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Atoms

The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb’s law between two opposite charge $+q 1,-q 2$ is modified to $=\frac{q_{1} q_{2}}{\left(4 \pi \epsilon_{0}\right)} \frac{1}{R_{0}^{2}}\left(\frac{R_{0}}{r}\right)^{\epsilon}, r \leq R_{0} \quad$ Calculate in such a case, the ground state energy of an $\mathrm{H}-$ atom, if $\varepsilon$

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Case 1: When $\varepsilon=0.1, \mathrm{R_O}=1 \mathrm{~A}$ $\mathrm R_1=8 \times 10^{-11}$ $\mathrm{M}=0.08 \mathrm{~nm}$ Velocity at ground level is given as $\mathrm v_1=1.44 \times 10^{6}...

The inverse square law in electrostatics is $|F|=\frac{e^{2}}{\left(4 \pi \epsilon_{0}\right)} r^{2}$ for the force between an electron and a proton. The $1 \mathrm{r}$ dependence of $|\mathbf{F}|$ can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass $\mathrm{mp}$ , force would be modified to $|F|=\frac{e^{2}}{\left(4 \pi \epsilon_{0}\right)} r^{2}\left[\frac{1}{r^{2}}+\frac{\lambda}{r}\right] \quad$ where $\lambda=\mathrm{m} \mathrm{cp} / \hbar$ and $h=\hbar / 2 \pi .$ Estimate the change in the ground state energy of an H-atom if $\mathrm{mp}$ were $10^{-6}$ times the mass of an electron.

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$m_{\mathrm{p}} c^{2}=10^{-6} \times$ electron mass $\times c^{2}$ $=10^{-6} \times 0.5 \mathrm{MeV}$ $\approx 10^{-6} \times 0.5 \times 1.6 \times 10^{-13}$ $\approx \mathrm{O} .8 \times 1...

In the Auger process, an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an $\mathrm{n}=4$ Auger electron emitted by Chromium by absorbing the energy from an $=2$ to $n=1$ transition.

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The energy in the nth state can be calculated as, $-\operatorname{Rch} Z^{2} / n^{2}=-13.6 z^{2} / n^{2} e V$ Where $R=$ Rydberg constant and $\mathrm{Z}=24$ Energy released is given as $\left(13.6...

Deuterium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in $1 \mathrm{H}$ and $2 \mathrm{H}$ . This is because the wavelength of transition depends to a certain extent on the nuclear mass. If nuclear motion is taken into account then the electrons and nucleus revolve around their common centre of mass. Such a system is equivalent to a single particle with a reduced mass $\mu$ , revolving around the nucleus at a distance equal to the electron-nucleus separation. Here $\mu=\mathrm{me} \mathrm{M} /(\mathrm{me}+\mathrm{M})$ where $\mathrm{M}$ is the nuclear mass and $\mathrm{me}$ is the electronic mass. Estimate the percentage difference in wavelength for the 1 st line of the Lyman series in $1 \mathrm{H}$ and $2 \mathrm{H}$ . (Mass of $1 \mathrm{H}$ nucleus is $1.6725 \times 10^{-27} \mathrm{~kg}$ , Mass of $2 \mathrm{H}$ nucleus is $3.3374 \times 10^{-27} \mathrm{~kg}$ , Mass of electron $=9.109 \times 10^{-31}$ kg.)

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The energy of an electron in the nth state is given by the expression, $E_ n=-\mu Z^{2} \mathrm{e}^{4} / 8 \varepsilon_ 0^{2} h^{2}\left(1 / n^{2}\right)$ For hydrogen atom we have, $\mu...

The first four spectral lines in the Lyman series of an H-atom are $\lambda=1218 \AA, 1028 \AA, 974.3 \AA$ and $951.4 \AA$ . If instead of Hydrogen, we consider Deuterium, calculate the shift in the wavelength of these lines.

