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Chapter 34: Problem 27

What's the maximum wavelength of light that can ionize hydrogen in its ground state? In what spectral region is this?

Short Answer

Expert verified
The maximum wavelength of light that can ionize hydrogen in its ground state is approximately \(91.21 nm\), which is in the ultraviolet spectral region.

Step by step solution

01

Set ionization energy

Observe that the ionization energy of hydrogen in its ground state is -13.6 eV. Considering energy is always conserved, this energy will be equal to the energy of the incoming photon that's capable of ionizing the hydrogen atom.
02

Convert energy to Joules

Change the ionization energy from eV to joules because the constants h and c are expressed in terms of Joule. To convert, use the appropriate conversion factor: 1 eV = 1.6 x \(10^{-19}\) joules. Therefore, the ionization energy E is -13.6 eV = -13.6 * 1.6 x \(10^{-19}\) J.
03

Calculate the maximum wavelength

Plug the given constants into the Equation \(E = \frac{hc}{\lambda}\) and solve for \(\lambda\) to find the maximum wavelength that can ionize hydrogen in its ground state. Therefore, \(\lambda = \frac{hc}{E}\). Using values: \(h = 6.63 x 10^{-34}\) Js, and \(c = 3.00 x 10^{8}\) m/s and \(E = 13.6 * 1.6 x 10^{-19}\) J, the calculation becomes: \(\lambda = \frac{6.63 x 10^{-34} Js x 3.00 x 10^{8} m/s}{13.6 x 1.6 x 10^{-19} J}\)
04

Identify the spectral region

The final wavelength falls in the ultraviolet region of the electromagnetic spectrum.

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Most popular questions from this chapter

Find the rate of photon production by (a) a radio antenna broadcasting \(1.0 \mathrm{kW}\) at \(89.5 \mathrm{MHz},\) (b) a laser producing \(1.0 \mathrm{mW}\) of 633-nm light, and (c) an X-ray machine producing 0.10-nm X rays with total power \(2.5 \mathrm{kW}\)

A 150 -pm X-ray photon Compton-scatters off an electron and emerges at \(135^{\circ}\) to its original direction. Find (a) the wavelength of the scattered photon and (b) the electron's kinetic energy.

Calculate the wavelengths of the first three lines in the Lyman series for hydrogen.

Find the kinetic energy of an initially stationary electron after a 0.10 -nm \(\mathrm{X}\) -ray photon scatters from it at \(90^{\circ}\)

A proton and electron have the same de Broglie wavelength. How do their speeds compare, assuming \(v \ll c\) for both?
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