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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

An electron in a hydrogen atom is in the ground state (1s). Calculate the probability of finding the electron within a Bohr radius \(\left(a_{0}=0.05295 \mathrm{nm}\right)\) of the proton. The ground state wave function for hydrogen is: \(\psi_{1 s}(r)=A_{1 s} e^{-r / a_{0}}=e^{-r / a_{0}} / \sqrt{\pi a_{0}^{3}}\).

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.)

Section 38.2 established that an electron, if observed in the ground state of hydrogen, would be expected to have an observed speed of \(0.0073 c .\) For what atomic charge \(Z\) would an innermost electron have a speed of approximately \(0.500 c,\) when considered classically?

An electron made a transition between allowed states emitting a photon. What physical constants are needed to calculate the energy of photon from the measured wavelength? (select all that apply) a) the Plank constant, \(h\) b) the basic electric charge, \(e\) c) the speed of light in vacuum, \(c\) d) the Stefan-Boltzmann constant, \(\sigma\)

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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