A recurring theme throughout the study of quantum physics has been the omnipresence ofPlanck’s constant h. This universal constant of the microscopic world first made itsappearance in 1900 in the study of the radiation of so-called blackbodies.A blackbody is a substance that absorbs radiation of all wavelengths and radiates in a continuousspectrum at all wavelengths. It is given the name blackbody because an object that absorbs light at allwavelengths appears black to the human eye.By the end of the 19th century, several properties of blackbody radiation had been established. First, thetotal intensity I (the average rate of radiation of energy per unit surface area) emitted from a blackbodywas shown to be proportional to the fourth power of its temperature_I=?T4.This is called the Stefan-Boltzmann law for a blackbody. The constant of proportionality ? is known asthe Stefan-Boltzmann constant and was determined to be ?=5.67Ã10?8W/(m2?K4). It had alsobeen discovered that the wavelength at which the radiation intensity was maximum varied inversely withtemperature. This result, known as the Wien displacement law, is written?mT=2.90Ã10?3m?K,where ?m is the wavelength with the greatest radiated intensity.One aspect of blackbody radiation that remained unexplained was the full wavelength dependence of theintensity of the radiation, I(?). In 1900, largely through trial and error, Max Planck formulated the followingequation that successfully explained the wavelength dependence of the intensity:I(?)=2?hc2?5(ehc/?kBT?1),where h is Planck’s constant, c is the speed of light in vacuum, and kB is Boltzmann’s constant.Planck justified his law by claiming that different modes of electromagnetic oscillations within the cavitycould only emit radiation in increments of energy equal to Planck’s constant h multiplied by thefrequency f. At first, Planck did not believe in that idea himself, but the revolutionary conceptofquantization (or “clumping”) of energy paved the way for the “quantum revolution” in physics.Part AConsider a blackbody that radiates with an intensity I1 at a room temperature ofintensity I2 will this blackbody radiate when it is at a temperature of 400K?300K. At whatExpress your answer in terms of I1.I2 =??????Part BAt what wavelength ?m would the intensity of blackbody radiation be at a maximum when the blackbodyis at 2900K?Express your answer in meters to two significant figures.?m = ?????? mPart CAn astronomer is trying to estimate the surface temperature of a star with a radius of 5.0Ã108m bymodeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the starat a distance of 2.5Ã1013m and found it to be equal to 0.055W/m2. Given this information, what isthe temperature of the surface of the star?Express your answer in kelvins to two significant digits.T = ????? K2.Exciting an Oxygen Molecule An oxygen molecule (O2) vibrates with an energy identical tothat of a single particle of mass m=1.340Ã10?26kg attached to a spring with a force constant of k =1215N/m . The energy levels of the system are uniformly spaced, as indicated in the figure( Figure 1) ,with a separation given by hf.Part AWhat is the vibration frequency of this molecule?Express your answer using four significant figures.f = ???? HzPart BHow much energy must be added to the molecule to excite it from one energy level to the next higherlevel?Express your answer using three significant figures.E = ??????? J3. A hydrogen atom, initially at rest, emits an ultraviolet photon with a wavelength of ? =124nm .Part AWhat is the recoil speed of the atom after emitting the photon?v = ???? m/s4. A blue-green photon (? = 488nm ) is absorbed by a free hydrogen atom, initially at rest.Part AWhat is the recoil speed of the hydrogen atom after absorbing the photon?v = ????? m/s5. An X-ray photon with a wavelength of 0.260nm scatters from a free electron at rest. Thescattered photon moves at an angle of 110? relative to its incident direction.Part AFind the initial momentum of the photon.p=??????kg?m/sPart BFind the final momentum of the photon.p=?????? kg?m/s
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