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Chapter 8: Problem 47

What must be the velocity of electrons if their associated wavelength is to equal the radius of the first Bohr orbit of the hydrogen atom?

Short Answer

Expert verified
The velocity of the electrons, at which the associated wavelength is equal to the radius of the first Bohr orbit of the hydrogen atom, is approximately \(1.4 \times 10^{6} m/s\).

Step by step solution

01

Understand the de Broglie Wavelength

According to de Broglie, every particle with momentum has a wavelength associated with it. The de Broglie wavelength is given by λ = h / mv where: λ is the wavelength, h is the Planck's constant which equal to \(6.626 \times 10^{-34} Js\), m is the mass of the electron which equals \(9.1 \times 10^{-31} kg\), and v is the velocity of the electron.
02

Bohr's Radius

The radius of the first Bohr orbit (also known as Bohr radius) for hydrogen is approximately 0.529 Å. Since we want the electron's wavelength to equal the first Bohr orbit radius, we set λ = 0.529 Å. However, to be consistent with our units, we must convert this to meters (m). 1 Å = \(1 \times 10^{-10} m\), so 0.529 Å = \(0.529 \times 10^{-10} m\)
03

Solve for Velocity

Now that we have the de Broglie wavelength and Bohr's radius, we set them equal and solve for velocity v: \(0.529 \times 10^{-10} m = \frac{6.626 \times 10^{-34} Js}{9.1 \times 10^{-31} kg*v}\). Solving for v we get \(v = \frac{6.626 \times 10^{-34} Js}{9.1 \times 10^{-31} kg*\times 0.529 \times 10^{-10} m}\), which when simplified equals approximately \(1.4 \times 10^{6} m/s\).

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