Atoms Test

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In an ordinary atom, as a first approximation, the motion of the nucleus can be ignored. In a positronium atom, a positron replaces the proton of hydrogen atom. The electron and positron masses are equal and therefore, the motion of the positron cannot be ignored. One must consider the motion of both electron and positron about their center of mass. A detailed analysis shows that formulae of Bohr's model apply to positronium atom provided that we replace $$m_e$$ by what is known as the reduced mass of the electron. For positronium, the reduced mass is $$m_e/2$$.

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Directions For Questions

The electrons in a $$H$$-atom kept at rest, jumps from the $$mth$$ shell to the $$nth$$ shell $$(m>n)$$. Suppose instead of emitting electromagnetic wave, the energy released is converted into kinetic energy of the atom. Assume Bohr's model and conservation of angular momentum are valid. Answer the following question

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An electron orbits a stationary nucleus of charge $$+ze$$, where $$z$$ is a constant and $$e$$ is the magnitude of electronic charge. It requires $$47.2\space eV$$ to excite the electron from the second Bohr orbit to the third Bohr Orbit.

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Hydrogen is the simplest atom of nature. There is one proton in its nucleus and an electron moves around the nucleus in a circular orbit. According to Niels Bohr, this electron moves in a stationary orbit. When this electron is in the stationary orbit, it emits no electromagnetic radiation. The angular momentum of the electron is quantized, i.e., $$mvr = (nh/2\pi)$$, where $$m$$ = mass of the electron, $$v$$ = velocity of the electron in the orbit, $$r$$ = radius of the orbit, and $$n = 1,2,3,...$$. When transition takes place from the $$Kth$$ orbit to the $$Jth$$ orbit, energy photon is emitted. If the wavelength of the emitted photon is $$\lambda$$, we find that $$\displaystyle\frac{1}{\lambda} = R\left[\displaystyle\frac{1}{J^2} - \displaystyle\frac{1}{K^2}\right]$$, where $$R$$ is the Rydberg's constant. On a different planet, the hydrogen atom's structure was somewhat different from ours. The angular momentum of electron was $$P = 2n(h/2\pi)$$, i.e., an even multiple of $$(h/2\pi)$$.

Answer the following questions regarding the other planet based on the above passage.

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A sample of hydrogen gas in its ground state is irradiated with photons of $$10.2\space eV$$ energies. The radiation from the above sample is used to irradiate two other samples of excited ionized $$He^+$$ and excited ionized $$Li^{2+}$$, respectively. Both the ionized samples absorb the incident radiation.

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