Question

Criticize or comment on the following statements: (a) The energy of a photon is inversely proportional to its wavelength. (b) The energy of the hydrogen electron is inversely proportional to the quantum number \(\ell\). (c) Electrons start to enter the fifth principal level as soon as the fourth is full.

Short answer

#Evaluation# Based on the analysis of the provided statements about photons and electrons, we can summarize the explanations and correctness of each statement as follows: (a) The energy of a photon is inversely proportional to its wavelength. This statement is true, as shown through the derivation using Planck's equation and the speed of light equation. (b) The energy of a hydrogen electron is proportional to the azimuthal quantum number 𝑙. This statement is false, as the energy of a hydrogen electron solely depends on the principal quantum number 𝑛, not 𝑙, according to the hydrogen atom energy levels equation. (c) Electrons wait for all the subshells of the fourth principal level to be fully occupied before entering the fifth principal level. This statement is not entirely accurate, as electrons follow the Aufbau Principle, Pauli Exclusion Principle, and Hund's Rule, which allow some subshells of the fifth principal level to be filled before the completion of all subshells in the fourth principal level.

Step-by-step solution

Statement (a)

The energy of a photon can be described by Planck's equation: $$E = h \cdot f$$ Where \(E\) is the energy, \(h\) is the Planck's constant \((6.63 \times 10^{-34} \, Js)\), and \(f\) is the frequency of the photon. Since the speed of light (\(c\)) can be described as the product of frequency and wavelength (\(\lambda\)): $$c = f \cdot \lambda$$ We can rearrange this equation and substitute it into Planck's equation: $$E = h \cdot \frac{c}{\lambda}$$ Therefore, the energy of a photon (\(E\)) is inversely proportional to its wavelength (\(\lambda\)). This statement is true.

Statement (b)

The energy of an electron in a hydrogen atom can be described by the equation: $$E_n = -\frac{13.6 \, eV}{n^2}$$ Where \(E_n\) is the energy, and \(n\) is the principal quantum number. Notice that the formula for the energy of an electron in a hydrogen atom does not depend on the quantum number \(\ell\). Instead, it solely depends on the principal quantum number \(n\). Therefore, this statement is false. The energy of the hydrogen electron is inversely proportional to the square of the principal quantum number \(n\), not \(\ell\).

Statement (c)

An atom's electron configuration is determined by the principles of quantum mechanics, with electrons gradually filling the energy levels following these main rules: the Aufbau Principle, Pauli Exclusion Principle, and Hund's Rule. According to the Aufbau Principle, electrons fill the lowest energy level available first. Energy levels are determined by the sum of the principal quantum number \(n\) and the azimuthal quantum number \(\ell\). For the fourth principal level (\(n = 4\)), possible values of \(\ell\) range from 0 to 3. For the fifth principal level (\(n = 5\)), possible values of \(\ell\) range from 0 to 4. It turns out that for \(n = 4\), not all subshells are entirely filled before electrons start to populate the fifth principal level. Specifically, the \(4d\) subshell (with \(n = 4\) and \(\ell = 2\)) is not completed before electrons start to fill the \(5s\) subshell (with \(n = 5\) and \(\ell = 0\)). Hence, this statement is not wholly accurate. Electrons do not necessarily wait for all the subshells of the fourth principal level to be entirely filled before entering the fifth principal level.

Key concepts

Planck's Equation

Understanding the energy of photons is critical in the realm of quantum mechanics chemistry, and Planck's equation plays a vital role in this understanding. Let's dive deep into the equation itself: \(E = h \times f\), where \(E\) is the energy of the photon, \(h\) is a fundamental constant known as Planck's constant (approximately \(6.63 \times 10^{-34} J\cdot s\)), and \(f\) is the frequency of the photon.

The key takeaway here is the relationship between energy and frequency. Higher frequency photons have more energy; these are often photons like X-rays or ultraviolet light. Conversely, lower frequency photons, such as those in the radio wave region, carry less energy. The notion that energy is inversely proportional to wavelength arises because as the wavelength increases, the frequency (and thus energy) decreases, due to the speed of light being a constant value. Planck's equation has far-reaching implications, extending to the explanation of the Photoelectric Effect, which supported the quantum theory of light.

Energy Levels in Atoms

When exploring energy levels in atoms, it is important to differentiate between the various quantum numbers that describe an electron's position and energy. In the case of a hydrogen atom, the energy level of an electron is quantified by the principal quantum number \(n\), which signifies which shell the electron occupies. The energy for any level \(n\) is given by \(E_n = -\frac{13.6 eV}{n^2}\).

This means the energy of an electron gets lower (more negative) as \(n\) increases, and the energy levels become closer together as \(n\) becomes larger. It's essential to highlight that the energy of an electron in a hydrogen atom does not depend on the azimuthal quantum number \(\ell\), which is incorrect as per the original exercise statement (b). This concept is fundamental in quantum chemistry because it illustrates how electrons occupy discrete energy levels within an atom, as opposed to classical physics which would predict a continuous range of possible energies.

Aufbau Principle

The Aufbau Principle is a guiding rule for predicting the electronic configurations of atoms during the filling process of atomic orbitals. In simple terms, it can be stated as: 'Electrons occupy the lowest available energy orbital first.' But it's not as straightforward as filling up each energy level one after another like pouring water into cups. Here's where the concept of subshells (designated by the quantum number \(\ell\)) comes into play.

For example, when looking at higher principal energy levels, such as the fourth (\(n = 4\)) and the fifth (\(n = 5\)), the orbitals don't fill sequentially by \(n\) value alone. Due to energy differences between different types of orbitals (s, p, d, f), you can have electrons entering the \(5s\) orbital before the \(4d\) orbital is completely filled. This order is determined by the sum of the principal quantum number and the azimuthal quantum number (\(n + \ell\)), which approximates the energy of the respective subshell. It is critical to understand the Aufbau Principle when predicting electron configurations and during the construction of the Periodic Table's familiar layout.