Atomic Model：練習問題

An atom of uranium is represented by the nuclide symbol \(^{238}_{92}\text{U}\). How many protons and neutrons are contained within the nucleus of this atom?

An electron in a hydrogen atom transitions from the \(n=3\) energy level to the \(n=2\) energy level. What is the wavelength of the photon emitted during this transition?

(Given: energy of an electron in hydrogen atom \(E_n = -\frac{13.6}{n^2}\) eV, Planck's constant \(h = 6.63 \times 10^{-34}\) J s, speed of light \(c = 3.00 \times 10^8\) m s\(^{-1}\), \(1\) eV \(= 1.60 \times 10^{-19}\) J)

Explain why Rutherford's planetary model of the atom was considered unstable according to classical electromagnetic theory.

An electron is accelerated from rest through a potential difference \( V \). If its resultant de Broglie wavelength is \( \lambda \), derive an expression for \( V \) in terms of \( \lambda \), electron mass \( m_e \), Planck's constant \( h \), and elementary charge \( e \).

An electron in a hydrogen atom is in the state with principal quantum number \( n = 4 \). If it transitions to a lower energy state such that its angular momentum decreases by \( \frac{h}{\pi} \), determine the final principal quantum number and the energy of the emitted photon in eV.

Rutherford's model of the atom was developed following the results of the alpha-particle scattering experiment.
(a) Briefly describe the distribution of charge and mass in an atom according to Rutherford's construction.
(b) Explain why the observation that a very small fraction of alpha particles were deflected by more than \(90^{\circ}\) was essential in proposing the existence of a nucleus.
(c) State one major limitation of Rutherford's model in accounting for the stability of an atom according to classical electromagnetic theory.

In a Transmission Electron Microscope (TEM), electrons are accelerated from rest through a high potential difference \(V\) to achieve a very short wavelength.
(a) Derive an expression for the de Broglie wavelength (\(\lambda\)) of the electrons in terms of \(V\), the electron mass \(m_e\), and the elementary charge \(e\).
(b) Calculate the wavelength of an electron accelerated through a potential difference of \(50\) kV.
(c) Explain, using the Rayleigh criterion (\(\theta \approx 1.22\frac{\lambda}{d}\)), why the TEM can resolve much smaller details than an optical microscope using visible light.
(Given: \(m_e = 9.11 \times 10^{-31}\) kg, \(e = 1.60 \times 10^{-19}\) C, \(h = 6.63 \times 10^{-34}\) J s)