Problem 27 - College Physics

Question

\bullet A triply ionized beryllium ion, \(\mathrm{Be}^{3+}\) (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom, except that the nuclear charge is four times as great. (a) What is the ground-level energy of \(\mathrm{Be}^{3+} ?\) How does this compare with the ground-level energy of the hydrogen atom? (b) What is the ionization energy of Be \(^{3+} ?\) How does this compare with the ionization energy of the hydrogen atom? (c) For the hydrogen atom, the wavelength of the photon emitted in the transition \(n=2\) to \(n=1\) is 122 nm. (See Example 28.6 . What is the wavelength of the photon emitted when a \(\mathrm{Be}^{3+}\) ion undergoes this transition? (d) For a given value of \(n,\) how does the radius of an orbit in \(\mathrm{Be}^{3+}\) compare with that for hydrogen?

Step-by-Step Solution

Answer

(a) -217.6 eV, 16 times more negative. (b) 217.6 eV, 16 times greater. (c) 7.57 nm, much shorter. (d) 4 times smaller radius.

Step 1: Understanding the Nuclear Charge

Beryllium has a nuclear charge of 4 because its atomic number is 4. In the triply ionized state, it acts like a hydrogen atom with a nuclear charge of 4.

Step 2: Calculating Ground-Level Energy of Hydrogen

The ground-level energy for a hydrogen atom is given by the formula: \[ E_n = - rac{13.6 \, ext{eV}}{n^2} \]For hydrogen, at ground level \( n = 1 \), so:\[ E_1 = - 13.6 \, ext{eV} \]

Step 3: Calculating Ground-Level Energy of Be³⁺

For \(\mathrm{Be}^{3+}\), the ground level energy is scaled by the square of the nuclear charge (\(Z = 4\)): \[ E_n = -13.6 \, ext{eV} \times Z^2 / n^2 \]For \( n = 1 \), \[ E_1 = -13.6 \, ext{eV} \times 4^2 = -13.6 \, ext{eV} \times 16 = -217.6 \, ext{eV} \]

Step 4: Comparing Ground-Level Energies

The ground-level energy of \(\mathrm{Be}^{3+}\) is \(-217.6 \, \text{eV}\), which is 16 times more negative than that of hydrogen \(-13.6 \, \text{eV}\).

Step 5: Finding Ionization Energy of Be³⁺

The ionization energy is the energy required to remove the electron from the ground state, which is equal to the absolute value of the ground-level energy:\[ E_{ ext{ionization}} = 217.6 \, ext{eV} \]

Step 6: Comparing Ionization Energies

The ionization energy of \(\mathrm{Be}^{3+}\) is 217.6 eV, 16 times greater than that of hydrogen, which is 13.6 eV.

Step 7: Wavelength of Photon Emission in Be³⁺ Transition

The energy of the photon emitted in a transition from \(n=2\) to \(n=1\) for \(\mathrm{Be}^{3+}\) is calculated as:\[ \Delta E = E_1(1) - E_2(2) = -217.6 (1^2) - (-217.6 \times (1/4)) \]\[ \Delta E = -217.6 + 54.4 = 163.2 \, \text{eV} \]Using the formula \( E = \frac{hc}{\lambda} \), solve for \( \lambda \):\[ \lambda = \frac{hc}{\Delta E} \]Using \( h = 4.135667696\times 10^{-15} \, \text{eV}\cdot\text{s}, \) and \( c = 3.00\times 10^8 \, \text{m/s}\),\[ \lambda \approx \frac{4.135667696 \times 10^{-15} \times 3.00 \times 10^8}{163.2} = 7.57 \, \text{nm} \]

Step 8: Comparing Photon Wavelengths

The wavelength of light emitted by \(\mathrm{Be}^{3+}\) during the \(n=2\) to \(n=1\) transition is 7.57 nm, which is significantly shorter than the 122 nm for hydrogen.

Step 9: Comparing Orbital Radii for Be³⁺ and Hydrogen

For a given \(n\), the orbital radius in hydrogen is given by:\[ r_n = n^2 a_0 / Z \]where \(a_0\) is the Bohr radius.Since \(\mathrm{Be}^{3+}\) has a \(Z = 4\), the radius is four times smaller:\[ r_{\text{Be}^{3+}} = \frac{r_{\text{Hydrogen}}}{4} \]

Key Concepts

Triply Ionized

A triply ionized beryllium ion, denoted as \( \mathrm{Be}^{3+} \), is a beryllium atom that has lost three of its four electrons. This leaves it with a single electron, similar to a hydrogen atom, wherein both now only have one orbiting electron.

