Question

The quantum-mechanical treatment of the hydrogen atom gives this expression for the wave function, \(\psi,\) of the \(1 s\) orbital: $$ \Psi=\frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} e^{-r l a_{0}} $$ where \(r\) is the distance from the nucleus and \(a_{0}\) is \(52.92 \mathrm{pm}\). The probability of finding the electron in a tiny volume at distance \(r\) from the nucleus is proportional to \(\psi^{2}\). The total probability of finding the electron at all points at distance \(r\) from the nucleus is proportional to \(4 \pi r^{2} \psi^{2}\). Calculate the values (to three significant figures) of \(\psi, \psi^{2},\) and \(4 \pi r^{2} \psi^{2}\) to fill in the following table and sketch a plot of each set of values versus \(r .\) Compare the latter two plots with those in Figure \(7.17 \mathrm{~A}\).

Short answer

Calculate specific values and use Tables

Step-by-step solution

Understand the Wave Function \(\frac{\text{1}}{\text{\textbackslash pi}} \text{\textbackslash left(}\frac{\text{1}}{\text{a\textsubscript{0}}}\text{\textbackslash right)\textsuperscript{3/2} e\textsuperscript{-r a\textsubscript{0}}\)}

\(ψ = \frac{1}{\text{\textbackslash sqrt{\text{\textbackslash pi}}}} \text{\textbackslash left( \text{\text{:}} \frac{1}{a\textsubscript{0}} \text{\text{} \text{:} \text{\text{} \text{superscript\text{} 3/2\text{} \text:{} e\text{} \text{}:-r\text{} a\textsubscript {0}\}\right)}\) is the wave function for the 1s orbital.

Calculate \( ψ \) at Specific Values of \( r \)

Using the wave function to find \(ψ\), select values for \(r\) such as 0, 0.1, 0.2, ..., up to 1 in multiples of 0.1 \( a\textsubscript {0}\).

Calculate \( ψ^{2} \)

Square the calculated \(ψ\) values: \( ψ^{2} = \text{ ψ} \textsuperscript{2} \)

Calculate \( 4 \ pi r \textsuperscript{2} ψ^{2} \)

Use: \( 4 \ pi \ r^{2} ψ^{2} \)

Key concepts

Wave Function

In quantum mechanics, the wave function, represented by \(\frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} e^{-r / a_{0}}\), is a mathematical description of the quantum state of a system. This specific wave function is for the ground state (1s orbital) of the hydrogen atom.

The wave function, \(\psi\), contains all the information about the electron's position in the 1s orbital. The variable \(r\) represents the distance between the electron and the nucleus, while \(a_{0}\) (Bohr radius) is approximately 52.92 pm, a characteristic length scale for the hydrogen atom.

Keep in mind, the value of the wave function itself is not physically observable. Instead, it's the square of this wave function that delivers observable quantities, specifically the probability distribution of the electron's position.

Electron Probability Distribution

The electron probability distribution is a concept derived from the square of the wave function, \(\psi^{2}\). It indicates the likelihood of finding an electron within a given tiny volume surrounding point \(r\) in space.

Since \(\psi\) holds complex values, squaring it, \(\psi^{2} = \psi \cdot \psi^{*}\), results in a real, non-negative value that corresponds to this probability.

This distribution shows regions where the electron is more likely to be found. For the 1s orbital, the probability density is highest at the nucleus and decreases as we move away, exhibiting an exponential decay.

• Closer to the nucleus: higher \(\psi^{2}\) values.
• Farther from the nucleus: lower \(\psi^{2}\) values.

The function \(4 \pi r^{2} \psi^{2}\) extends the probability to a spherical shell at distance \(r\), accounting for the volume elements in spherical coordinates.

1s Orbital

The 1s orbital is the simplest type of atomic orbital in quantum mechanics. It's spherical in shape, surrounding the nucleus, and is defined by the principal quantum number, \(n=1\), and angular momentum quantum number, \(l=0\). There are no angular nodes, making it a perfectly symmetric sphere around the nucleus.

The shape and size of the 1s orbital are characterized by the wave function we discussed earlier. It determines where the electron is most likely to reside around the nucleus in the hydrogen atom.

The notable features of the 1s orbital include:

• Highest electron probability density at \(r=0\) (at the nucleus).
• Exponential decrease in probability as you move away from the nucleus.
• A spherical symmetry because it only depends on \(r\) and not on \(\theta\) or \(\phi\) angular coordinates.

Understanding the 1s orbital helps in grasping more complex orbitals and the overall quantum behavior of electrons in atoms.