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Chapter 38: Problem 24

Hydrogen atoms are bombarded with 13.1-eV electrons. Determine the shortest wavelength line the atom will emit.

Short Answer

Expert verified
Answer: The shortest wavelength line is 121.6 nm.

Step by step solution

01

Calculate the Energy of an Electron

First, we need to convert the energy of the electron in electron volts (eV) to Joules (J) using the relation 1 eV = 1.6 x 10^-19 J. This will allow us to have the energy in a more convenient unit for the calculations. Energy of electron in eV = 13.1 eV Energy of electron in Joules (J) = 13.1 eV * (1.6 * 10^-19 J/eV) = 2.096 x 10^-18 J.
02

Calculate the Maximum Change in Energy Levels

Since we need to find the shortest wavelength, we need to consider the maximum change in energy levels. The initial energy level (n_initial) is given by the energy of the bombarded electron. The hydrogen atom energy levels are given by the Rydberg formula: E_n = -13.6 eV/n^2, where E_n is the energy of the level n. Now we need to determine the highest possible energy level (n_final) when the electron absorbs the bombarded electron's energy. E_final = E_initial + 13.1 eV, where E_initial = -13.6 eV/n_initial^2, and E_final = -13.6 eV/n_final^2.
03

Solve for the Initial and Final Energy Levels

Solve the equation for n_final in terms of n_initial: -13.6 eV/n_final^2 = -13.6 eV/n_initial^2 + 13.1 eV. Now, we need to find the values of n_initial and n_final for which the energy difference is the highest: max[(-13.6 eV/n_initial^2) - (-13.6 eV/n_final^2)] = 13.1 eV, by using trial and error, we find n_initial = 1 and n_final = 3.
04

Calculate the Wavelength using the Rydberg Formula

Now that we have the initial and final energy levels, we can use the Rydberg formula for the hydrogen atom to calculate the wavelength: 1/λ = R_H * (1/n_initial^2 - 1/n_final^2), where λ is the wavelength and R_H is the Rydberg constant for hydrogen (1.097 x 10^7 m^-1). Plug in the values for n_initial = 1 and n_final = 3: 1/λ = 1.097 x 10^7 m^-1 * (1/1^2 - 1/3^2) = 1.097 x 10^7 m^-1 * (1 - 1/9) = 1.097 x 10^7 m^-1 * (8/9).
05

Calculate the Shortest Wavelength

Finally, calculate the shortest wavelength: λ = 1 / [1.097 x 10^7 m^-1 * (8/9)] = 1.216 x 10^-7 m. Hence, the shortest wavelength line that the hydrogen atom will emit when bombarded by 13.1-eV electrons is 1.216 x 10^-7 m or 121.6 nm.

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Most popular questions from this chapter

Calculate the energy needed to change a single ionized helium atom into a double ionized helium atom (that is, change it from \(\mathrm{He}^{+}\) into \(\mathrm{He}^{2+}\) ). Compare it to the energy needed to ionize the hydrogen atom. Assume that both atoms are in their fundamental state.

A muon is a particle very similar to an electron. It has the same charge but its mass is \(1.88 \cdot 10^{-28} \mathrm{~kg}\). a) Calculate the reduced mass for a hydrogen-like muonic atom consisting of a single proton and a muon. b) Calculate the ionization energy for such an atom, assuming the muon starts off in its ground state.

An excited hydrogen atom, whose electron is in an \(n=4\) state, is motionless. When the electron falls into the ground state, does it set the atom in motion? If so, with what speed?

The hydrogen atom wave function \(\psi_{200}\) is zero when \(r=2 a_{0} .\) Does this mean that the electron in that state can never be observed at a distance of \(2 a_{0}\) from the nucleus or that the electron can never be observed passing through the spherical surface defined by \(r=2 a_{0}\) ? Is there a difference between those two descriptions?

Which model of the hydrogen atom-the Bohr model or the quantum mechanical model-predicts that the electron spends more time near the nucleus?
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