Problem 33 - General Chemistry: Principles and Modern Applications

Question

What electron transition in a hydrogen atom, starting from the orbit \(n=7,\) will produce light of wavelength \(410 \mathrm{nm} ?\)

Step-by-Step Solution

Answer

The electron transition in a hydrogen atom that will produce light of wavelength \(410nm\) will end in the ground state with principal quantum number \(n_f=2\).

Step 1: Conversion of Wavelength to Energy

Convert the given wavelength \(410 \mathrm{nm}\) to energy. Energy is related to wavelength by the following relation, \[E = \frac{{hc}}{λ}\] where \(h = 6.62607015 \times 10^{-34} m^2kgs^{-1}\) is Planck’s constant, \(λ = 410nm = 410 \times 10^{-9}m\) is the wavelength of light, and \(c = 3 \times 10^8 ms^{-1}\) is the speed of light.

Step 2: Calculation of Energy

Plugging in the values into the energy formula \[E = \frac{{(6.62607015 \times 10^{-34} m^2kgs^{-1}) \times (3 \times 10^8 ms^{-1})}}{{410 \times 10^{-9}m}} = 4.85 \times 10^{-19} joules\] This value represents the energy difference between two levels in the hydrogen atom.

Step 3: Implementation of Rydberg's Formula

According to the Rydberg’s formula for the energy levels of hydrogen, we have \[∆E = 13.6eV \times \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\] where ∆E is the energy difference, \(n_i\) and \(n_f\) are the initial and final (unknown) quantum numbers.

Step 4: Conversion of Energy Units of Energy Change

Before equating the energy change to the separated energy levels, convert the energy from joules to eV (electron volts). \(1 Joule = 6.242 \times 10^{18} eV\). So, the energy of photon in eV would be \[ E = 4.85 \times 10^{-19} Joules \times 6.242 \times 10^{18} eV/Joule = 3.02 eV \] This is the change in energy.

Step 5: Calculation of Final Quantum Number

Substitute this value and the initial quantum number in the Rydberg’s formula for the hydrogen's energy levels and solve for final quantum level, \(n_f\). \[ 3.02eV = 13.6eV \times \left( \frac{1}{n_f^2} - \frac{1}{7^2} \right) \] Solving for \(n_f\) gives \(n_f=2\) as the principal quantum number for the final state.

Key Concepts

Rydberg's Formula

Rydberg's formula is a critical tool in understanding electron transitions in hydrogen atoms. It describes the relationship between the energy difference of electron orbits and their quantum numbers. The formula: \[ \Delta E = 13.6 eV \times \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \] is especially useful in calculating the energy needed for an electron to move between different energy levels. Here, \(\Delta E\) represents the energy change, while \(n_i\) and \(n_f\) denote the initial and final quantum numbers respectively.Understanding Rydberg's formula helps us predict the energy transitions and the resultant spectral lines in hydrogen spectra.Ul>

Predict spectral lines from electron transitions.

Calculate energy differences between orbits.

The formula is rooted in the principles of quantum mechanics and provides valuable insights into the properties of hydrogen spectra.

Quantum Numbers

Quantum numbers are like addresses for electrons within an atom, defining their position and energy level. In the context of the hydrogen atom, the principal quantum number, \(n\), is crucial as it signifies the electron shell or energy level. Quantum numbers dictate the stability, energy, and geometry of electron orbits.
There are four types of quantum numbers:

• Principal quantum number (\(n\)): Determines the energy of the orbit.
• Angular momentum quantum number (\(l\)): Defines the shape of the orbit.
• Magnetic quantum number (\(m_l\)): Specifies the orientation of the orbit.
• Spin quantum number (\(m_s\)): Describes the electron's spin direction.

Each electron in an atom is described by a unique set of these quantum numbers, aiding in the prediction of chemical properties and the form of spectral lines.

Hydrogen Atom

The hydrogen atom is often considered the simplest atom and serves as a fundamental building block in quantum physics and chemistry. It consists of one proton and one electron, with its simple structure making it a perfect candidate to study quantum mechanics. Key points about hydrogen atoms include:

• The electron can inhabit only certain discrete energy levels or orbits.
• When transitions occur between these levels, specific amounts of energy are absorbed or emitted, usually as electromagnetic radiation.

The hydrogen atom's simplicity allows it to be fully described by quantum numbers, giving clear insight into phenomena like spectral lines and atomic absorption/emission.

Energy Levels

The concept of energy levels is fundamental to understanding electron transitions in a hydrogen atom. These levels are like floors in a building, with each level representing a definite energy state where an electron can exist. When electrons move between these predefined levels, they either absorb or release energy. Important aspects of energy levels:

• Energy levels are quantized, meaning they exist only in specific states.
• An electron jumping from a higher to lower energy level releases energy, often in the form of light.
• The energy difference between levels corresponds to the color of light in the visible spectrum or other electromagnetic radiation.

Energy levels underpin the atomic spectra, making them pivotal for analyzing the electronic structure of atoms.

Wavelength to Energy Conversion

Converting wavelength into energy is crucial when analyzing electron transitions. This conversion is governed by the equation: \[ E = \frac{hc}{\lambda} \] where \(E\) is energy, \(h\) is Planck’s constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the light.This formula provides insight into how light relates to electron transitions:

• The shorter the wavelength, the greater the energy.
• This conversion is key to understanding the energy emitted or absorbed during electron transitions.

By converting wavelengths to energy, scientists can decipher the intricate details of atomic emissions and absorptions, unraveling the atomic structure and processes.