Atomic orbitals may be combined to form molecular orbitals. In such orbitals, there is a nonzero probability of finding an electron on any of the atoms that contribute to that molecular orbital. Consider an electron that is confined in a molecular orbital that extends over two adjacent carbon atoms. The electron can move freely between the two atoms. The C-C distance is \(139 \mathrm{pm}\). (a) Using the one-dimensional particle-in-the-box model, calculate the energy required to promote an electron from the \(n=1\) to the \(n=2\) level, assuming that the length of the box is determined by the distance between two carbon atoms. (b) To what wavelength of radiation does this correspond? (c) Repeat the calculation for a linear chain of 1000 carbon atoms. (d) What can you conclude about the energy separation between energy levels as the size of the atom chain increases?

The particle-in-a-box model is a fundamental concept in quantum mechanics that provides significant insight into the behaviour of electrons in molecules, especially when it comes to understanding electronic transitions. This simplistic model assumes that a particle, such as an electron, is confined to moving within a one-dimensional box with infinitely high walls. The reality is far more complex, but this abstraction allows us to use simple mathematics to describe the electron's energy levels.

According to this model, the energy levels (\( E_n \) are quantized, meaning the particle can only occupy certain energy states. The formula to calculate these energy states is given by \( E_n = \frac{n^2h^2}{8m_eL^2} \) where \( n \) is the principal quantum number, \( h \) is Planck’s constant, \( m_e \) is the electron mass, and \( L \) is the length of the box. The restrictions of the model mean that as \( n \) increases, so does the energy, but the difference between each level decreases.

Applying this to chemistry, the 'box' can represent the space within which an electron is confined — for example, within a bond between two carbon atoms. Therefore, understanding this model is essential for predicting the electronic behaviour within molecules, which directly impacts the physical and chemical properties of substances.

Quantum mechanics plays an integral role in chemistry as it provides a theoretical framework for understanding the behaviour of atoms and molecules at the subatomic level. The principles of quantum mechanics explain the discrete energy levels of electrons, the shapes and orientations of atomic and molecular orbitals, and the phenomena of electron tunneling and entanglement.

In the realm of chemistry, the particle-in-a-box model is a simple yet powerful application of quantum mechanics. It elucidates how electrons constrained within the potential well of a molecular orbital can only have certain energy values. This quantization arises because the electron’s wavelength must fit neatly within the box, allowing only specific standing wave patterns. When electrons transition between these energy levels, they absorb or emit photons with energy corresponding to the difference between the states. This transition is fundamental to spectroscopy, a tool widely used in chemistry to investigate the structure and composition of substances. The model is a stepping stone that ultimately leads to more complex and accurate descriptions of molecular behaviour, such as those given by molecular orbital theory.

Electronic transitions are vital to the understanding of molecular behaviour and the interaction of molecules with light. When an electron moves between quantized energy levels, it either absorbs or emits a photon, leading to an electronic transition. The energy \( \Delta E \) associated with this transition is precisely the difference between the initial and final energy states of the electron.

In the context of the problem, the energy required to promote an electron from \( n=1 \) to \( n=2 \) (the first excited state) is calculated using the particle-in-a-box model. This computed energy corresponds to the energy of the photon that an electron would need to absorb to undergo this transition. The relationship between the energy of a photon and the wavelength of the corresponding radiation is given by \( E = \frac{hc}{\lambda} \) where \( c \) is the speed of light. Solving for wavelength, \( \lambda \), highlights the inverse relationship between a photon's energy and its wavelength.

Understanding how the wavelength relates to electron transitions allows chemists to tap into various analytical techniques such as UV-Visible spectroscopy, which probes these transitions to garner information about molecular structure and electronic environments. The particle-in-a-box problem typically ends with a reflection on the trend that, as the 'box' size increases (as with a longer carbon chain), the energy separation between levels decreases, leading to a spectrum of closer-spaced wavelengths.