Problem 26
(a) For one-electron ions with nuclear charge Z, what is the speed of the electron in a Bohr-model orbit labeled with \(n\)? Give your answer in terms of \(v_1\), the orbital speed for the \(n\) = 1 Bohr orbit in hydrogen. (b) What is the largest value of Z for which the \(n\) = 1 orbital speed is less than 10\(\%\) of the speed of light in vacuum?
Problem 29
(a) An atom initially in an energy level with \(E\) = -6.52 eV absorbs a photon that has wavelength 860 nm. What is the internal energy of the atom after it absorbs the photon? (b) An atom initially in an energy level with \(E\) = -2.68 eV emits a photon that has wavelength 420 nm. What is the internal energy of the atom after it emits the photon?
Problem 31
Using a mixture of CO\(_2\), N\(_2\), and sometimes He, CO\(_2\) lasers emit a wavelength of 10.6 \(\mu\)m. At power outputs of 0.100 kW, such lasers are used for surgery. How many photons per second does a CO\(_2\) laser deliver to the tissue during its use in an operation?
Problem 32
Pulsed dye lasers emit light of wavelength 585 nm in 0.45-ms pulses to remove skin blemishes such as birthmarks. The beam is usually focused onto a circular spot 5.0 mm in diameter. Suppose that the output of one such laser is 20.0 W. (a) What is the energy of each photon, in eV? (b) How many photons per square millimeter are delivered to the blemish during each pulse?
Problem 33
How many photons per second are emitted by a 7.50-mW CO\(_2\) laser that has a wavelength of 10.6 \(\mu\)m?
Problem 34
Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers 0.25 \(\mu\)m thick using pulses lasting 12.0 ns from a laser beam of wavelength 193 nm. Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. (a) In what part of the electromagnetic spectrum does this light lie? (b) What is the energy of a single photon? (c) If a 1.50-mW beam is used, how many photons are delivered to the lens in each pulse?
Problem 35
A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5\(s\) state to the number in a 3\(p\) state at (a) 300 K; (b) 600 K; (c) 1200 K? The energies of these states, relative to the ground state, are E\(_{5s}\) = 20.66 eV and E\(_{3p}\) = 18.70 eV. (d) At any of these temperatures, the rate at which a neon gas will spontaneously emit 632.8-nm radiation is quite low. Explain why.
Problem 38
Determine \(\lambda_m\) , the wavelength at the peak of the Planck distribution, and the corresponding frequency \(f\), at these temperatures: (a) 3.00 K; (b) 300 K; (c) 3000 K.
Problem 39
Radiation has been detected from space that is characteristic of an ideal radiator at \(T\) = 2.728 K. (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?
Problem 40
The shortest visible wavelength is about 400 nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?