However, unlike hydrogen, which has a nuclear charge of 1 due to its single proton, the beryllium ion retains a nuclear charge associated with its atomic number, 4, due to its four protons. This larger nuclear charge affects how the electron behaves in the atom, making it different in energy levels and other physical properties compared to hydrogen.

Nuclear Charge

Nuclear charge is the total charge of protons in the nucleus. For \( \mathrm{Be}^{3+} \), the nuclear charge is 4 because it has four protons. This is fundamentally different from hydrogen, which has a nuclear charge of 1.

The nuclear charge determines the strength of the attraction between the nucleus and the electron. A higher nuclear charge means a stronger pull on the electron, which influences several atomic properties, like the amount of energy needed to remove the electron (ionization energy) and the size of the electron's orbit around the nucleus (orbital radius).

• Lower nuclear charge means a weaker pull.
• Higher nuclear charge results in a stronger pull.

Ground-Level Energy

The ground-level energy of an atom’s electron is the energy it has in its lowest energy state, or when \( n = 1 \). For hydrogen, this energy is \(-13.6 \, \text{eV}\).

When calculating this for \( \mathrm{Be}^{3+} \), we account for its higher nuclear charge. The ground-level energy becomes \(-217.6 \, \text{eV}\), which is 16 times more negative than that of hydrogen.

This larger negative value indicates a stronger hold on the electron, requiring more energy to remove it from the atom or move it to a higher energy level.

Ionization Energy

Ionization energy is the energy required to completely remove an electron from an atom in its ground state. Thus, for hydrogen, it takes \( 13.6 \, \text{eV} \), mirroring its ground-level energy but as a positive value.

For \( \mathrm{Be}^{3+} \), because of the higher nuclear charge, the ionization energy is much greater at \( 217.6 \, \text{eV} \). This indicates a stronger attraction to the electron.

In essence, ionization energy is a direct reflection of how tightly the electron is bound to the atom:

• Higher ionization energy implies a more tightly bound electron.
• Lower ionization energy suggests a more loosely held electron.

Photon Emission

Photon emission occurs when an electron transitions between energy levels within an atom, releasing energy in the form of a photon. When transitioning from \( n=2 \) to \( n=1 \) in hydrogen, a photon with a wavelength of 122 nm is emitted.

In the case of \( \mathrm{Be}^{3+} \), the energy difference between \( n=2 \) and \( n=1 \) is much greater due to the higher nuclear charge. The emitted photon has a shorter wavelength of 7.57 nm. This indicates that \( \mathrm{Be}^{3+} \) releases more energy during its electron transition compared to hydrogen.

Orbital Radius

The orbital radius of an electron in an atom describes the average distance from the nucleus during a particular energy level, or shell \( n \). For hydrogen, this follows Bohr’s model with the formula \( r_n = n^2 a_0 \), where \( a_0 \) is the Bohr radius.

For \( \mathrm{Be}^{3+} \), with its nuclear charge \( Z = 4 \), the formula modifies to \( r_n = \frac{n^2 a_0}{Z} \). This results in an orbital radius that is four times smaller than for hydrogen’s equivalent shell, reflecting the stronger pull of the nucleus on the electron.

When considering atomic structure:

• Higher nuclear charge decreases orbital radius.
• Lower nuclear charge increases orbital radius.

Hydrogen Atom Comparison

Comparing \( \mathrm{Be}^{3+} \) to the hydrogen atom helps in understanding atomic physics due to their similar single-electron setup. However, several differences are noteworthy:

• Nuclear charge in \( \mathrm{Be}^{3+} \) is four times that of hydrogen.
• \( \mathrm{Be}^{3+} \) shows significantly stronger hold on its electron.
• Ground-level and ionization energies in \( \mathrm{Be}^{3+} \) are 16 times greater than in hydrogen.
• Photon emissions in \( \mathrm{Be}^{3+} \) result in much shorter wavelengths due to higher energy releases.
• The orbital radius in \( \mathrm{Be}^{3+} \) is reduced due to a stronger nuclear attraction.

Understanding these differences illustrates how an increased nuclear charge influences atomic properties, making \( \mathrm{Be}^{3+} \) a more tightly bound and energetically distinct system despite its hydrogen-like electron setup.