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Chapter 27: Problem 51

Calculate, according to the Bohr model, the speed of the electron in the ground state of the hydrogen atom.

Short Answer

Expert verified
Answer: The speed of an electron in the ground state of a hydrogen atom, according to the Bohr model, is approximately 2.188 * 10^6 m/s.

Step by step solution

01

Remember the Bohr model formula for the speed of an electron in an atom

According to the Bohr model, the speed of an electron in an orbit is given by the formula: v = (Z * e^2) / (2 * epsilon_0 * h * n) where: v is the speed of the electron, Z is the atomic number of the element, e is the elementary charge (1.602 * 10^{-19} C), epsilon_0 is the vacuum permittivity (8.854 * 10^{-12} C^2/N m^2), h is the Planck's constant (6.626 * 10^{-34} Js), and n is the principal quantum number (energy level).
02

Set the values for hydrogen in the ground state

Since we are working with a hydrogen atom, its atomic number Z = 1. The ground state corresponds to the lowest energy level, so n = 1. Now we can plug in these values into the formula: v = (1 * 1.602 * 10^{-19}) / (2 * 8.854 * 10^{-12} * 6.626 * 10^{-34} * 1)
03

Calculate the speed of the electron

By calculating the above expression, we get: v = (1.602 * 10^{-19}) / (2 * 8.854 * 10^{-12} * 6.626 * 10^{-34}) v ≈ 2.188 * 10^6 m/s So, the speed of the electron in the ground state of the hydrogen atom, according to the Bohr model, is approximately 2.188 * 10^6 m/s.

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Most popular questions from this chapter

Suppose that you have a glass tube filled with atomic hydrogen gas (H, not \(\mathrm{H}_{2}\) ). Assume that the atoms start out in their ground states. You illuminate the gas with monochromatic light of various wavelengths, ranging through the entire IR, visible, and UV parts of the spectrum. At some wavelengths, visible light is emitted from the \(\mathrm{H}\) atoms. (a) If there are two and only two visible wavelengths in the emitted light, what is the wavelength of the incident radiation? (b) What is the largest wavelength of incident radiation that causes the \(\mathrm{H}\) atoms to emit visible light? What wavelength(s) is/are emitted for incident radiation at that wavelength? (c) For what wavelengths of incident light are hydrogen ions \(\left(\mathrm{H}^{+}\right)\) formed?
(a) Light of wavelength 300 nm is incident on a metal that has a work function of 1.4 eV. What is the maximum speed of the emitted electrons? (b) Repeat part (a) for light of wavelength 800 nm incident on a metal that has a work function of \(1.6 \mathrm{eV} .\) (c) How would your answers to parts (a) and (b) vary if the light intensity were doubled?
A photon passes near a nucleus and creates an electron and a positron, each with a total energy of \(8.0 \mathrm{MeV} .\) What was the wavelength of the photon?
What happens to the energies of the characteristic x-rays when the potential difference accelerating the electrons in an x-ray tube is doubled?
A \(100-\) W lightbulb radiates visible light at a rate of about $10 \mathrm{W} ;$ the rest of the EM radiation is mostly infrared. Assume that the lightbulb radiates uniformly in all directions. Under ideal conditions, the eye can see the lightbulb if at least 20 visible photons per second enter a dark-adapted eye with a pupil diameter of \(7 \mathrm{mm}.\) (a) Estimate how far from the source the lightbulb can be seen under these rather extreme conditions. Assume an average wavelength of 600 nm. (b) Why do we not normally see lightbulbs at anywhere near this distance?
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