Problem 152 - Chemistry

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 location of the node of the \(2 s\) wave function from the nucleus.

Step-by-Step Solution

Answer

The node of the \(2 s\) wave function from the nucleus is at a distance of \(1.058 \times 10^{-9}\) m.

Step 1: Set the wave function equal to zero

The expression of the wave function is given as \(\Psi_{2 s}=\frac{1}{\sqrt{2 a_{0}^{3}}}(1-\frac{\rho}{2})e^{-\rho/2}\). To find the node, this function should be equal to zero. Therefore, we set \(\Psi_{2 s}\) equal to zero and consider the radial part.

Step 2: Solve for \(\rho\)

When setting \(\Psi_{2 s}\) to zero, note that \(\Psi_{2 s}\) equals zero whenever the expression in the parenthesis equals zero. Therefore, we solve the equation \(1-\frac{\rho}{2} = 0\), which gives \(\rho = 2.0\).

Step 3: Convert \(\rho\) to \(r\)

Now, given that \(\rho = Z(r / a_{0})\) where \(Z = 1\) for hydrogen and that \(a_{0} = 0.529 \cdot 10^{9}\) m, \(r\) can be determined by solving the equation \( r = \rho a_{0} = 2.0 \cdot 0.529 \cdot 10^{-9}\) m.

Key Concepts

Wave Function

The wave function is a fundamental concept in quantum mechanics. It describes the probability amplitude of a particle's position in space. For the hydrogen atom, the wave function captures the behavior of an electron around the nucleus.
In our exercise, the wave function for the 2s orbital is a mathematical expression that determines how likely it is to find an electron at a given point. This expression includes variables like \(\rho\) and the Bohr radius.

• It's usually denoted by the Greek letter Psi (\(\Psi\)).
• For specific orbitals, the wave function will have nodes or points where the probability of finding the electron is zero.
• Eigenfunctions of the Schrödinger equation for different orbitals are solutions that provide these wave functions.

Understanding the wave function helps predict the behavior of electrons in atoms.

2s Orbital

The 2s orbital is one of the hydrogen atom's possible orbitals, specifically corresponding to the principal quantum number n = 2. Unlike the 1s orbital, which is a lower energy level, the 2s orbital allows the electron to be found further away from the nucleus more often.
The shape and size of this orbital are determined by the radial wave function \( \Psi_{2s} \):

• This orbital has a spherical symmetry.
• It possesses a node, which is a region where there is zero probability of finding the electron.
• The node in a 2s orbital is a feature that results from the wave function becoming zero.

Computing the location of this node can provide insights into the electron's behavior and energy.

Bohr Radius

The Bohr radius is a physical constant that represents the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is a key component in quantum mechanics and is represented by \(a_0\).

• The value is approximately 0.529 nm.
• It serves as a unit for measuring atomic distances, particularly in hydrogen-like atoms.
• The Bohr model uses these distances to predict electron positions and energy levels.

The Bohr radius is crucial in determining the value of \(\rho\) and thus the exact node location for different quantum states or orbitals.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the complex interactions of tiny particles like electrons and photons. It sets the framework for understanding phenomena at the atomic scale, where classical physics doesn't apply.
In quantum mechanics, elements such as wave functions and orbitals are used to explain the behavior and energy of particles.

• Employs mathematical equations, like the Schrödinger equation, to describe systems.
• Introduces the concept of quantization, where energy levels are discrete and specific.
• Allows for the understanding of atom structures, chemical bonds, and material properties.

The concepts of nodes and probability distributions emerge here, helping to resolve the characteristics of electrons in atoms and their subsequent reactions and bonds.