Question

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

Short answer

In the given electronic transitions, energy is absorbed when transitioning from n=2 to n=6 and from n=6 to n=9. Energy is emitted when transitioning from an orbit of radius 4.76 Å to one of radius 0.529 Å.

Step-by-step solution

Compare Initial and Final Principal Quantum Numbers

In this case, the electron transitions from n=2 to n=6. Since the final principal quantum number (n=6) is greater than the initial (n=2), the electron moves to a higher energy orbit.

Determine If Energy is Emitted or Absorbed

When moving to a higher energy orbit, the electron absorbs energy. Therefore, during the transition from n=2 to n=6, energy is absorbed. #b)_Transition_from_4.76_Å_to_0.529_Å#

Relate Orbital Radius to Principal Quantum Number

The orbital radius (r) of hydrogen is related to the principal quantum number (n) by the formula: \(r = n^2a_0\), where \(a_0\) is the Bohr radius (approximately 0.529 Å).

Calculate Initial and Final Principal Quantum Numbers

Given the initial radius of 4.76 Å and the final radius of 0.529 Å, we can find the initial and final principal quantum numbers by rearranging the equation: Initial: \(n_1 = \sqrt{r_1/a_0} = \sqrt{4.76\text{ Å} / 0.529\text{ Å}} \approx 3\) Final: \(n_2 = \sqrt{r_2/a_0} = \sqrt{0.529\text{ Å} / 0.529\text{ Å}} = 1\)

Determine If Energy is Emitted or Absorbed

Since the final principal quantum number (n=1) is lower than the initial (n=3), the electron moves to a lower energy orbit. Therefore, during the transition from a radius of 4.76 Å to 0.529 Å, energy is emitted. #c)_Transition_from_n=6_to_n=9#

Compare Initial and Final Principal Quantum Numbers

In this case, the electron transitions from n=6 to n=9. Since the final principal quantum number (n=9) is greater than the initial (n=6), the electron moves to a higher energy orbit.

Determine If Energy is Emitted or Absorbed

When moving to a higher energy orbit, the electron absorbs energy. Therefore, during the transition from n=6 to n=9, energy is absorbed. In conclusion, for the given electronic transitions: a) Energy is absorbed. b) Energy is emitted. c) Energy is absorbed.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as n, is a fundamental concept in quantum mechanics which describes the size and energy level of an electron's orbit in an atom. Think of it as the 'address' of the electron within the atomic structure. Higher values of n correspond to larger orbits and hence electrons further away from the nucleus, with increased energy. The further an electron is from the nucleus, the more energy it has.

When looking at electronic transitions, such as an electron moving from n=2 to n=6, the principal quantum number increases. This means the electron is absorbing energy to move to a higher energy state. Alternatively, when an electron falls to a lower energy state, like moving from n=3 to n=1, the principal quantum number decreases and the energy is released in the form of electromagnetic radiation, typically observed as light.

Bohr Model of the Hydrogen Atom

The Bohr model of the hydrogen atom, proposed by Niels Bohr in 1913, is a simple, yet powerful way to visualize atomic structure and electron transitions within hydrogen - the most simple atom. This model posits that electrons orbit the nucleus in distinct paths called orbits or shells with fixed sizes and energies, very much like planets around the Sun.

In the context of the Bohr model, the energy levels are quantized, meaning the electron can only exist in certain allowed orbits, corresponding to certain principal quantum numbers. It uses the Bohr radius a₀ as the unit for these orbit sizes, which correspond to the lowest energy state, n=1. Transitions between these fixed orbits involve discrete changes in energy, either absorbed when moving to larger orbits or emitted when moving to smaller ones. This discrete nature explains the characteristic emission or absorption spectra observed for hydrogen.

Energy Absorption and Emission

The process of energy absorption and emission in an atom happens when an electron makes a transition between different energy levels or orbits. When an electron jumps to a higher energy level, it absorbs a specific amount of energy, which matches the energy difference between the initial and final levels. Conversely, an electron emits energy when it falls to a lower energy level.

Determining whether energy is absorbed or emitted can be deduced by comparing the principal quantum numbers of an electron’s initial and final orbits. An increase from n=2 to n=6, or n=6 to n=9, signifies absorption. A decrease, from an orbit of radius 4.76 Å to 0.529 Å, which correspond to n=3 to n=1, indicates emission. The emitted or absorbed energy can manifest as different forms of electromagnetic radiation, including visible light, with the exact frequency of the radiation depending on the energy difference between the electron's initial and final state.