Quantum Mechanics 101 · Unit 1: The Classical World Cracks

Quantum Academy · Chapter 3: Blackbody Radiation & Planck's Desperate Act

The ultraviolet catastrophe, and the 1900 guess that started a revolution

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Instructions. Try every question without the lesson first, then check. Use b = 2.898×10⁻³ m·K, σ = 5.670×10⁻⁸ W/(m²·K⁴), h = 6.626×10⁻³⁴ J·s, k = 1.381×10⁻²³ J/K, c = 2.998×10⁸ m/s and 1 eV = 1.602×10⁻¹⁹ J. Show every step in the calculations: in my lab, a number without its reasoning scores nothing. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Blackbody
2.Thermal radiation
3.Thermal equilibrium
4.Spectral radiance
5.Wien's displacement law
6.Stefan-Boltzmann law
7.Emissivity
8.Mode
9.Equipartition theorem
10.Ultraviolet catastrophe
11.Boltzmann factor
12.Planck's constant

A.P = εσA·T⁴: the total power a surface radiates grows as the fourth power of its absolute temperature.
B.An ideal object that absorbs all electromagnetic radiation falling on it, at every wavelength and angle; in thermal equilibrium its emission spectrum depends only on its temperature.
C.λₘₐₓ·T ≈ 2.898×10⁻³ m·K: a blackbody's peak wavelength is inversely proportional to its absolute temperature.
D.The ratio of a surface's thermal emission to that of a blackbody at the same temperature; symbol ε, between 0 and 1.
E.The power a surface emits per unit area, per unit solid angle and per unit wavelength (or frequency) interval: the height of a blackbody curve.
F.The classical rule that in thermal equilibrium each independent quadratic energy term holds an average of kT/2, so each oscillator averages kT.
G.The classical prediction, from the Rayleigh-Jeans law, that blackbody radiation grows without limit at short wavelengths and has infinite total energy.
H.Electromagnetic radiation emitted by matter because of its temperature, produced by the thermal jiggling of charged particles.
I.The constant h, exactly 6.62607015×10⁻³⁴ J·s, that sets the size of an energy packet through E = hf.
J.The relative probability e^(−E/kT) that a system in thermal equilibrium at temperature T is found in a state of energy E.
K.One allowed standing-wave pattern of a confined wave, such as the radiation trapped in a cavity.
L.A state in which a body has a single steady temperature and exchanges no net energy with its surroundings.

Part B Questions

1.
Which property defines an ideal blackbody?Circle one
1. AIt is coated with the darkest available pigment
2. BIt absorbs all electromagnetic radiation that falls on it, at every wavelength and angle
3. CIt emits radiation at a single wavelength set by its temperature
4. DIt reflects visible light but absorbs infrared
2.
A blackbody at 6,000 K would look black to your eye.Circle one
TrueFalse
3.
In 1900 Planck proposed that light travels through space as a stream of particles.Circle one
TrueFalse
4.
Wien's displacement law states that a blackbody's peak wavelength is inversely proportional to its absolute .Fill in the blank
5.
Paul Ehrenfest's 1911 name for the classical prediction of unlimited short-wavelength radiation is the catastrophe.Fill in the blank
6.
Four pieces of the same steel glow in a dark forge. Which is hottest?Circle one
1. AThe one glowing dull red
2. BThe one glowing orange
3. CThe one glowing white
4. DThey must all be the same temperature, because colour depends only on the steel
7.
In the Rayleigh-Jeans calculation, which assumption leads directly to infinite total energy?Circle one
1. AEach of the unlimited number of cavity modes holds an average energy kT, whatever its frequency
2. BThe cavity walls are made of metal
3. CLight travels at the finite speed c
4. DThe number of modes falls at high frequency
8.
The cosmic microwave background has a blackbody spectrum at 2.725 K. At what wavelength does it peak, and in which part of the spectrum is that?Show your working
Working
Answer:
9.
Human skin is at about 306 K (33 °C). Where does its thermal emission peak, and why do thermal cameras work in the 8 to 14 μm band?Show your working
Working
Answer:
10.
An electric stove element has a radiating surface area of 0.010 m² and glows at 1,000 K. Treating it as a blackbody (ε = 1), how much power does it radiate? How much would it radiate at 500 K?Show your working
Working
Answer:
11.
A lamp filament is heated from 2,000 K to 3,000 K. Find its peak wavelength at each temperature, and the factor by which its radiated power increases.Show your working
Working
Answer:
12.
One of Planck's oscillators has a frequency of 5.00×10¹⁴ Hz. What is the size of one energy packet in joules and in electron-volts, and how many such packets make up 1.00 J?Show your working
Working
Answer:
13.
In two or three sentences, explain why quantizing oscillator energy in packets of hf removes the ultraviolet catastrophe.Short answer
14.
Why is Max Planck often called a reluctant revolutionary?Short answer

Part C Mad scientist challenge bonus

Weigh Planck's constant with a thermometer, as Planck did in 1900. Planck's law predicts two results you have met: the Wien constant is b = hc/(4.965k), and the Stefan-Boltzmann constant is σ = 2π⁵k⁴/(15c²h³). Experiment gives b = 2.898×10⁻³ m·K and σ = 5.670×10⁻⁸ W/(m²·K⁴), and c = 2.998×10⁸ m/s. (a) Use b to find the ratio h/k. (b) Rewrite the formula for σ so that k is expressed in terms of σ, c and h/k, and compute k. (c) Find h. (d) With the gas constant R = 8.314 J/(mol·K), compute Avogadro's number, N = R/k. (e) Planck's 1900 inputs were b = 2.94×10⁻³ m·K and σ ≈ 5.30×10⁻⁸ W/(m²·K⁴), and he obtained h = 6.55×10⁻³⁴ J·s and k = 1.346×10⁻²³ J/K. By what percentage was each of his results off, and why?

Working

Memory aid

Mnemonic Wien shifts, Stefan soars, Planck prices; red is the warm-up, white is the blaze. Picture a furnace with three gauges. Turn up the heat and Wien shifts the peak to shorter wavelengths, λₘₐₓ = b/T. Stefan soars: the power climbs as T⁴. Planck prices: every packet of energy at frequency f costs hf, while the thermal budget is only about kT, so high frequencies are priced out of reach. On any blackbody problem, recite the three in order. Shift finds the colour or the temperature, soar finds the power, and price explains why the curve falls at short wavelengths instead of exploding. The tail line guards against the myth: red is the first, coolest glow; white means hotter.

In one line Hot objects glow with a temperature-set spectrum; classical physics predicted infinite ultraviolet energy until Planck's packets, E = hf, priced high frequencies out.