Problem 2 - Applied Physics

Question

Find the energy of an electron in the \(n=3\) energy level in a hydrogen atom.

Step-by-Step Solution

Answer

The energy of an electron at \(n=3\) is \(-1.51 \, \text{eV}\).

Step 1: Understanding the Energy Level Formula for Hydrogen

The energy of an electron in a hydrogen atom for a given energy level \(n\) is given by the formula:\[E_n = - \frac{13.6 \, \text{eV}}{n^2}\]This formula calculates the energy in electron volts (eV) where 13.6 eV is the ionization energy of hydrogen from its ground state.

Step 2: Substitute the Energy Level Value

For the question, we need to find the energy at the energy level \(n=3\). Substitute \(n=3\) into our formula:\[E_3 = -\frac{13.6 \, \text{eV}}{3^2}\]This simplifies to calculate the energy of the electron at this level.

Step 3: Calculate the Energy Value

Now calculate the energy by simplifying the expression:\[E_3 = -\frac{13.6 \, \text{eV}}{9} = -1.51 \, \text{eV}\]Thus, the energy of the electron in the \(n=3\) energy level is \(-1.51 \, \text{eV}\).

Key Concepts

Energy Calculation

In quantum mechanics, calculating the energy levels of electrons in atoms is fundamental to understanding atomic structure. For the hydrogen atom, each electron resides in specific energy levels, denoted by the quantum number \(n\). The formula for determining the energy of an electron at a given level \(n\) in a hydrogen atom is:\[E_n = - \frac{13.6 \, \text{eV}}{n^2}\]This equation is derived from the Bohr model of the hydrogen atom and gives the energy in electron volts (eV). Here, \(13.6\) eV represents the ionization energy, which is the minimal energy needed to remove an electron completely from the hydrogen atom when it is in its ground state.

• The negative sign indicates that energy must be added to the atom to free the electron.
• As the quantum number \(n\) increases, the energy level rises closer to zero, which means the electron has more energy and is less tightly bound to the nucleus.

Electron Volts

Electron volts (eV) are a convenient unit of energy used frequently in atomic and subatomic physics. It is defined as the amount of kinetic energy gained or lost by an electron when it moves through an electric potential difference of one volt.

• One electron volt equals \(1.602 \, \times 10^{-19}\) joules.
• This unit allows scientists and students to work with the relatively small energy amounts associated with atomic particles effectively.
• In the context of an atom, using electron volts simplifies calculating, comparing, and understanding energy levels and transitions between them.

Ultimately, using electron volts makes it easier to understand the scale of energies at the atomic and subatomic levels, compared to the much larger SI unit joules.

Ionization Energy

Ionization energy is the quantity of energy needed to remove an electron from an atom or ion in its gaseous state. For hydrogen, the ionization energy from its ground state is \(13.6\) electron volts (eV).

• This value is important because it represents the energy required to completely remove the electron, thus ionizing the atom.
• In a multi-electron atom, this energy would vary due to the various interactions between electrons and the nucleus, but hydrogen is the simplest case with just one electron.

Knowing the ionization energy helps in understanding the strength of the electron’s bond and allows us to predict how an electron might behave under different conditions, such as the energy changes when moving between energy levels. This is crucial for applications such as spectroscopy, where light is used to study the energy levels of electrons in atoms.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of very small particles, like electrons, at the atomic and subatomic levels. It contrasts with classical physics by introducing concepts such as quantization of energy, wave-particle duality, and uncertainty principles.

• Electrons exist in discrete energy levels or shells around a nucleus, which is a concept fundamentally described by quantum mechanics.
• Observations at this level reveal that electrons show both particle-like and wave-like properties.
• The mathematical modeling used in quantum mechanics, such as Schrödinger's equation, helps to predict the energy levels and behaviors of these particles.

This field forms the basis for understanding how energy levels are calculated, like those for hydrogen atoms, and how atoms absorb and emit light as electrons transition between these levels.