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Chapter 39: Problem 59

What must be the temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wavelength can excite the electron in the Bohr-model hydrogen atom from the ground level to the \(n\) = 4 energy level?

Short Answer

Expert verified
The temperature must be approximately 29,810 K.

Step by step solution

01

Find the Energy Required for Transition

Using the Bohr model, the energy required to excite an electron from the ground state (n=1) to the fourth energy level (n=4) is given by the formula: \(E = -13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\) eV. Here, \(n_1 = 1\) and \(n_2 = 4\). Substitute these values into the formula to find the energy. We get: \(E = -13.6 \left(\frac{1}{1^2} - \frac{1}{4^2}\right) = 12.75\) eV.
02

Convert Energy to Wavelength

Using the photon energy equation \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} Js\) and \(c\) is the speed of light \(3 \times 10^8 m/s\), convert the energy from eV to joules \( (1 eV = 1.602 \times 10^{-19} J) \). Calculate the wavelength. \(12.75 \ eV \) becomes \(2.043 \times 10^{-18} J\) and \(\lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{2.043 \times 10^{-18}} = 9.72 \times 10^{-8} m\).
03

Apply Wien's Displacement Law

Wien's Displacement Law relates the peak wavelength \(\lambda_{max}\) of radiation from a blackbody to its temperature \(T\) as \(\lambda_{max} T = 2.898 \times 10^{-3} m\cdot K\). Solve for temperature \(T\) using the wavelength from Step 2. Substitute \(\lambda_{max} = 9.72 \times 10^{-8} m\), \(T = \frac{2.898 \times 10^{-3}}{9.72 \times 10^{-8}} K = 29810 \ K\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bohr model and Energy Levels
The Bohr model of the atom is a revolutionary concept that helps explain how electrons are arranged around an atomic nucleus. Imagine it like a mini solar system where electrons orbit the nucleus in distinct levels or shells. These shells are named by numbers such as n=1, n=2, and so on. Each of these levels has a specific energy.
  • The ground state is the lowest energy level, which is n=1.
  • To move an electron from a lower to a higher energy level, energy must be added; this is called excitation.
In this exercise, the electron moves from n=1 to n=4. The energy needed for this transition is calculated by considering the differences in energy for each level. The formula used is:\[ E = -13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV} \] Substituting n=1 and n=4 into this equation gives us the energy required to complete the excitation.
Photon Energy and Wavelength
Photon energy plays a crucial role in understanding how light interacts with matter. A photon is essentially a packet of light energy, and its energy is inversely proportional to its wavelength. This is expressed by the equation:\[ E = \frac{hc}{\lambda} \] where \(E\) is the energy of the photon, \(h\) is Planck's constant, and \(c\) is the speed of light.
  • Planck's constant \(h = 6.626 \times 10^{-34} \text{ Js}\).
  • The speed of light \(c = 3 \times 10^8 \text{ m/s}\).
When an electron is excited, it often absorbs or emits a photon corresponding to these energies. In Step 2 of the solution, we convert the energy from electron volts to joules and use this relationship to find the wavelength of light that has this energy.
Understanding Wien's Displacement Law
Wien's Displacement Law gives us a wonderful way to connect temperature and the wavelength of peak emission in blackbody radiation. Blackbodies are ideal emitters of thermal radiation. The law is articulated through the formula:\[ \lambda_{max} T = 2.898 \times 10^{-3} \text{ m}\cdot\text{K} \] Here, \(\lambda_{max}\) is the wavelength at which the emission is strongest, and \(T\) is the temperature.
  • According to Wien's Law, as temperature increases, the peak wavelength decreases, meaning hotter blackbodies emit light at shorter wavelengths.
  • This principle helps astrophysicists determine the temperature of distant stars by observing their light.
Thus, by using the wavelength derived from photon energy, this law lets us calculate the temperature needed for an ideal blackbody.

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Most popular questions from this chapter

(a) What accelerating potential is needed to produce electrons of wavelength 5.00 nm? (b) What would be the energy of photons having the same wavelength as these electrons? (c) What would be the wavelength of photons having the same energy as the electrons in part (a)?

An electron has a de Broglie wavelength of 2.80 \(\times\) 10\(^{-10}\) m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is 24,000 K and that it radiates energy at a total rate of 1.0 \(\times\) 10\(^{25}\) W. Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius B? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more \(total\) energy per second, the hot Sirius B or the (relatively) cool sun with a surface temperature of 5800 K? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature \(T\) and a diameter 3.0 times that of the hotter star. (a) What is the temperature of the hotter star in terms of \(T\) ? (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

The radii of atomic nuclei are of the order of 5.0 \(\times\) 10\(^{-15}\) m. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. (37.39), to obtain an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron separated by 5.0 \(\times\) 10\(^{-15}\) m. On the basis of your result, could there be electrons within the nucleus? (\(Note\): It is interesting to compare this result to that of Problem 39.72.)
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