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Chapter 31: Problem 22

Initially, an electron is in the \(n=3\) state of hydrogen. If this electron acquires an additional \(1.23 \mathrm{eV}\) of energy, what is the value of \(n\) in the final state of the electron?

Short Answer

Expert verified
The final value of \(n\) is 7.

Step by step solution

01

Understand the Energy Levels in Hydrogen

The energy level of an electron in a hydrogen atom is given by the formula \( E_n = -13.6 \frac{1}{n^2} \) eV, where \( n \) is the principal quantum number. Each energy level has a negative energy value that becomes less negative (approaching zero) as \( n \) increases.
02

Calculate the Initial Energy

For the initial state, \( n = 3 \). Using the energy formula, the energy of the electron in the \( n = 3 \) state is \( E_3 = -13.6 \frac{1}{3^2} \) eV. Calculate this to find \( E_3 = -13.6 \times \frac{1}{9} = -1.51 \) eV.
03

Determine the Final Energy

The electron acquires an additional \( 1.23 \) eV of energy. Therefore, the final energy becomes the sum of its initial energy and the gained energy: \( E_f = E_3 + 1.23\, \text{eV} = -1.51 + 1.23 = -0.28\, \text{eV}.\)
04

Find the Final Energy Level

We need to find the principal quantum number \( n_f \) where \( E_f = -13.6 \frac{1}{n_f^2} \). Set \( -0.28 = -13.6 \frac{1}{n_f^2} \), which gives \( n_f^2 = \frac{13.6}{0.28} \). Solve this for \( n_f \). \( n_f^2 = 48.57 \) means \( n_f \approx 7.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Quantum Number
Understanding the principal quantum number is crucial for grasping the concept of electron energy levels in a hydrogen atom. In atomic physics, each electron in an atom is assigned a principal quantum number, denoted by the symbol "n". This number determines the electron's energy level and its average distance from the nucleus.
  • The principal quantum number is an integer with a minimum value of 1.
  • As "n" increases, the electron's energy becomes less negative, indicating higher energy levels.
  • It also affects the electron's orbital radius; larger "n" values mean the electron is further from the nucleus.
In the original problem, the electron starts at the third energy level, represented by the principal quantum number, "n = 3". This information is vital as it sets the stage for calculating energy changes during electron transitions.
Electron Transition
When an electron gains or loses energy, it makes a transition between energy levels. This process is known as an electron transition, which can result in absorption, emission, or excitation, depending on whether the electron moves to a higher or lower level.
  • Electron transitions are quantized, meaning they can only occur in discrete amounts defined by the energy difference between two levels.
  • A transition to a higher energy level occurs when the electron absorbs energy.
  • In contrast, releasing energy will lead to a transition to a lower energy state.
In the given problem, the electron moves from the initial state when it absorbs 1.23 eV and jumps to a higher energy level. The task is to identify the new energy level, signified by the final principal quantum number.
Energy Absorption
Energy absorption plays a pivotal role in exciting electrons to higher energy states. This occurs when the electron takes in photons or thermal energy sufficient to jump to a higher energy level.
  • The original energy state of an electron is determined using the formula: \( E_n = -13.6 \frac{1}{n^2} \text{ eV} \).
  • When energy is absorbed, the electron's total energy increases, moving it closer to zero in terms of eV.
  • This change leads to the electron being in a higher orbital around the nucleus.
In this exercise, the electron initially at \( n = 3 \) absorbs 1.23 eV, leading to an energy change which results in its new state being determined as "n = 7". The calculation shows how strategic energy intake alters the energy levels and quantum state of the electron.

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Most popular questions from this chapter

Do you expect the ionization energy of sodium (Na) to be greater than, less than, or equal to the ionization energy of lithium (Li)? Explain.

Find the energy of the photon required to excite a hydrogen atom from the \(n=2\) state to the \(n=5\) state.

Find an expression for the de Broglie wavelength of an electron in the \(n\)th state of the hydrogen atom.

Consider two different states of a hydrogen atom. In state I the maximum value of the magnetic quantum number is \(m_{\ell}=3 ;\) in state II the corresponding maximum value is \(m_{\ell}=2 .\) Let \(L_{1}\) and \(L_{\mathrm{II}}\) represent the magnitudes of the orbital angular momentum of an electron in states I and II, respectively. (a) Is \(L_{1}\) greater than, less than, or equal to \(L_{\mathrm{II}}\) ? Explain. (b) Calculate the ratio \(L_{\mathrm{I}} / L_{\mathrm{II}}\).

Rydberg Atoms There is no limit to the size a hydrogen atom can attain, provided it is free from disruptive outside influences. In fact, radio astronomers have detected radiation from large, so-called "Rydberg atoms" in the diffuse hydrogen gas of interstellar space. (a) Find the smallest value of \(n\) such that the Bohr radius of a single hydrogen atom is greater than 8.0 microns, the size of a typical single-celled organism. (b) Find the wavelength of radiation this atom emits when its electron drops from level \(n\) to level \(n-1\). (c) If the electron drops one more level, from \(n-1\) to \(n-2,\) is the emitted wavelength greater than or less than the value found in part (b)? Explain.
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