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Chapter 27: Problem 36

How much energy must be supplied to a hydrogen atom to cause a transition from the ground state to the \(n=4\) state?

Short Answer

Expert verified
Answer: The energy required for the transition is 12.75 eV.

Step by step solution

01

Energy level formula for hydrogen atom

The formula to calculate the energy levels of the hydrogen atom is: \(E_n = -\frac{13.6 eV}{n^2}\), where \(E_n\) is the energy of the nth energy level and n is the principal quantum number.
02

Calculate the energy of the ground state (n=1)

Using the energy level formula, we can calculate the energy of the ground state (n=1): \(E_1 = -\frac{13.6 eV}{1^2} = -13.6 eV\)
03

Calculate the energy of the n=4 state

Now, we can calculate the energy of the n=4 state using the same formula: \(E_4 = -\frac{13.6 eV}{4^2} = -\frac{13.6 eV}{16} = -0.85 eV\)
04

Find the energy difference between the two states

To find the energy required for the transition, we need to calculate the energy difference between the two states: \(\Delta E = E_4 - E_1 = (-0.85 eV) - (-13.6 eV) = 12.75 eV\)
05

Final answer

The energy that must be supplied to cause a transition from the ground state to the n=4 state in a hydrogen atom is 12.75 eV.

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Most popular questions from this chapter

In a color TV tube, electrons are accelerated through a potential difference of \(20.0 \mathrm{kV} .\) Some of the electrons strike the metal mask (instead of the phosphor dots behind holes in the mask), causing x-rays to be emitted. What is the smallest wavelength of the \(x\) -rays emitted?
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By directly substituting the values of the fundamental constants, show that the Bohr radius \(a_{0}=\hbar^{2} /\left(m_{\mathrm{e}} k e^{2}\right)\) has the numerical value \(5.29 \times 10^{-11} \mathrm{m}.\)
Follow the steps outlined in this problem to estimate the time lag (predicted classically but not observed experimentally) in the photoelectric effect. Let the intensity of the incident radiation be $0.01 \mathrm{W} / \mathrm{m}^{2} .\( (a) If the area of the atom is \)(0.1 \mathrm{nm})^{2},$ find the energy per second falling on the atom. (b) If the work function is 2.0 eV, how long would it take (classically) for enough energy to fall on this area to liberate one photoelectron? (c) Explain briefly, using the photon model, why this time lag is not observed.
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