Question

The wave function for the \(2 s\) orbital in the hydrogen atom is $$ \psi_{2 s}=\frac{1}{\sqrt{2 a_{0}^{3}}}\left(1-\frac{\rho}{2}\right) e^{-\rho / 2} $$ where \(a_{0}\) is the value of the radius of the first Bohr orbit, equal to \(0.529 \mathrm{nm} ; \rho\) is \(Z\left(r / a_{0}\right) ;\) and \(r\) is the distance from the nucleus in meters. Calculate the distance from the nucleus (in \(\mathrm{nm}\) ) of the node of the \(2 s\) wave function.

Short answer

The node of the \(2s\) wave function is at 1.058 nm from the nucleus.

Step-by-step solution

Understand the Problem

The problem asks us to find the distance from the nucleus where the wave function for the \(2s\) orbital has a node. A node refers to a point where the wave function, \(\psi_{2 s}\), is zero.

Identify the Node Condition

The wave function will be zero at a node. The equation given is: \[ \psi_{2 s} = \frac{1}{\sqrt{2 a_{0}^{3}}}\left(1-\frac{\rho}{2}\right) e^{-\rho / 2} \] Set the term inside the parentheses equal to zero, because the exponential part never reaches zero. So we have:\[ 1 - \frac{\rho}{2} = 0 \]

Solve for \(\rho\)

Rearrange the equation from Step 2:\[ \frac{\rho}{2} = 1 \]Multiply both sides by 2:\[ \rho = 2 \]

Relate \(\rho\) to \(r\)

Given the relation \( \rho = Z\left(\frac{r}{a_{0}}\right) \) and knowing we are dealing with a hydrogen atom where \( Z = 1 \), substitute into the equation:\[ 2 = \frac{r}{a_{0}}\] Solve for \(r\):\[ r = 2a_{0} \]

Substitute \(a_0\) Value to Find \(r\) in nm

Substitute the given value of \(a_0 = 0.529 \mathrm{nm}\) into the equation:\[ r = 2 \times 0.529 \mathrm{nm} \]Calculate the distance:\[ r = 1.058 \mathrm{nm} \]

Key concepts

Wave Function

In quantum chemistry, the wave function is a fundamental concept that describes the quantum state of a particle or system of particles. It encapsulates all information about a system. For the hydrogen atom, wave functions are solutions to the Schrödinger equation, allowing us to predict the probability distribution of an electron.

The wave function, denoted by \( \psi \), is a mathematical tool used to calculate the likelihood of a particle's properties. For instance, in a hydrogen atom's \(2s\) orbital, the wave function is expressed as:

• \( \psi_{2s} = \frac{1}{\sqrt{2 a_0^3}}\left(1-\frac{\rho}{2}\right) e^{-\rho/2} \)

This expression demonstrates the complex interplay of variables that govern a particle's probable location and energy state. Understanding wave functions is essential for analyzing electronic structures and predicting chemical behavior.

Hydrogen Atom

The hydrogen atom is the simplest atom, consisting of just one electron bound to a proton nucleus. Despite its simplicity, it plays a crucial role in the study of quantum mechanics.

It serves as a model for understanding atomic structures and electron behavior in more complex atoms. Quantum theory is pivotal to understanding the hydrogen atom since classical mechanics fails to explain the stability of its electron orbitals.

• The hydrogen atom was instrumental in developing key concepts like energy quantization and the introduction of quantum numbers, which define electron orbitals.

By delving into the hydrogen atom's electronic configurations, scientists can extrapolate principles applicable to more complex chemical systems.

2s Orbital

In atomic theory, orbitals describe regions of space where there is a high probability of finding an electron. The \(2s\) orbital, in particular, is a type of atomic orbital within the hydrogen atom.

It represents the second lowest energy level of an electron in the hydrogen atom with the principal quantum number \(n = 2\). Unlike the \(1s\) orbital, the \(2s\) orbital features one radial node, which influences how electrons are distributed spatially around a nucleus.

• Electrons in the \(2s\) orbital possess more energy compared to those in the \(1s\) orbital.
• They also have a higher average distance from the nucleus.

Understanding these characteristics helps in visualizing electron configurations and chemical bonding processes.

Node

Nodes are critical concepts in quantum chemistry, representing points or surfaces where the probability density of finding an electron is zero. In simpler terms, they are regions within an orbital where the electron presence is null.

Identifying nodes is important for interpreting wave functions and determining an electron's distribution within an atom.

• In the \(2s\) orbital of a hydrogen atom, there is one radial node where the wave function equals zero.

The presence of a node in the \(2s\) orbital indicates a change in the sign of the wave function and is associated with the electron's radial distance from the nucleus. Quantifying these nodes is crucial for describing an atom's electronic structure.

Bohr Radius

The Bohr radius is a constant that represents the most probable distance between the nucleus and the electron in a hydrogen atom's ground state. It is named after Niels Bohr, who introduced this concept through his Bohr model.

The Bohr radius, denoted as \( a_0 \), provides a scale for measuring atomic distances and orbitals. It is magnitude is approximately \(0.529 \text{ nm} \).

• It functions as a reference value when calculating the radius of orbitals for the hydrogen atom.
• Understanding its implications helps in evaluating other atomic systems.

By anchoring computations in the Bohr radius, scientists can compare atomic sizes and scrutinize how electrons orbit in various atoms, enriching their comprehension of atomic interactions.