Berlin PTR, 1898) “Black Body” radiation at 100°C is 0.0731 watts/cm 2 greater than at 0°C. steam ice water

24. Otto Lummer & Ernst Pringsheim (PTR, Feb.1900) max T = 2940  m K max T = Const Wilhelm Wien (PTR, 1893) 1.782  m Wien (PTR, 1896)

25. Equations Planck (Oct. 19, 1900) Wien (PTR, 1896) Lummer & Pringsheim (PTR, Feb. 1900) Thiesen (PTR, Feb. 1900) “simple form…more likely to indicate the possibility of a general significance” Why are high frequency oscillators underpopulated? ?

26. "We treat E however - and this is the most significant point in the whole calculation - as composed of specific number of identical finite parts and make use for that purpose of the constant of nature h = 6.55 x 10 -27 [erg x sec]. This constant multiplied by the common frequency of the oscillators [within a given family] gives the energy element  in ergs.” Planck’s “Desperation” Hypothesis (Dec. 14, 1900)

27. Planck’s Hypothesis (Dec. 14, 1900) Forced to choose between putting lots of energy into a degree of freedom, or none at all, statistics will choose none at all - until the temperature becomes high enough that localizing so large an amount of energy is not very unlikely. Planck’s “Desperation” Hypothesis (Dec. 14, 1900) *(h, when is large) *

28. k and h Planck Empirical E( ) Planck Theoretical E( ) Lummer and Pringsheim’s max T = 2940  m K gave h/k Kurlbaum's 0.0731 watts/cm 2 difference in radiant heat between 100°C and 0°C gave k 4 /h 3 Gave h within 1%, k within 2.5% (Avogadro-Lohschmidt ; Faraday ; electron charge)

29. A critical comparison of the two processes would be of interest, Lord Rayleigh (1905) but not having succeeded in following Planck’s reasoning, I am unable to undertake it.

30. to Wilhelm Wien 1911 Nobel Prize in Physics for work on Black-Body Radiation for his Displacement Law & his Spectral Distribution Law (~valid at short wavelengths) “The problem now became to bridge the gap between these two laws [Wien & Rayleigh]… It was Planck who solved this problem; as far as we are aware, his formula provides the long sought-after connecting link…” (Nobel Presentation Address) as far as we are aware

31. March 1905 Generalization that light energy is intrinsically quantized (not just when it enters or leaves a molecular oscillator) Albert Einstein (age 26) April 1906 May 1905 Theory of Brownian Motion June 1905 Special Relativity - SpaceTime Sept. 1905 E = mc 2 Experte III. Klasse  Experte II. Klasse

32. Max Planck (1858-1947) Nobel Prize 1918 Succeeded (1928) by Erwin Schrödinger President Kaiser Wilhelm Society 1930-1937 Brought Einstein to Berlin (1914) “Reluctant Revolutionary” ; 1945-1946