A triply ionized beryllium ion, Be³⁺ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. What is the ground-level energy of Be³⁺? How does this compare to the ground-level energy of the hydrogen atom?

Verified step by step guidance

1. The energy levels of a hydrogen-like atom are given by the formula: $E=-\frac{Z^2}{R}/n^2$, where $Z$ is the atomic number, $R$ is the Rydberg constant (approximately 13.6 eV for hydrogen), and $n$ is the principal quantum number.
2. For the ground state, $n=1$. Substituting $n=1$ into the formula simplifies it to: $E=-Z^{2}R$.
3. For the Be3+ ion, the nuclear charge $Z$ is 4 (since beryllium has an atomic number of 4). Substituting $Z=4$ into the formula gives: $E=-4^{2}R$, which simplifies to $E=-16R$.
4. The ground-level energy of the hydrogen atom is $-R$. Comparing this to the ground-level energy of Be3+, we see that the energy of Be3+ is 16 times more negative than that of hydrogen, due to the $Z^2$ dependence.
5. Thus, the ground-level energy of Be3+ is $-16R$, which is 16 times the ground-level energy of the hydrogen atom in magnitude but more negative.

Key Concepts

Hydrogen-like Atoms

Hydrogen-like atoms are systems that consist of a nucleus with a positive charge and a single electron orbiting around it. The energy levels of these atoms can be described using the Bohr model, which states that the energy levels are quantized and depend on the atomic number. For hydrogen, the energy levels are given by the formula E_n = -13.6 eV/n², where n is the principal quantum number. This concept is crucial for understanding the energy levels of ions like Be3+.

Nuclear Charge

Nuclear charge refers to the total charge of the nucleus, which is determined by the number of protons it contains. In the case of Be3+, the nuclear charge is +4 due to its four protons. This increased nuclear charge affects the energy levels of the electrons, leading to a more negative energy state compared to hydrogen, which has a nuclear charge of +1. Understanding nuclear charge is essential for calculating the energy levels of multi-electron ions.

Energy Level Comparison

Comparing energy levels between different atoms or ions involves analyzing how the energy states change with varying nuclear charges and electron configurations. For hydrogen, the ground state energy is -13.6 eV, while for Be3+, the energy can be calculated using the modified formula E_n = -Z² * 13.6 eV/n², where Z is the nuclear charge. This comparison highlights how increased nuclear charge leads to lower (more negative) energy levels, indicating a stronger attraction between the nucleus and the electrons.