Question

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=6,(\mathbf{b})\) from an orbit of radius 4.76\(\hat{\mathrm{A}}\) to one of radius \(0.529 \mathrm{A},(\mathbf{c})\) from the \(n=6\) to the \(n=9\) state.

Short answer

In summary, for the given electronic transitions in hydrogen: (a) From n=2 to n=6: Energy is absorbed (b) From an orbit of radius 4.76 Å to one of radius 0.529 Å (n=3 to n=1): Energy is emitted (c) From the n=6 to the n=9 state: Energy is absorbed

Step-by-step solution

(Transition from n=2 to n=6)

The electron is transitioning from n=2 to n=6, which means it is going from a lower energy level to a higher energy level. Therefore, energy will be absorbed during this transition.

(Transition from 4.76 Å to 0.529 Å)

First, we need to find the principal quantum numbers corresponding to the given radii. Using the radius formula: \(r_n = a_0n^2\) For 4.76 Å: \(\frac{4.76}{0.529} = n_1^2\) \(n_1^2 = 9\) \(n_1 = 3\) For 0.529 Å: \(\frac{0.529}{0.529} = n_2^2\) \(n_2^2 = 1\) \(n_2 = 1\) The electron is transitioning from n=3 to n=1, which means it is going from a higher energy level to a lower energy level. Therefore, energy will be emitted during this transition.

(Transition from n=6 to n=9)

The electron is transitioning from n=6 to n=9, which means it is going from a lower energy level to a higher energy level. Therefore, energy will be absorbed during this transition. In summary, the energy is absorbed during transitions (a) and (c) and emitted during transition (b).

Key concepts

Energy Absorption and Emission

When discussing the behaviors of electrons within an atom, energy absorption and emission are key phenomena to understand. In a simplified model of the hydrogen atom, electrons orbit the nucleus in specific energy levels. As these electrons move between the levels, energy is either absorbed or emitted, which is fundamental to the electron transition process.

An electron absorbs energy to jump to a higher energy level, such as when moving from the second energy level (\(n=2\)) to the sixth (\(n=6\)). Conversely, an electron emits energy when it drops to a lower energy level, for example, descending from the third energy level (\(n=3\)) to the ground state (\(n=1\)). The energy change associated with these transitions corresponds directly to the frequency of light absorbed or emitted. This forms the basis for the study of atomic spectra, providing valuable information about the structure of atoms.

Quantum Numbers

Quantum numbers are the address of an electron in an atom, providing information about its location and behavior. There are four quantum numbers: the principal quantum number (\(n\)), azimuthal or angular momentum quantum number (\(l\)), magnetic quantum number (\(m_l\)), and spin quantum number (\(m_s\)).

These numbers arise from the mathematical solutions to the Schrödinger equation for the hydrogen atom and define properties such as the electron's energy level, shape, orientation, and spin of an orbit. The principal quantum number identifies the energy level of an electron; the azimuthal quantum number denotes the shape of the orbital; the magnetic quantum number specifies the orientation of the orbital within a magnetic field; and the spin quantum number tells us the direction of the electron's spin.

Energy Level and Orbital Shape

The value of the principal quantum number ranges from 1 to infinity and determines the size of the electron cloud. The azimuthal quantum number can take any integer value from 0 to \(n-1\) and is linked to the electron's angular momentum, affecting the shape of its probability cloud, or orbital.

Principal Quantum Number

The principal quantum number, symbolized by (\(n\)), is integral to understanding electronic transitions in atoms, especially hydrogen. It defines the energy level of an electron in an atom and, by extension, its relative distance from the nucleus. As (\(n\) increases, the electron occupies orbits further from the nucleus, with higher energy and larger radii.

In any energy absorption or emission event, it's the change in the principal quantum number that indicates if an electron is moving to a higher or lower energy level. For example, if an electron moves from the first energy level (\(n=1\)) to the third energy level (\(n=3\)), the principal quantum number increases, meaning the electron has absorbed energy. The reverse, a decrease in the principal quantum number, signals energy emission.

Determining Energy Exchange

By using the principal quantum number, one can calculate the specific energy involved in transitions between levels. This is vividly demonstrated in the provided textbook exercises, where they employ the Bohr radius and the principal quantum number to predict whether energy is absorbed or emitted in a transition.