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Chapter 8: Problem 13

What is \(\Delta E\) for the transition of an electron from \(n=6\) to \(n=3\) in a Bohr hydrogen atom? What is the frequency of the spectral line produced?

Short Answer

Expert verified
The energy change \(\Delta E\) for the transition of an electron from \(n=6\) to \(n=3\) in a Bohr hydrogen atom is -1.133 eV or -1.815x10^-19 Joules. The frequency of the spectral line produced is 2.74 x 10^{14} Hz.

Step by step solution

01

Calculate Energy at Level 6

First, calculate the energy of the electron at energy level \(n=6\) using the formula \(E_n = -13.6/ n^2\). Here, \(n = 6\), so \(E_{n=6} = -13.6/6^2 = -0.3778\) eV
02

Calculate Energy at Level 3

Next, calculate the energy of the electron at energy level \(n=3\) using the same formula. Here, \(n = 3\). Hence, \(E_{n=3} = -13.6/3^2 = -1.511\) eV
03

Calculate ∆E

Now calculate the change in energy, \(\Delta E \), which is the energy difference between the two levels. \(\Delta E = E_{n=3} - E_{n=6} = -1.511 - (-0.3778) = -1.133\) eV
04

Convert Energy to Joules

We will convert the energy difference from electron Volts (eV) to Joules (J) to utilize it further: 1eV = \(1.602\times10^{-19}\) J. Thus \(\Delta E = -1.133 \times 1.602\times10^{-19} = -1.815\times10^{-19}\) Joules.
05

Calculate Frequency

Now, calculate the frequency of the spectral line produced using the equation \(E = hf\), where \(h\) is Planck's constant. Solving for \(f\), we get \(f = \Delta E / h\). Given \(h = 6.626\times10^{-34}\) Js, \(f = -1.815\times10^{-19} / 6.626\times10^{-34} = 2.74 \times 10^{14}\) Hz.

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Most popular questions from this chapter

Briefly describe each of the following ideas or phenomena: (a) atomic (line) spectrum; (b) photoelectric effect; (c) matter wave; (d) Heisenberg uncertainty principle; (e) electron spin; (f) Pauli exclusion principle; (g) Hund's rule; (h) orbital diagram; (i) electron charge density; (j) radial electron density.

Although Einstein made some early contributions to quantum theory, he was never able to accept the Heisenberg uncertainty principle. He stated, "God does not play dice with the Universe." What do you suppose Einstein meant by this remark? In reply to Einstein's remark, Niels Bohr is supposed to have said, "Albert, stop telling God what to do." What do you suppose Bohr meant by this remark?

The following electron configurations correspond to the ground states of certain elements. Name each element. (a) \([\mathrm{Ar}] 3 d^{10} 4 s^{2} 4 p^{3} ;\) (b) \([\mathrm{Ne}] 3 s^{2} 3 p^{4} ;\) (c) \([\mathrm{Ar}] 3 d^{1} 4 s^{2}\) (d) \([\mathrm{Kr}] 4 d^{6} 5 s^{2} ;\) (e) \([\mathrm{Xe}] 4 f^{12} 6 s^{2}\)

What is the expected ground-state electron configuration for each of the following elements? (a) tellurium; (b) cesium; (c) selenium; (d) platinum; (e) osmium; (f) chromium.

Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.
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