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Let the reduced masses of electrons of hydrogen and deuteriur be $\mu _{H}$ and $\mu _{D}$ The fixed mass series for hydrogen and deuterium be $n_i$ and $n_f$ $R_h / R_d=\mu _H / \mu _D$ Reduced...

What is the minimum energy that must be given to an $\mathrm{H}$ atom in ground state so that it can emit an Hy line in Balmer series? If the angular momentum of the system is conserved, what would be the angular momentum of such Hy photon?

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Energy required for the transition from $n=1$ to $n=5$ can be calculated as, $E=E _5-E_ 1=\left(-13.6 / 5^{2}\right)-\left(-13.6 / 1^{2}\right)=13.06 \mathrm{eV}$ Angular momentum is given as change...

Show that the first few frequencies of light that are emitted when electrons fall to the $n$ th level from levels higher than $\mathrm{n}$ are approximate harmonics (i.e. in the ratio $1: 2: 3 \ldots$ ) when $\mathrm{n}>>1$ .

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The difference between the two atoms is used to indicate the frequency of any line in the hydrogen spectrum series. $\mathrm{f_{cm}}=\mathrm{cRZ}^{2}\left[1 /(\mathrm{n}+\mathrm{p})^{2}-1 /...

Using the Bohr model, calculate the electric current created by the electron when the H-atom is in the ground state.

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Let the velocity of the electron be $v$ Number of revolutions per unit time is given as $\mathrm{f}=2 \mathrm{ma}_{0} / \mathrm{V}$ The electric current is given as $I=Q$ When change $Q$ flows in...

Assume that there is no repulsive force between the electrons in an atom but the force between positive and negative charges is given by Coulomb’s law as usual. Under such circumstances, calculate the ground state energy of a He-atom.

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For He nucleus we have, $Z=2$ and in ground state, $n=1$ As a result, $E n=-13.6 Z^{2} / n^{2} e V=-54.4 \mathrm{eV}$

Positronium is just like an H-atom with the proton replaced by the positively charged anti-particle of the electron (called the positron which is as massive as the electron). What would be the ground state energy of positronium?

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The positronium has the lowest energy of -6.8 electron volts. The greatest energy level of positronium is -1.7 electron volt, which is the next highest energy level after n = 1.

Consider two different hydrogen atoms. The electron in each atom is in an excited state. Is it possible for the electrons to have different energies but the same orbital angular momentum according to the Bohr model?

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Because the value of n in two hydrogen atoms differs, their angular momentum differs as well. The angular momentum, according to Bohr's model, is given as L = nh/2π

The mass of H atom is less than sum of the masses of a proton and electron. Why is this?

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Protons and neutrons (and other particles) are held together by something called the strong nuclear force (strong interaction). This force diminishes the system's overall energy, which is dissipated...

Let $E_{n}=\frac{-1}{8 \epsilon_{0}^{2}} \frac{m e^{4}}{n^{2} h^{2}} \quad$ be the energy of the nth level of H-atom. If all the H-atoms are in the ground state and radiation of frequency $(E_2-E_1)/h$ falls on it
(a) it will not be absorbed at all
(b) some of the atoms will move to the first excited state
(c) all atoms will be excited to the $n=2$ state
(d) no atoms will make a transition to the $\mathrm{n}=3$ state

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The correct options are: (b) some of the atoms will move to the first excited state (d) no atoms will make a transition to the $n=3$ state

The Balmer series for the H-atom can be observed
(a) if we measure the frequencies of light emitted when an excited atom falls to the ground state
(b) if we measure the frequencies of light emitted due to transitions between excited states and the first excited state
(c) in any transition in a $\mathrm{H}$ -atom
(d) as a sequence of frequencies with the higher frequencies getting closely packed

Atoms, CBSE, Class 12, NCERT Exemplar, Physics

The correct options are: (b) if we measure the frequencies of light emitted due to transitions between excited states and the first excited state (d) as a sequence of frequencies with the higher...