Problem 31 - General Chemistry: Principles and Modern Applications

Question

For the Bohr hydrogen atom determine (a) the radius of the orbit \(n=4\) (b) whether there is an orbit having a radius of \(4.00 \AA\) (c) the energy level corresponding to \(n=8\) (d) whether there is an energy level at \(-2.5 \times 10^{-17} \mathrm{J}\)

Step-by-Step Solution

Answer

(a) The radius for the \(n=4\) orbit is \(8.464 \, \AA\). (b) No, there is no orbit having a radius of \(4.00 \, \AA\). (c) The energy level corresponding to \(n=8\) is \(-3.425 \times 10^{-20} \, \mathrm{J}\). (d) Yes, there is an energy level at \(-2.5 \times 10^{-17} J\), which corresponds to \(n=3\).

Step 1: Calculation of radius for \(n=4\)

Use the given formula for radius \(r_n = n^2 \times r_1\). Substitute \(n=4\) and \(r_1 = 0.529 \, \AA\) into the formula to find \(r_4\).

Step 2: Check for an orbit with radius \(4.00 \, \AA\)

Rearrange the radius formula as \(n= \sqrt{\frac{{r}}{{r_1}}}\), where \(r\) is the given radius. Substitute \(r = 4.00 \, \AA\) and \(r_1 = 0.529 \, \AA\) into the formula. If \(n\) is a whole number, then there exists an orbit with that radius.

Step 3: Calculation of energy level for \(n=8\)

Use the given formula for energy levels \(E_n= -\frac{{R_H}}{{n^2}}\). Substitute \(n=8\) and \(R_H = 2.18 \times 10^{-18} \mathrm{J}\) into the formula to find \(E_8\).

Step 4: Check for an energy level at \(-2.5 \times 10^{-17} \, \mathrm{J}\)

Rearrange the energy formula as \(n= \sqrt{\frac{{-R_H}}{{E}}}\), where \(E\) is the given energy. Substitute \(E = -2.5 \times 10^{-17} \, \mathrm{J}\) and \(R_H = 2.18 \times 10^{-18} \mathrm{J}\) into the formula. If \(n\) is a whole number, then there exists an energy level with that value.

Key Concepts

Atomic Orbitals

The concept of atomic orbitals is central to understanding the structure of atoms, particularly within the Bohr model. In this framework, electrons inhabit discrete energy levels or orbits around the nucleus. Each orbit signifies a specific energy level associated with an electron. The Bohr hydrogen atom model, while simplified, illustrates how an electron in a ground state can occupy an orbit closest to the nucleus. More distinctively, every orbit correlates to a particular quantum number, denoted by \(n\).

These orbitals are critical as they determine where electrons are most likely to be found. Here are some key points:

• Electrons can exist only in certain energy levels or orbits.
• Each orbit is associated with a quantum number, \(n\), which can take integer values such as 1, 2, 3, etc.
• The size and energy of an electron's orbit increase with higher quantum numbers.

While it is straightforward in hydrogen with a single electron, atomic orbitals become more complex with more electrons, requiring quantum mechanics' more robust theories. However, in the hydrogen atom, these orbitals give a glimpse into the organized nature of atomic structures.

Quantum Energy Levels

Quantum energy levels describe the distinct levels of energy that an electron in an atom can have. For a hydrogen atom modeled by Bohr, each energy level corresponds to a specific orbit. The energy level is quantified by the principal quantum number \(n\), where the energy of each level is determined by the formula:\[E_n = -\frac{R_H}{n^2}\]Here, \(R_H\) is the Rydberg constant, approximately \(2.18 \times 10^{-18} \, \text{J}\).

Key features of quantum energy levels include:

• The lowest energy level (ground state) corresponds to \(n=1\).
• Higher quantum numbers indicate higher energy levels and orbits further from the nucleus.
• The energy becomes less negative with increasing \(n\), indicating greater freedom and energy for the electron.
• Transitions between these levels involve absorption or emission of energy, characterizing spectral lines.

Understanding these energy levels is crucial as they form the basis for atomic emission and absorption spectra, providing insights into atomic structure and behavior.

Atomic Radius Calculation

In the Bohr model, the atomic radius represents the distance from the nucleus to the orbit of an electron in a given energy level. Calculating the radius of an orbit follows the formula: \[r_n = n^2 \times r_1\]where \(r_1\) is the radius of the first orbit (approximately \(0.529 \, \text{Å}\) for hydrogen). Each subsequent orbit for a given quantum number \(n\) yields a larger radius due to the factor \(n^2\).

Some insights about atomic radius calculations include:

• The atomic radius increases quickly with higher \(n\) values, as it scales with the square of the quantum number.
• Calculating whether a specific radius corresponds to an atomic orbit involves rearranging the formula to solve for \(n\), ensuring \(n\) is a whole number for a valid orbit.
• Atomic radius provides insights into the space an electron will occupy, influencing how atoms interact and bond.

The straightforward calculations in the Bohr model give foundational understanding of the size and scale of atoms and the behavior of their electrons.