Question

Does the electron probability distribution for \(1 \mathrm{~s}\) and \(2 \mathrm{~s}\) support or refute the Bohr picture of the shell?

Short answer

In summary, the electron probability distribution for the 1s and 2s orbitals refutes the Bohr model's picture of the shell structure. The Bohr model describes electrons orbiting in discrete shells, while the probability density functions for the 1s and 2s orbitals show a continuous distribution of electron positions that do not have clearly defined boundaries. Additionally, the presence of radial nodes in the 2s orbital contradicts the Bohr model, which does not account for such instances.

Step-by-step solution

Understand the Bohr model

The Bohr model, proposed by Niels Bohr in 1913, is a historical model for the hydrogen atom. It consists of a nucleus at the center and electrons orbiting around the nucleus in specific, fixed orbits called shells. Each shell can accommodate a certain number of electrons, and these electrons can only move between shells by absorbing or emitting energy in the form of discrete quanta called photons.

Analyze the electron probability distribution for 1s and 2s orbitals

To analyze the electron probability distribution for the 1s and 2s orbitals, we need to look into their respective wave functions and the square of the wave functions (Ψ^2) as they represent the probability density of finding an electron in a specific location in the orbital. For the 1s orbital, the wave function is given by: \[\Psi_{1s}(r) = \dfrac{1}{\sqrt{\pi}} \cdot \dfrac{1}{a_0^{3/2}} e^{-r/a_0}\] And for the 2s orbital, the wave function is given by: \[\Psi_{2s}(r) = \dfrac{1}{4\sqrt{2\pi}} \cdot \dfrac{1}{a_0^{3/2}} \left(2-\dfrac{r}{a_0}\right) e^{-r/2a_0}\] Here, \(a_0\) is the Bohr radius.

Calculate the probability density functions for 1s and 2s orbitals

The probability density function is the square of the wave function, Ψ^2. For the 1s orbital, the probability density function is: \[\Psi_{1s}^2(r) = \dfrac{1}{\pi a_0^3} e^{-2r/a_0}\] For the 2s orbital, the probability density function is: \[\Psi_{2s}^2(r) = \dfrac{1}{32 \pi a_0^3} \left(2-\dfrac{r}{a_0}\right)^2 e^{-r/a_0}\]

Compare the probability density functions with the Bohr model

From the probability density functions of the 1s and 2s orbitals, it can be observed that the electron density decreases as the radial distance from the nucleus increases. However, there is no clearly defined boundary for the electron's position as it is in the Bohr model. In the case of the 2s orbital, the probability density function has a radial node (r > 0), which indicates that there is a non-zero probability of finding the electron between two regions. This phenomenon contradicts the Bohr model, which suggests no such probability for the electrons in the set orbits.

Summarize the findings

In summary, the electron probability distribution for the 1s and 2s orbitals do not support the Bohr model's picture of the shell structure. The probability density functions indicate a continuous distribution of electron position, directly contradicting the Bohr model, which illustrates discrete orbits. Furthermore, the presence of radial nodes in the 2s orbital contradicts the Bohr model, which does not account for such instances.

Key concepts

Electron Probability Distribution

When studying the arrangement of electrons around an atom, one must consider the electron probability distribution. This concept refers to the likelihood of finding an electron in a particular area around an atom's nucleus.

Unlike the Bohr model, which predicts that electrons traverse fixed orbits, in the quantum model, electrons are described as being distributed around the nucleus in a probabilistic manner. This is graphically depicted by what we call probability clouds or electron clouds, which are denser where the likelihood of finding an electron is higher. The quantum model doesn't offer a precise path like a planet's orbit around the sun but instead provides regions called atomic orbitals where an electron is most likely to be found.

Wave Functions

Wave functions are at the heart of quantum mechanics, acting as mathematical descriptions of the quantum states of particles such as electrons. Each wave function, denoted by the Greek letter Psi (\r\text{\(\Psi\)}) is unique to a given particle and contains all the information about that particle's state.

An electron's wave function, for instance, determines the spatial distribution of the electron around the nucleus. The absolute square of this wave function (\(\Psi^2\)) then gives us the probability density function for finding the electron at a particular point in space. This key concept refutes the classical idea of an electron's precise path and instead emphasizes a cloudy, probabilistic pattern.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces a radical departure from classical physics, particularly when it comes to understanding electron behavior in atoms.

The principles of quantum mechanics tell us that particles such as electrons have properties of both particles and waves (wave-particle duality). This duality explains why electrons exhibit behavior that is sometimes particle-like, such as collisions, and other times wave-like, with interference patterns. Quantum mechanics uses complex numbers in wave functions to predict the probability of an electron's position and energy, offering a much more nuanced and accurate portrayal of atomic structure than the Bohr model.

Atomic Orbitals

Atomic orbitals are regions around the nucleus within which electrons are most likely to be found, according to the quantum model of the atom. These orbitals are defined by the solutions to the Schrödinger equation, which describe the wave-like behavior of electrons.

Unlike the circular orbits of the Bohr model, atomic orbitals have intricate shapes—such as spherical (s orbitals), dumbbell-shaped (p orbitals), and more complex structures like d and f orbitals. Each of these shapes corresponds to probability distributions and is a visual representation of where an electron is likely to be based on the wave function's square. Electrons do not 'orbit' the nucleus in these orbitals but are rather distributed in a way that the probability of finding an electron decreases with distance from the nucleus.

Probability Density Functions

Probability density functions (PDFs) are one of the most decisive elements distinguishing the Bohr model from the quantum model in the study of atomic behavior. For an electron in an atomic orbital, the PDF represents how the electron's probability distribution is spread out in space.

The calculation of the PDF involves squaring the modulus of the wave function (\(\Psi^2\)), which gives the probability per unit volume of finding the electron at a given distance from the nucleus. This concept may seem abstract, but it is what allows scientists to make predictions about an electron's behavior without knowing its exact location. Interestingly, the quantum model's PDFs sometimes show areas of zero probability, known as nodes, which is an aspect that doesn't appear in the simplified Bohr model. The existence of nodes in orbitals, such as the 2s orbital, is a clear indicator that the electron probability distribution provided by the quantum model offers a more complex and accurate picture than Bohr’s shell model.