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. (a) 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. Identify the formula for the energy levels of a hydrogen-like atom, which is given by E_n = -\frac{Z^2 \times 13.6 \text{ eV}}{n^2}, where Z is the atomic number (nuclear charge) and n is the principal quantum number.
2. For the Be3+ ion, substitute Z = 4 (since beryllium has a nuclear charge of +4) and n = 1 (for the ground state, which is the lowest energy level) into the formula.
3. Calculate the energy for the ground state of Be3+ using the substituted values.
4. For comparison, calculate the ground state energy of a hydrogen atom (H) by substituting Z = 1 and n = 1 into the same formula.
5. Compare the calculated energy of Be3+ with that of the hydrogen atom to understand how the increased nuclear charge affects the energy levels.

Key Concepts

Hydrogen-like Atoms

Hydrogen-like atoms are systems with a single electron orbiting a nucleus with a positive charge. The energy levels of these atoms can be described using the Bohr model, where the energy levels depend on the atomic number and the principal quantum 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, resulting in a more negative energy value compared to hydrogen, which has a nuclear charge of +1. Understanding nuclear charge is essential for comparing the energy levels of different ions.

Energy Level Comparison

The energy levels of an atom or ion can be compared by examining the formula for energy levels in hydrogen-like systems. For Be3+, the energy levels can be calculated using E_n = -Z² * 13.6 eV/n², where Z is the nuclear charge. This means that the ground state energy of Be3+ will be four times more negative than that of hydrogen, indicating that it is more tightly bound. This comparison is key to understanding the differences in energy states between hydrogen and triply ionized beryllium.