Problem 43 - Fundamentals of Physics

Question

In the ground state of the hydrogen atom, the electron has a total energy of \(-13.6 \mathrm{eV}\). What are (a) its kinetic energy and (b) its potential energy if the electron is one Bohr radius from the central nucleus?

Step-by-Step Solution

Answer

(a) Kinetic energy is -4.53 eV, (b) Potential energy is -9.06 eV.

Step 1: Understand the relationship between energies

The total energy (\(E\)) of the electron in the ground state of a hydrogen atom is given as \(-13.6 \: \mathrm{eV}\). For a hydrogen atom, the relationship between total energy, kinetic energy (\(K\)), and potential energy (\(U\)) is such that \(E = K + U\). Additionally, by the Virial Theorem, \(K = -\frac{1}{2} U\) for a system bound by a central force such as in the hydrogen atom.

Step 2: Express kinetic energy using total energy

Since the total energy \(E = K + U\) and according to the Virial Theorem \(K = -\frac{1}{2} U\), we can substitute \(U = 2K\) into the expression \(E = K + U\), giving \(E = K + 2K = 3K\). Thus, \(K = \frac{E}{3}\).

Step 3: Substitute and solve for kinetic energy

Substitute the known value of the total energy into the equation from Step 2: \(K = \frac{-13.6 \: \mathrm{eV}}{3} = -4.53 \: \mathrm{eV}\).

Step 4: Calculate potential energy using kinetic energy

Since the potential energy \(U = 2K\) and we have found \(K = -4.53 \: \mathrm{eV}\), thus \(U = 2 \times (-4.53 \: \mathrm{eV}) = -9.06 \: \mathrm{eV}\).

Key Concepts

Bohr model

The Bohr model is a crucial concept for understanding the hydrogen atom. It was proposed by Niels Bohr in 1913 and represented a significant advancement in atomic theory. This model describes an atom with a dense nucleus surrounded by electrons that revolve in discrete orbits. Each orbit corresponds to a specific energy level, and the energy levels are quantized.
In the context of the hydrogen atom, the electron moves in a circular path around the nucleus at a fixed radius, called the Bohr radius. This radius is a fundamental constant (\(a_0 = 0.529 imes 10^{-10}\) meters) which represents the most probable distance of the electron from the nucleus in its ground state.

The Bohr model helps explain why atoms emit or absorb light at specific wavelengths, correlating these wavelengths with the differences in energy levels. The transition of an electron from a higher energy orbit to a lower one results in the emission of light, and vice versa for absorption. This model laid the groundwork for further developments in quantum mechanics.

kinetic energy

Kinetic energy in the context of the hydrogen atom is the energy associated with the motion of the electron around the nucleus. In classical terms, kinetic energy (\(K\)) for an object with mass (\(m\)) moving at velocity (\(v\)) is given by \(K = \frac{1}{2}mv^2\). However, for electrons bound within atoms like in the hydrogen atom, this is understood in quantum terms.

For the electron in the hydrogen atom, kinetic energy is linked to the overall energy level of the electron. Using the relationships derived from the Bohr model, in the ground state of the hydrogen atom, the kinetic energy of the electron can be calculated from the total energy (\(E\)) and the potential energy (\(U\)). As per the solution to the exercise, the formula is \(K = \frac{E}{3}\).

This means that if you know the total energy of an electron in the ground state (such as \(-13.6 \, \text{eV}\)), you can calculate the kinetic energy to be \(-4.53 \, \text{eV}\). This kinetic energy relates directly to the movement of the electron in its orbit.

potential energy

Potential energy within the hydrogen atom comes from the electrostatic force between the negatively charged electron and the positively charged nucleus. This interaction is typically represented by Coulomb's law. The potential energy (\(U\)) in a bound system like the hydrogen atom, is negative, reflecting the attractive force between opposite charges.
For the hydrogen atom, in its ground state, the relationship between total energy (\(E\)), kinetic energy (\(K\)), and potential energy follows from \(E = K + U\).

Using the Virial Theorem, this relationship simplifies to \(U = 2K\), allowing us to compute potential energy once we know the kinetic energy. In our exercise, since the kinetic energy (\(K\)) was calculated as \(-4.53 \, \text{eV}\), the potential energy (\(U\)) becomes \(2 \times (-4.53 \, \text{eV}) = -9.06 \, \text{eV}\).
This negative sign reflects the binding nature of the electron to the nucleus. This understanding reinforces key ideas of energy conservation and quantum mechanical calculations in atomic physics.

Virial Theorem

The Virial Theorem is a versatile and powerful concept in physics, applicable to various systems including atomic, molecular, and astrophysical scenarios. In the context of the hydrogen atom, it provides a profound insight into the energy relationships of bound systems.
The Virial Theorem states that, for a stable, bound system governed by inverse-square forces (like Coulomb forces in atoms), the average kinetic energy (\(K\)) is related to the average potential energy (\(U\)) through the equation \(K = -\frac{1}{2}U\).

This relationship allows us to deduce that for every unit of potential energy a system possesses, twice the magnitude will manifest as kinetic energy, but of the opposite sign when considering bound states.
In applying this to a hydrogen atom, knowing the total energy allows us to determine both kinetic and potential energies using these relationships, simplifying complex quantum calculations into manageable pieces of information. Understanding the Virial Theorem thus bridges the observations of energy states derived from purely theoretical physics with measurable quantities.