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Chapter 27: Problem 39

How much energy is required to ionize a hydrogen atom initially in the \(n=2\) state?

Short Answer

Expert verified
Answer: The energy required to ionize a hydrogen atom initially in the \(n=2\) state is \(3.4\,\text{eV}\).

Step by step solution

01

Review the Rydberg Formula

The Rydberg formula for hydrogen is given by: $$E_n = -\frac{Z^2R_H}{n^2}$$ where \(E_n\) is the energy of the electron in the hydrogen atom, \(n\) is the principal quantum number (1, 2, 3, ...), \(Z\) is the atomic number (1 for hydrogen), and \(R_H\) is the Rydberg constant, approximately \(13.6\,\text{eV}\).
02

Calculate the initial energy level

Using the Rydberg formula with \(n = 2\), we find the energy of the hydrogen atom in the initial state: $$E_2 = -\frac{1^2(13.6\,\text{eV})}{2^2} = -\frac{13.6\,\text{eV}}{4} = -3.4\,\text{eV}$$
03

Calculate the ionization energy level

To ionize the hydrogen atom, the electron must be removed from the atom completely, which corresponds to the energy level at infinity, \(E_\infty\). When \(n\) approaches infinity, the energy level becomes 0: $$E_{\infty} = 0$$
04

Calculate the difference between initial and ionization energy levels

To find the energy required to ionize the hydrogen atom initially in the \(n=2\) state, we subtract the initial energy level from the ionization energy level: $$E_{ionization} = E_{\infty} - E_2 = 0 - (-3.4\,\text{eV}) = 3.4\,\text{eV}$$ So, the required energy to ionize a hydrogen atom initially in the \(n=2\) state is \(3.4\,\text{eV}\).

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Most popular questions from this chapter

What is the cutoff frequency for an x-ray tube operating at \(46 \mathrm{kV} ?\)
A 640 -nm laser emits a 1 -s pulse in a beam with a diameter of $1.5 \mathrm{mm} .\( The rms electric field of the pulse is \)120 \mathrm{V} / \mathrm{m} .$ How many photons are emitted per second? [Hint: Review Section \(22.6 .]\)
What is the minimum potential difference applied to an x-ray tube if \(x\) -rays of wavelength \(0.250 \mathrm{nm}\) are produced?
(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?
Show that the cutoff frequency for an x-ray tube is proportional to the potential difference through which the electrons are accelerated.
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