Problem 23 - College Physics

Question

Light of wavelength \(59 \mathrm{nm}\) ionizes a hydrogen atom that was originally in its ground state. What is the kinetic energy of the ejected electron?

Step-by-Step Solution

Answer

The kinetic energy of the ejected electron is approximately \( 1.784 \times 10^{-17} \mathrm{J} \).

Step 1: Identify the Given Values

We are given that the light has a wavelength \( \lambda = 59 \mathrm{nm} \) (convert to meters: \(59 \times 10^{-9} \mathrm{m} \)). The electron was originally in the ground state of the hydrogen atom.

Step 2: Calculate Energy of the Photon

The energy of a photon is calculated using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \(6.626 \times 10^{-34} \mathrm{J \cdot s}\), \( c \) is the speed of light \(3 \times 10^8 \mathrm{m/s}\) and \( \lambda \) is the wavelength. Substitute these values to find \( E \).

Step 3: Calculate Energy of Ground State of Hydrogen

The energy of the electron in the ground state of hydrogen is \( -13.6 \mathrm{eV} \). Convert this energy into joules: \(-13.6 \mathrm{eV} \times 1.602 \times 10^{-19} \mathrm{J/eV}\).

Step 4: Utilize Conservation of Energy

Using the conservation of energy principle, the kinetic energy of the ejected electron is given by \( KE = E_{\text{photon}} - E_{\text{initial}} \). We use the energy of the incoming photon and subtract the energy of the ground state of the hydrogen atom.

Step 5: Calculate Kinetic Energy

Substitute the calculated values for \( E_{\text{photon}} \) and \( E_{\text{initial}} \) into the formula \( KE = E_{\text{photon}} - E_{\text{initial}} \). Convert and simplify the result of \( KE \) to obtain it in joules.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest and most basic element in the universe. It is composed of a single proton in the nucleus and one electron orbiting around it. This simplicity makes hydrogen the perfect starting point for studying atomic physics and fundamental principles of quantum mechanics. In its ground state, the electron occupies the lowest energy level possible.
In this state, the electron is closest to the nucleus, and the potential energy is minimized. This ground state energy level is crucial when calculating interactions such as those seen in the photoelectric effect, where light energy can eject an electron from an atom.

Kinetic Energy

Kinetic energy (KE) is the energy of motion. When an electron is ejected from an atom, it gains kinetic energy, allowing it to move freely. In the context of the photoelectric effect, the kinetic energy of an ejected electron can be found using the principle of energy conservation.

• The formula for kinetic energy is given by: \( KE = E_{\text{photon}} - E_{\text{initial}} \).
• Here, \(E_{\text{photon}}\) represents the energy of the incoming light, and \(E_{\text{initial}}\) is the initial energy of the electron (from the ground state).

The energy consequence ensures that the energy added by the photon is transformed into kinetic energy of the ejected electron.

Wavelength Calculation

To calculate the energy of a photon, it is essential to understand the role of wavelength in photon energy. The energy of a photon is inversely proportional to its wavelength.
The formula used is: \[ E = \frac{hc}{\lambda} \]

• Here, \(h\) stands for Planck's constant \(6.626 \times 10^{-34} \text{J}\cdot\text{s}\).
• \(c\) is the speed of light, approximately \(3 \times 10^8 \text{m/s}\).
• \(\lambda\) is the wavelength of the light in meters.

By substituting the given wavelength into this formula, you can calculate the energy of the incoming photon involved in ionizing the atom.

Energy Conversion

Energy conversion is a fundamental concept in physics that occurs when energy changes from one form to another. In the photoelectric effect, energy from a photon is converted into kinetic energy of an electron, plus any threshold energy needed to release the electron.
This conversion can be expressed mathematically:

• The energy from the photon \( E_{\text{photon}} \) is used to overcome the electron's binding energy \( E_{\text{initial}} \).
• Any excess photon energy becomes the kinetic energy \( KE \) of the ejected electron.
• This results in: \( E_{\text{photon}} = KE + E_{\text{initial}} \).

Understanding these conversions ensures a better grasp of how energy transfer works in quantum mechanics.