Question

The radius of the \(n=1\) orbit in the hydrogen atom is $$ a_{0}=0.053 \mathrm{nm} $$ a) Compute the radius of the \(n=6\) orbit. How many times larger is this compared to the \(n=1\) radius? b) If an \(n=6\) electron relaxes to an \(n=1\) orbit (ground state), what is the frequency and wavelength of the emitted radiation? What kind of radiation was emitted (visible, infrared, etc.)? c) How would your answer in (a) change if the atom was a singly ionized helium atom \(\left(\mathrm{He}^{+}\right),\) instead?

Short answer

#tag_title# Step 10: Compare the radii of the n=6 orbit in a hydrogen atom and a singly ionized helium atom #tag_content# Divide the radius of the n=6 orbit in a singly ionized helium atom by the radius of the n=6 orbit in a hydrogen atom to find out how their sizes relate to each other.

Step-by-step solution

Formula for orbit radius in an atom

According to the Bohr model, the radius of the n-th orbit in an atom can be expressed as: $$ r_n = a_0 n^2 / Z $$ where \(r_n\) is the radius of the n-th orbit, \(a_0\) is the Bohr radius, n is the principal quantum number, and Z is the atomic number.

Compute the radius of the n=6 orbit in a hydrogen atom

Using the formula for the radius of an electron orbit, we can plug in the given values for a hydrogen atom: $$ r_6 = \frac{0.053 \mathrm{nm} \times (6^2)}{1} $$ Calculate \(r_6\) to obtain the radius of the n=6 orbit in a hydrogen atom.

Compare the radii of the n=6 orbit and the n=1 orbit

Divide the radius of the n=6 orbit by the radius of the n=1 orbit (which is given as \(a_0\)) to find out how many times larger the n=6 orbit is compared to the n=1 orbit.

Formula for energy levels in an atom

According to the Bohr model, the energy of an electron in an n-th orbit can be expressed as: $$ E_n = -\frac{13.6 \mathrm{eV}}{n^2} $$ where \(E_n\) is the energy of the n-th orbit, and n is the principal quantum number.

Calculate the energy difference between n=6 and n=1 orbits

Use the formula for energy levels to calculate the energy difference between an electron in the n=6 and n=1 orbits.

Calculate the frequency of the emitted radiation

The energy difference can be expressed in terms of the emitted photon's frequency as follows: $$ E = h\nu $$ where \(E\) is the energy difference, \(h\) is the Planck constant, and \(\nu\) is the frequency. Rearrange the formula to solve for the frequency and use the calculated energy difference to obtain the frequency of the emitted radiation.

Calculate the wavelength of the emitted radiation

The frequency and wavelength of a photon are related through the speed of light as follows: $$ c = \lambda \nu $$ where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(\nu\) is the frequency. Rearrange the formula to solve for the wavelength and use the calculated frequency to obtain the wavelength of the emitted radiation.

Determine the type of radiation

Compare the calculated wavelength to the ranges for various types of electromagnetic radiation (visible, infrared, ultraviolet, etc.) to determine the type of radiation emitted when an electron relaxes from the n=6 orbit to the n=1 orbit.

Compute the radius of the n=6 orbit in a singly ionized helium atom

Using the formula for the radius of an electron orbit, we can plug in the given values for a singly ionized helium atom (note that helium has an atomic number of 2): $$ r'_6 = \frac{0.053 \mathrm{nm} \times (6^2)}{2} $$ Calculate \(r'_6\) to obtain the radius of the n=6 orbit in a singly ionized helium atom.

Key concepts

Bohr Radius

The Bohr radius (\(a_0\)) is a fundamental constant that represents the smallest average distance between an electron and the nucleus in a hydrogen atom in its ground state. It is approximately 0.053 nanometers, or 52.9 picometers. According to Niels Bohr's theory, electrons orbit the nucleus at certain discrete distances, or orbits, with the ground state being the closest one.

The equation to determine the radius of the n-th electron orbit, as derived from the Bohr model, is \( r_n = a_0 n^2 / Z \), where \( n \) is the principal quantum number corresponding to the orbit's energy level, and \( Z \) is the atomic number, which is 1 for hydrogen. Each orbit beyond the first is progressively larger, showing that electrons can occupy areas further from the nucleus at higher energy states.

Quantum Number

The principal quantum number, symbolized by \( n \), defines the energy level of an electron in an atom. This number is integral and non-zero, starting from 1 for the ground state, and increasing as energy levels get further from the nucleus. Each energy level, or shell, can hold a certain maximum number of electrons, which impacts the electron arrangement in atoms.

As the quantum number increases, so does the electron's energy and the size of the orbit. The quantum number plays a crucial role in an atom's electronic structure and determines properties such as the atom's ionization energy and chemical reactivity.

Energy Levels in Atoms

In the Bohr model of the atom, the energy levels represent the fixed energies that an electron can have while orbiting the nucleus. The energy of an electron is quantized, meaning it can only exist in these specific energy states and can jump between levels by absorbing or emitting specific amounts of energy, in the form of photons.

The equation for determining the energy of an electron at a particular energy level is \(-\frac{13.6 \mathrm{eV}}{n^2}\), where 13.6 eV is the ionization energy of hydrogen, and \( n \) again represents the principal quantum number. When an electron changes level, the energy difference between the initial and final orbits equals the energy of the emitted or absorbed photon.

Wavelength of Emitted Radiation

When an electron transitions between energy levels, it emits or absorbs a photon with a specific wavelength. This wavelength correlates to the energy difference between the initial and final states. The relationship between the wavelength of radiation (\( \lambda \)), its frequency (\( u \)), and the speed of light (\( c \)) is given by \( c = \lambda u \).

Using the energy difference calculated from the Bohr model and the known value of Planck's constant, one can find the frequency and subsequently the wavelength of the emitted photon. Matching the computed wavelength to known electromagnetic spectrum ranges (such as visible light, infrared, etc.) can identify the type of radiation produced during electron transitions.