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Chapter 7: Q 7.38 (page 293)

It's not obvious from Figure 7.19 how the Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(ϵ) for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

Short Answer

Expert verified

The function is:

u(ϵ)=8π(hc)3ϵ3eϵ/kT-1

Step by step solution

01

Given information

The Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(ϵ) for T = 3000 K and T = 6000 K

02

Explanation

The photon's Planck spectrum is given as:

u(ϵ)=8π(hc)3ϵ3eϵ/kT-1(1)

This function must be plotted at temperatures of T= 3000 K and T = 6000 K, with the constants in eV supplied by:

h=4.136×10-15eV·sk=8.62×10-5eV/Kc=3.00×108m/s

Using python to plot the function and the code is:

The graph is:

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Most popular questions from this chapter

Starting from equation 7.83, derive a formula for the density of states of a photon gas (or any other gas of ultra relativistic particles having two polarisation states). Sketch this function.

Use the results of this section to estimate the contribution of conduction electrons to the heat capacity of one mole of copper at room temperature. How does this contribution compare to that of lattice vibrations, assuming that these are not frozen out? (The electronic contribution has been measured at low temperatures, and turns out to be about40% more than predicted by the free electron model used here.)

The sun is the only star whose size we can easily measure directly; astronomers therefore estimate the sizes of other stars using Stefan's law.

(a) The spectrum of Sirius A, plotted as a function of energy, peaks at a photon energy of2.4eV, while Sirius A is approximately 24times as luminous as the sun. How does the radius of Sirius A compare to the sun's radius?

(b) Sirius B, the companion of Sirius A (see Figure 7.12), is only role="math" localid="1647765883396" 3%as luminous as the sun. Its spectrum, plotted as a function of energy, peaks at about7eV. How does its radius compare to that of the sun?

(c) The spectrum of the star Betelgeuse, plotted as a function of energy, peaks at a photon energy of 0.8eV, while Betelgeuse is approximately10,000times as luminous as the sun. How does the radius of Betelgeuse compare to the sun's radius? Why is Betelgeuse called a "red supergiant"?

For a system of bosons at room temperature, compute the average occupancy of a single-particle state and the probability of the state containing 0,1,2,3bosons, if the energy of the state is

(a) 0.001eVgreater than μ

(b) 0.01eVgreater than μ

(c) 0.1eVgreater than μ

(d) 1eVgreater than μ

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) 1eVless than μ

(b) 0.01eVless than μ

(c) equal to μ

(d) 0.01eVgreater than μ

(e) 1eVgreater thanμ
See all solutions

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