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Chapter 38: Problem 56

What is the ionization energy of a hydrogen atom excited to the \(n=2\) state?

Short Answer

Expert verified
Answer: The ionization energy of a hydrogen atom excited to the \(n=2\) state is \(3.4\,\text{eV}\).

Step by step solution

01

Identify the hydrogen atomic energy level equation

The energy levels of a hydrogen atom can be calculated using the following equation: \(E_n = -\frac{13.6\,\text{eV}}{n^2}\) where \(n\) is the principal quantum number, and the energy \(E_n\) is in electron volts (eV).
02

Calculate the energy of the hydrogen atom at the \(n=2\) state

Using the energy level equation, we can calculate the energy of the hydrogen atom at the \(n=2\) state: \(E_2 = -\frac{13.6\,\text{eV}}{2^2} = -\frac{13.6\,\text{eV}}{4} = -3.4\,\text{eV}\)
03

Calculate the energy of the ionized state

When the hydrogen atom is ionized, the electron is completely removed and the energy of the atom is \(0\,\text{eV}\).
04

Calculate the ionization energy

The ionization energy is the difference in energy between the \(n=2\) state and the ionized state: Ionization Energy = \(E_\text{ionized} - E_2 = 0\,\text{eV} - (-3.4\,\text{eV}) = 3.4\,\text{eV}\) So, the ionization energy of a hydrogen atom excited to the \(n=2\) state is \(3.4\,\text{eV}\).

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Most popular questions from this chapter

A beam of electrons is incident upon a gas of hydrogen atoms. What minimum speed must the electrons have to cause the emission of light from the \(n=3\) to \(n=2\) transition of hydrogen?

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?

The binding energy of an extra electron when As atoms are doped in a Si crystal may be approximately calculated by considering the Bohr model of a hydrogen atom. a) Show the ground energy of hydrogen-like atoms in terms of the dielectric constant and the ground state energy of a hydrogen atom. b) Calculate the binding energy of the extra electron in a Si crystal. (The dielectric constant of Si is about 10.0 , and the effective mass of extra electrons in a Si crystal is about \(20.0 \%\) of that of free electrons.)

An 8.00 -eV photon is absorbed by an electron in the \(n=2\) state of a hydrogen atom. Calculate the final speed of the electron.

The radial wave function for hydrogen in the \(1 s\) state is given by \(R_{1 s}=A_{1} e^{-r / a_{0}}\) a) Calculate the normalization constant \(A_{1}\). b) Calculate the probability density at \(r=a_{0} / 2\). c) The \(1 s\) wave function has a maximum at \(r=0\) but the \(1 s\) radial density peaks at \(r=a_{0} .\) Explain this difference.
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