Question

Place the following transitions of the hydrogen atom in order from shortest to longest wavelength of the photon emitted: \(n=5\) to \(n=3, n=4\) to \(n=2, n=7\) to \(n=4\), and \(n=3\) to \(n=2\).

Short answer

The order of the transitions, from shortest to longest wavelength of the emitted photon, is: \(n=3\) to \(n=2\), \(n=4\) to \(n=2\), \(n=5\) to \(n=3\), and \(n=7\) to \(n=4\).

Step-by-step solution

Understand the Rydberg Formula

To find the wavelength of the emitted photons, we will use the Rydberg formula: \(\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\) where: - \(\lambda\) is the wavelength of the emitted photon - \(R_H\) is the Rydberg constant for hydrogen, approximately equal to \(1.097 \times 10^7 m^{-1}\) - \(n_1\) and \(n_2\) are the principal quantum numbers of the two energy levels with \(n_1 < n_2\).

Calculate the wavelength for each transition

Now, we need to calculate the emitted photon wavelength for each given transition using the Rydberg formula. For \(n=5\) to \(n=3\): \(\frac{1}{\lambda_1} = R_H \left(\frac{1}{3^2} - \frac{1}{5^2}\right)\) For \(n=4\) to \(n=2\): \(\frac{1}{\lambda_2} = R_H \left(\frac{1}{2^2} - \frac{1}{4^2}\right)\) For \(n=7\) to \(n=4\): \(\frac{1}{\lambda_3} = R_H \left(\frac{1}{4^2} - \frac{1}{7^2}\right)\) For \(n=3\) to \(n=2\): \(\frac{1}{\lambda_4} = R_H \left(\frac{1}{2^2} - \frac{1}{3^2}\right)\)

Compare the wavelengths

To find the order of the given transitions of the hydrogen atom based on the wavelength, we need to compare the calculated wavelengths. Remember that a smaller value of \(1/\lambda\) means a longer wavelength. We can notice that: - \(1/\lambda_4 > 1/\lambda_2 > 1/\lambda_1 > 1/\lambda_3\) So the order of the transitions, from shortest to longest wavelength of the emitted photon, is: \(n=3\) to \(n=2\), \(n=4\) to \(n=2\), \(n=5\) to \(n=3\), and \(n=7\) to \(n=4\)

Key concepts

Rydberg Formula

The Rydberg Formula is a fundamental equation in atomic physics used to determine the wavelength of light emitted or absorbed when an electron transitions between energy levels in a hydrogen atom. It can be represented as: \[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] Here, \( \lambda \) stands for the wavelength of the photon emitted, and \( R_H \) is the Rydberg constant specifically for hydrogen.

• This formula helps us understand the quantized nature of energy levels in atoms.
• The principal quantum numbers \( n_1 \) and \( n_2 \) represent the initial and final energy states of the electron, respectively.
• \( n_1 \) must be less than \( n_2 \) since energy levels are numbered from the closest level to the nucleus to those further away.

By using this formula, we can calculate the specific wavelengths of light that a hydrogen atom can emit or absorb, revealing the electronic structure of the atom.

Photon Wavelength

The concept of photon wavelength is crucial in understanding how light interacts with atoms. Wavelength (\( \lambda \)) refers to the distance between successive crests of a wave and determines the color of the light when it falls within the visible spectrum. In the context of hydrogen atom transitions:

• When an electron transitions from a higher energy orbit to a lower one, a photon of a specific wavelength is emitted.
• This emitted photon's wavelength is directly related to the energy difference between the two levels.
• Energy is inversely proportional to wavelength according to the relation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) is the speed of light.

In practical terms, shorter wavelengths correspond to higher energy transitions and vice versa. Understanding photon wavelength allows us to predict the spectral lines of hydrogen, which are visible in its emission spectrum.

Principal Quantum Numbers

Principal quantum numbers (\( n \)) in quantum mechanics specify the energy level of an electron in an atom. These numbers are integers and define the shell that an electron occupies around the nucleus. They are crucial in calculating the energy and position of electrons:

• Each principal quantum number corresponds to a new energy level or shell, with the first level (\( n=1 \)) closest to the nucleus.
• Higher values of \( n \) indicate shells further away from the nucleus, which have higher energy levels.
• Principal quantum numbers are vital in determining the photon wavelengths emitted or absorbed when an electron transitions between these levels.

In transitioning from a higher \( n \) to a lower \( n \), the electron emits a photon. Understanding these numbers helps in predicting the electron configuration and energy transitions in an atom.

Rydberg Constant

The Rydberg Constant (\( R_H \)) is a physical constant involved in various calculations related to the spectral lines of hydrogen. Its approximate value is \( 1.097 \times 10^7 \text{ m}^{-1} \), and it is essential for determining the wavelengths of photons emitted or absorbed by a hydrogen atom:

• It represents the limiting value of the highest wavenumber of any photon that can be emitted or absorbed.
• The Rydberg Constant is specific to hydrogen, but similar constants can be defined for other elements.
• This constant is crucial in the Rydberg Formula, which is used to calculate the wavelengths of spectral lines.

Being fundamental in spectral analysis, the Rydberg Constant allows physicists to decode information about atomic structures and assess how electrons interact within an atom.