In the course of developing his model, Bohr arrived at the...

In the course of developing his model, Bohr arrived at the following formula for the radius of the electron's orbit: $r_{n}=$ $n^{2} h^{2} \varepsilon_{0} / \pi m_{0} e^{2},$ where $m_{c}$ is the electron's mass, $e$ is its charge, and $\varepsilon_{0}$ is a constant related to charge attraction in a vacuum. Given that $m_{\mathrm{z}}=9.109 \times 10^{-31} \mathrm{~kg}, e=1.602 \times 10^{-19} \mathrm{C},$ and $\varepsilon_{0}=8.854 \times 10^{-12} \mathrm{C}^{2} / \mathrm{J} \cdot \mathrm{m}$ calculate the following: (a) The radius of the first $(n=1)$ orbit in the $\mathrm{H}$ atom (b) The radius of the tenth $(n=10)$ orbit in the $\mathrm{H}$ atom

In the course of developing his model, Bohr arrived at the following formula for the radius of the electron's orbit: $r_{n}=$ $n^{2} h^{2} \varepsilon_{0} / \pi m_{0} e^{2},$ where $m_{c}$ is the electron's mass, $e$ is its charge, and $\varepsilon_{0}$ is a constant related to charge attraction in a vacuum. Given that $m_{\mathrm{z}}=9.109 \times 10^{-31} \mathrm{~kg}, e=1.602 \times 10^{-19} \mathrm{C},$ and $\varepsilon_{0}=8.854 \times 10^{-12} \mathrm{C}^{2} / \mathrm{J} \cdot \mathrm{m}$
calculate the following:
(a) The radius of the first $(n=1)$ orbit in the $\mathrm{H}$ atom
(b) The radius of the tenth $(n=10)$ orbit in the $\mathrm{H}$ atom

Key Concepts

Bohr Atomic Model

The Bohr atomic model posits that electrons orbit the nucleus in discrete, quantized orbits rather than following arbitrary paths. This model introduced the idea that electrons can only exist in certain allowed orbits, with quantized energy levels corresponding to different principal quantum numbers. The model successfully explained the spectral lines of hydrogen by associating the emission or absorption of light with transitions between these quantized energy states.

Quantization of Angular Momentum

A core element of the Bohr model is the quantization of angular momentum. According to this principle, the angular momentum of an electron in orbit can only take on discrete values, typically expressed as integral multiples of Planck's constant divided by 2?. This quantization leads directly to discrete orbit radii and energy levels, with the radius of each orbit being proportional to the square of the principal quantum number (n²).

Orbit Radius Formula

The formula for the radius of an electron's orbit in the Bohr model, given by r_n = (n² h² ??) / (? m? e²), is derived by equating the centripetal force necessary to maintain an electron in a circular orbit with the Coulombic force of attraction between the negatively charged electron and the positively charged nucleus. This equation highlights the dependence of the orbit's radius on both fundamental physical constants and the square of the quantum number, illustrating the quantized structure of the atom.

Transcript

Please give Ace some feedback