Question

Ions with a single electron such as He \(^{+}, \mathrm{Li}^{2+},\) and \(\mathrm{Be}^{3+}\) are described by the \(\mathrm{H}\) atom wave functions with \(Z / a_{0}\) substituted for \(1 / a_{0},\) where \(Z\) is the nuclear charge. The 1 s wave function becomes \(\psi(r)=1 / \sqrt{\pi}\left(Z / a_{0}\right)^{3 / 2} e^{-Z r / a_{0}},\) Using this result, compare the mean value of the radius \((r)\) at which you would find the 1 s electron in \(\mathrm{H}, \mathrm{He}^{+}, \mathrm{L}_{1}^{2+}\), and \(\mathrm{Be}^{3+}\).

Short answer

The mean values of the 1s electron radius for the given ions are: H (\(\frac{1}{2}a_{0}\)), He+ (\(\frac{1}{4}a_{0}\)), Li2+ (\(\frac{1}{6}a_{0}\)), and Be3+ (\(\frac{1}{8}a_{0}\)). As the nuclear charge (Z) increases, the mean value of the 1s electron radius decreases, indicating that the electron is found closer to the nucleus. This demonstrates the increased attraction between the electron and the nucleus as the number of protons increases.

Step-by-step solution

Understand the given wavefunction and its variables

The 1s wave function for ions with a single electron is given as: \( \psi(r)=\frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_{0}}\right)^{3/2} e^{-\frac{Zr}{a_{0}}} \) Here, \(r\) is the radial distance from the nucleus, \(Z\) is the nuclear charge, and \(a_{0}\) is the Bohr radius.

Write down the expectation value formula for the radius operator

The expectation value of the radius operator can be expressed as: \(\langle r \rangle = \int_{0}^{\infty} r^2 R_{1s}(r) r R_{1s}(r) dr\) Here, \(R_{1s}(r)\) is the radial part of the 1s wave function, which can be obtained by multiplying \(\psi(r)\) by the spherical harmonics \(\langle Y_{00} | Y_{00} \rangle\): \(R_{1s}(r) = \sqrt{\frac{1}{4\pi}} \psi(r)\)

Calculate the expectation value of radius for each ion

First, substitute \(R_{1s}(r)\) into the expectation value formula: \(\langle r \rangle = \int_{0}^{\infty} r^2 \left( \sqrt{\frac{1}{4\pi}} \psi(r) \right) r \left( \sqrt{\frac{1}{4\pi}} \psi(r) \right) dr\) Next, substitute the wavefunction \(\psi(r)\) for each ion and evaluate the integral: For H (\(Z = 1\)): \(\langle r \rangle_{H} = \int_{0}^{\infty} r^2 \left( \sqrt{\frac{1}{4\pi}} \frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3/2} e^{-\frac{r}{a_{0}}} \right) r \left( \sqrt{\frac{1}{4\pi}} \frac{1}{\sqrt{\pi}}\left(\frac{1}{a_{0}}\right)^{3/2} e^{-\frac{r}{a_{0}}} \right) dr = \frac{1}{2}a_{0}\) Similar calculations for He+ (\(Z=2\)), Li2+ (\(Z=3\)) and Be3+ (\(Z=4\)) give the expectation values: \(\langle r \rangle_{He^{+}} = \frac{1}{4}a_{0}\) \(\langle r \rangle_{Li^{2+}} = \frac{1}{6}a_{0}\) \(\langle r \rangle_{Be^{3+}} = \frac{1}{8}a_{0}\)

Compare the mean values of the radius for the four ions

Now that we have calculated the expectation values of the radius for each ion, we can compare them: - H: \(\frac{1}{2}a_{0}\) - He+: \(\frac{1}{4}a_{0}\) - Li2+: \(\frac{1}{6}a_{0}\) - Be3+: \(\frac{1}{8}a_{0}\) As the nuclear charge (\(Z\)) increases, the mean value of the 1s electron radius decreases, indicating that the electron is found closer to the nucleus. This shows how the electron is being more attracted to the nucleus as the number of protons increases.

Key concepts

Quantum Mechanical Wavefunctions

The wavefunction in quantum mechanics is a crucial concept that describes the probability amplitude for a particle's position or state. One can think of it as a mathematical object that encapsulates all the information about a system's state. For hydrogenic ions, which consist of a single electron orbiting a nucleus, the wavefunction takes a specific form that reflects the nature of the atom or ion in question.

When dealing with the 1s orbital, the simplest case, the wavefunction is spherically symmetric and describes the distribution of the electron's position around the nucleus. If we apply this to a hydrogenic ion, the 1s wavefunction is altered by the presence of a different nuclear charge, represented by the variable Z.

In the given exercise, the wavefunction for the ion's 1s electron is modified to include Z, which is the number of protons in the nucleus. As a result, the wavefunction suggests that the electron's presence is more concentrated closer to the nucleus for ions with a higher nuclear charge. This concept is fundamental to understanding not only the structure of atoms and ions but also their chemical and physical behaviors.

Nuclear Charge Effect on Electron Radius

The nuclear charge of an ion directly influences the radius at which an electron is likely to be found. The relationship between the two is an inverse one: as the nuclear charge increases, the average radius of an electron's orbit decreases. This occurs because a higher nuclear charge corresponds to a stronger electrostatic attraction exerted on the electron, pulling it closer to the nucleus.

In the exercise, we compare hydrogenic ions with increasing nuclear charges. The higher the charge, denoted by Z, the smaller the expectation value for the electron radius becomes. Thus, an ion such as Be3+, with a +3 charge, has a much smaller mean radius for its 1s electron compared to a hydrogen atom (H), which has a +1 charge.

Understanding the effects of the nuclear charge on electron radius is fundamental in atomic physics because it helps explain why atoms and ions behave differently, including their sizes, ionization energies, and spectral lines, among other characteristics.

Expectation Values in Quantum Mechanics

In quantum mechanics, expectation values are a statistical means to predict the average outcome of a given property if a measurement were to be performed many times on identically prepared systems. For instance, the expectation value of the radius (r) for an electron in a given state offers an averaged measure of how far the electron is from the nucleus over many observations.

The calculation involves an integral over all space of the probability density function for the electron's position times the radius itself. This probability density is found by squaring the radial part of the wavefunction. As we see in the exercise solution, the expectation values for the radius decrease with increasing nuclear charge, reflecting the tighter confinement of the electron due to the stronger nuclear attraction.

These values provide not just predictive insights but also help explain why certain chemical and physical properties of elements and ions are the way they are, linking the abstract quantum description with tangible phenomena.

Bohr Model Adaptations for Different Ions

While the Bohr model is an earlier atomic theory that predates quantum mechanics, it is still useful for visualizing certain aspects of atomic structure and has been adapted to describe ions with a single electron, such as He+, Li2+, and Be3+. These adaptations involve adjusting the model's parameters, such as the orbit radius, to account for the increased nuclear charge in different ions.

In the context of the Bohr model, the orbit radius for an electron is inversely proportional to the nuclear charge (Z). The larger the charge, the smaller the radius, which aligns with what we find when looking at the quantum mechanical expectation values. The adjustments for different ions show the predictive nature of the Bohr model, despite its limitations, and where it provides a bridge to understanding more complex quantum mechanical descriptions.

These adapted models illuminate how an increased nuclear charge compacts the electron orbit, an understanding which is significant in grasping not just the structure, but also the reactivity of ions.