# Statistical Mechanics

## Presentation on theme: "Statistical Mechanics"— Presentation transcript:

Statistical Mechanics
Planck radiation law Wien’s Displacement Law Stefan-Boltzmann Law Apr. 11 go to slide 11. The laws of thermodynamics: You can’t get anything without working for it. The best you can accomplish by work is to break even. You can’t break even.

Clarification of testable material in chapter 9
skip section 9.7 skip section 9.8 do sections skip section 9.11. “Testable” sections from chapter 9: 1 through 6, 9, 10.

9.6 Planck Radiation Law In the last section we derived the Rayleigh-Jeans formula for the spectral energy density for blackbody radiation. We found that it failed miserably. What's wrong with the Rayleigh-Jeans formula? Rayleigh divided up the energy into kT/2 for each degree of freedom. This is fine for a continuous energy distribution. We now know that it doesn't work for harmonic oscillators, where energy is quantized in units of hf. Therefore, our energy density was wrong.

Our density of standing wave modes,
as calculated above is OK. To get the correct energy density of photons in the cavity, we multiply our density of modes by the energy of each mode and the probability as a function of T that a mode is occupied. Warning—two different f ’s; might be better to use  for frequency here.

Wien's displacement law
The last equation on the previous slide is called the Planck radiation formula, and it works very well. Plug it into Mathcad and try it some time. Note that  was set equal to 0 in Bose-Einstein distribution function, which was used in the equation above.  is a constant which, mathematically, “says” that the number of particles is conserved. If we don't require conservation of particles, we set =0. Wien's displacement law You can calculate the wavelength at the peak of the energy density by expressing u as a function of wavelength and taking the derivative.

Use c = f  to write u = u() and set 0 = du /d to find  for maximum
Use c = f  to write u = u() and set 0 = du /d to find  for maximum. The result is Wien's displacement law:  at which u is maximum! not maximum  in spectrum! Wien's displacement law tells which way the peak in the blackbody spectrum shifts as the temperature changes.

But the Rayleigh-Jeans formula blows up as f   and as
You can write a formula for u(f) as we did, or for u(). The plots look similar: But the Rayleigh-Jeans formula blows up as f   and as   0 (can be visually confusing!). Also, the peak in the spectrum shifts towards higher f as T increases, but to lower . In these plots, B corresponds to our u and  corresponds to our f.

Homework problem The brightest part of the spectrum of the star Sirius is located at a wavelength of 290 nm. What is the surface temperature of Sirius? Aren’t stars hotter than this? Yes—interior temperatures, and temperatures a short distance above the surface, reach millions of degrees. Aren’t stars hotter than this?

Applications: Portable, variable temperature blackbody radiation source for precision calibration of radiation thermometers. Portable thermal imager.

Anybody here have children?
Whey you do, they come with a sickness guarantee. They are guaranteed to get sick. Then you have to take their temperature. Sure. Ever tried to get a screaming 2-year old to hold a thermometer in his (her) mouth? Taking a sick toddler’s temperature is a 2-parent job. Enough said. Sure. Or you can get one of these. Worth every penny of the \$35-40 it costs. No, I am not affiliated with any thermometer company!

The Stefan-Boltzmann Law
The total energy density inside the cavity is found by integrating the energy density over all frequencies. a constant The result is proportional to T4. The Stefan-Boltzmann law says R=eσT4, where R is the radiated energy per second per unit area, and e is the emissivity of the radiating surface, and σ is a constant (see Beiser for the value of σ).

Homework problem If a certain blackbody radiates 1 kW when its temperature is 500 ºC, at what rate will it radiate when its temperature is 750 ºC? some of the other problems for this material are more complex, but are still fair game for the exam don’t forget!

Global Warming? It is a fact that human activity has put enormous amounts of “extra” CO2 into the atmosphere since the beginning of the industrial revolution. It is also a fact that CO2 is a greenhouse gas. There exists extensive data which suggests the earth is warming. Much of the data is subject to interpretation (i.e., one might say “yes, warming,” or one might say “coincidence, not proof”).

My sense is that “most” scientists believe something is truly going on
My sense is that “most” scientists believe something is truly going on. Some very emphatically. Others less so. What will be the result of increasing earth temperature? We don’t know. There will be change. Perhaps dramatic. Perhaps not in the direction we might predict. How do you feel about change? Global warming is intimately connected to blackbody radiation and the Stefan-Boltzmann law. Opinion: those running our government need to learn more physics.

Consider designing a cryostat or furnace
Consider designing a cryostat or furnace. What things would you worry about? What would you do to make the cryostat or furnace work well?

Fall 2003: Something to ponder while I hand out quizzes…