According to the Bohr model, the speed of the electron in...

According to the Bohr model, the speed of the electron in the ground state of singly ionized helium $\left(\mathrm{He}^{+}\right.$, with $Z=2$ ) is $4.4 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Use this information to find the speed of an electron in the first excited state of triply ionized beryllium $\left(\mathrm{Be}^{3+}\right.$ with $Z=4$ ).

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Key Concepts

Bohr Model

The Bohr model describes the atom with electrons moving in fixed, circular orbits around the nucleus, where each orbit corresponds to a specific energy level. This model introduces the concept of quantized orbits to explain atomic spectra and the stability of atoms, despite its limitations compared to modern quantum mechanics.

Quantized Angular Momentum

In the Bohr model, an electron’s angular momentum is quantized, meaning it can only take on discrete values given by n multiplied by a fundamental unit of angular momentum. This quantization condition is essential because it restricts the electron to specific, allowed orbits, and it directly influences the energy and the orbital speed of the electron.

Dependence on Nuclear Charge and Principal Quantum Number

According to the Bohr model, the speed of an electron in its orbit is directly proportional to the nuclear charge (Z) and inversely proportional to the principal quantum number (n), following the relation v ? Z/n. This concept is crucial for understanding how the electron velocity changes in different ions and for electrons in varying energy levels, making it possible to compare systems like He+ and Be3+ by adjusting for differences in Z and n.

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