Question

The energy of the electron in a hydrogen atom has a negative sign for all possible orbits because (a) when the electron is attracted by the nucleus and is present in orbit \(n\), the energy is emitted and its energy is lowered. (b) when the electron is attracted by the nucleus and is present in orbit \(n\), the energy is absorbed and its energy is increased. (c) when the electron is repelled by the nucleus, the energy is released and its energy is lowered. (d) None of these.

Short answer

The correct answer is (a) because the electron emits energy when it transitions to a lower energy orbit, which is a negative energy level, meaning its total energy reduces and thus is represented by a negative sign.

Step-by-step solution

- Understanding the Concept of Energy in a Hydrogen Atom

The energy level of an electron in a hydrogen atom is determined by the orbit in which the electron is present. According to the Bohr model, these energy levels are quantized, meaning the electron can only exist in certain discrete energy levels which are negative. This is because the energy is defined relative to zero level energy, which corresponds to the electron being at an infinite distance from the nucleus. In any bound state, where the electron is in an orbit around the nucleus, the energy will be less than zero, indicating a negative value.

- Analyzing Option (a)

Option (a) suggests that when the electron is in any orbit 'n', it implies that the electron has emitted energy to come to that state (coming from a higher energy level to a lower one), which leads to a decrease in its energy level resulting in a more negative value. This is consistent with the Bohr model, as the energy levels become progressively negative as the electron transitions to orbits closer to the nucleus.

- Dismissing Option (b)

Option (b) states that the energy is absorbed when the electron is attracted by the nucleus and its energy is increased. However, this describes a process where the electron moves to a higher energy orbit (from a lower n to a higher n), not why all energy levels are negative. Therefore, (b) does not explain the nature of energy levels in hydrogen.

- Dismissing Option (c)

Option (c) suggests that the electron is repelled by the nucleus, leading to a decrease in energy. This is incorrect because the fundamental force between the nucleus and the electron is attractive due to their opposite charges. Therefore, (c) is not a valid explanation for the negative energy levels.

- Dismissing Option (d)

Option (d) states 'None of these', which would suggest that none of the given options correctly explain why the energy of an electron in a hydrogen atom is always negative. However, as previously established, option (a) does provide a correct explanation. Therefore, (d) is not the correct answer.

Key concepts

Bohr Model

The Bohr model, proposed by Niels Bohr in 1913, revolutionized our understanding of atomic structure by introducing a new concept that electrons orbit the nucleus in distinct paths. Imagining the atom to resemble a mini solar system, the Bohr model focuses on the hydrogen atom, the simplest atom with a single electron orbiting a single proton.

According to the Bohr model, the electron can only occupy certain orbits or 'shells' at fixed distances from the nucleus without losing energy in the form of radiation. These orbits correspond to specific energy levels, and the electron must gain or release energy in discrete amounts to jump from one level to another—a concept known as quantization. The most stable and lowest energy state of an electron in a hydrogen atom is defined as the ground state.

Orbits and Energy Levels

In the Bohr model's framework, the energy levels are determined by the quantized orbits, meaning electrons can't have energy values between these levels. These stable orbits help prevent the electron from spiraling inward and collapsing into the nucleus, which classical physics could not explain.

Transitions and Photons

When an electron jumps to a higher energy orbit, it must absorb energy; when it transitions to a lower orbit, it emits energy. This energy is exchanged in the form of photons, with the photon's energy directly proportional to the difference between the two energy levels involved in the transition.

Quantized Energy Levels

Quantized energy levels are a fundamental aspect of the Bohr model that underpins our understanding of atomic physics. These discrete levels are like the rungs on a ladder, where the electron can only be on a rung, not in between. This quantization explains why only certain wavelengths of light are emitted or absorbed by atoms, leading to the characteristic spectral lines for each element.

Quantum Jumps

Electrons can 'jump' between these energy levels, absorbing or emitting energy. These jumps are discontinuous and happen instantaneously, which means the electron does not exist in a transition state between energy levels.

Understanding the Negative Values

Why are these energy levels negative? It's a matter of defining the zero point. In the Bohr model, the zero energy level is set when the electron is completely free from the atomic nucleus—essentially at an infinite distance away. Energy levels within the atom are lower (less energy) than this free state, hence they are classified as negative.

Negative Energy Levels

The concept of negative energy levels in a hydrogen atom can sometimes be puzzling. However, they can be understood as an indicator of how much energy it would take to remove the electron from the atom—i.e., the ionization energy. Negative energy signifies that the electron is bound to the atom and has less energy than a free electron, and the magnitude of this negative energy represents how 'deeply' the electron is bound within the potential well of the nucleus.

Bound vs. Unbound States

An electron in a negative energy level is in a bound state, meaning it is attached to an atom and requires energy to be freed. Conversely, a positive energy level would mean the electron is in an unbound state, free from the nucleus and capable of escaping the atom's pull.

Relation to Stability

The more negative the energy level, the more stable the electron's orbit. At the lowest, most negative energy state—the ground state—the electron is most tightly held and is least likely to be perturbed by external energies. This stability arises from the attractive force between the negatively charged electron and the positively charged nucleus.