Chemistry

Chemistry

Problem

Two single-electron species are considered: hydrogen (Z=1Z=1) and singly-ionised helium (He+\text{He}^+, Z=2Z=2). a. (a) Derive an algebraic expression for the ratio vn (He+)vn (H)\dfrac{v_{n\,(\text{He}^+)}}{v_{n\,(\text{H})}} of the electron speeds in their respective nthn^{\text{th}} orbits. b. (b) Hence evaluate the ratio for n=2n=2. c. (c) State which electron moves faster in that orbit and give a physical explanation based on your result.
Intuitive explanation

Both atoms have just one electron, so their energy levels follow the same hydrogen-like formula—but the key difference is nuclear charge. The ZZ value in the denominator of the energy equation means He⁺ has a much stronger pull on its electron, making its orbitals smaller and its energy levels much more negative (lower). When comparing ionization energies or orbital sizes, remember that doubling the nuclear charge doesn't just double the effect; it scales quadratically for energy and inversely for radius.

Intuitive explanation

Both atoms have just one electron, so their energy levels follow the same hydrogen-like formula—but the key difference is nuclear charge. The ZZ value in the denominator of the energy equation means He⁺ has a much stronger pull on its electron, making its orbitals smaller and its energy levels much more negative (lower). When comparing ionization energies or orbital sizes, remember that doubling the nuclear charge doesn't just double the effect; it scales quadratically for energy and inversely for radius.

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