Problem 22 - College Physics

Question

\(\cdot\) For a hydrogen atom in the ground state, determine, in electron volts, (a) the kinetic energy of the electron, (b) the potential energy, (c) the total energy, (d) the minimum energy required to remove the electron completely from the atom. (e) What wavelength does a photon with the energy calculated in part (d) have? In what region of the electromagnetic spectrum does it lie?

Step-by-Step Solution

Answer

(a) 13.6 eV (KE), (b) -27.2 eV (PE), (c) -13.6 eV, (d) 13.6 eV, (e) 91.2 nm UV region.

Step 1: Understanding the Hydrogen Atom's Ground State

The hydrogen atom's ground state energy level is characterized by an electron in the lowest energy orbit. The total energy of an electron in the ground state of hydrogen can be expressed as \( E = -13.6 \, \text{eV} \). This is the energy needed to remove an electron from its orbit entirely, also known as the ionization energy.

Step 1: Calculate the Kinetic Energy

In quantum mechanics, for a hydrogen atom, the kinetic energy (KE) of the electron in the ground state can be found as the negative of the total energy, which is \( KE = -E = 13.6 \, \text{eV} \).

Step 2: Calculate the Potential Energy

The potential energy (PE) of the electron in a hydrogen atom is twice the total energy and is given by \( PE = 2E = 2(-13.6 \, \text{eV}) = -27.2 \, \text{eV} \).

Step 3: Calculate the Total Energy

The total energy of the electron in the hydrogen atom's ground state is already known from the introductory information, \( E = -13.6 \, \text{eV} \).

Step 4: Minimum Energy to Remove the Electron

The minimum energy required to remove the electron from the hydrogen atom (ionization energy) is equal to the magnitude of the total energy, \( 13.6 \, \text{eV} \).

Step 5: Calculate the Wavelength of the Photon

To find the wavelength (\( \lambda \)) of a photon with energy \( 13.6 \, \text{eV} \), we use the relation \( E = \frac{hc}{\lambda} \). Solving for \( \lambda \), we get \( \lambda = \frac{hc}{E} \). Using \( h = 4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s} \) and \( c = 2.998 \times 10^8 \, \text{m/s} \), \( \lambda = \frac{4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s} \cdot 2.998 \times 10^8 \, \text{m/s}}{13.6 \, \text{eV}} \approx 91.2 \, \text{nm} \).

Step 6: Determine the Region of the Electromagnetic Spectrum

The wavelength calculated is approximately 91.2 nm, which lies in the ultraviolet (UV) region of the electromagnetic spectrum.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum represents the range of all types of electromagnetic radiation. Radiation is energy that moves and spreads out as it travels. One way to understand the spectrum is to think of it as a range or continuum of light waves ordered by wavelength or frequency. Each type of electrical wave has a specific range of wavelengths and energy levels.

• Radio waves have the longest wavelengths and the least energy.
• On the other end of the spectrum, gamma rays have the shortest wavelengths and carry the most energy.

Each section of the spectrum corresponds to different applications and is detected by various sensors. For instance, the visible spectrum is the light that can be seen by the human eye, which is just a tiny portion of the entire spectrum. Ultraviolet light, with shorter wavelengths than visible light, is what causes sunburns. In the context of the hydrogen atom exercise, the calculated wavelength of a photon was found to be in the ultraviolet region.

Ionization Energy

Ionization energy is a crucial concept in understanding atoms, representing the energy required to remove an electron from an atom or ion in its gaseous state. This energy indicates how strongly an atom holds onto its electrons.

• For a hydrogen atom in its ground state, the ionization energy is 13.6 eV, which means it would take 13.6 electron volts to remove the electron completely from the atom.
• This value reflects the stability of the electron configuration.

If an electron is to be removed, it must overcome this energy barrier. In our hydrogen atom scenario, this involves a photon with sufficient energy hitting the atom to provide the electron with the needed energy to escape.

Photon Wavelength

Photon wavelength is a measure of the distance between two consecutive peaks of the electromagnetic wave associated with light. It is inversely proportional to the energy held by the photon, meaning that higher-energy photons have shorter wavelengths.

This is seen in the formula:\[ E = \frac{hc}{\lambda} \]where:

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant,
• \( c \) is the speed of light,
• and \( \lambda \) is the wavelength.

For the photon that ionizes the hydrogen atom, its wavelength was determined to be approximately 91.2 nm, placing it in the ultraviolet region. This ties back to the energy calculated for ionization and helps locate where this photon fits in the electromagnetic spectrum.

Quantum Mechanics

Quantum mechanics is the foundational theory in physics that describes nature at the smallest scales, such as particles like electrons in atoms. It introduces significant concepts that challenge classical physics and help explain atomic behavior.

• In quantum mechanics, energy levels are quantized, meaning electrons in atoms can only occupy specific energy states.
• The behavior of electrons is described by probability distributions rather than exact positions and velocities. This is best exemplified by the uncertainty principle.

In our exercise, when calculating the kinetic and potential energies of a hydrogen atom's electron, quantum mechanics allows us to use well-defined rules to determine these values accurately. It shows how an electron's kinetic energy, being the negative of its known total energy, can be evaluated using quantum principles.