A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneo

Question

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $ν_{1}$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $ν_{2}$ . The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that the positron has the same mass as that of the electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is _____. [2025]

Smart Achievers · Accepted Answer

(11.80)

$E_{1} = h ν_{1} = 13.6 + 10 = 23.6 eV$

Energy of positronium in ground state

$= - 13.6 \frac{μ}{m} {(\frac{Z}{n})}^{2} eV = - 13.6 \times \frac{1}{2} eV = - 6.8 eV$

$E_{2} = h ν_{2} = (10 - 5) + 6.8 = 11.8 eV$

$∴$ $Difference in energy = 23.6 - 11.8 = 11.8 eV$

Solution

Ans.
(11.80)

$E_{1} = h ν_{1} = 13.6 + 10 = 23.6 eV$

Energy of positronium in ground state

$= - 13.6 \frac{μ}{m} {(\frac{Z}{n})}^{2} eV = - 13.6 \times \frac{1}{2} eV = - 6.8 eV$

$E_{2} = h ν_{2} = (10 - 5) + 6.8 = 11.8 eV$

$∴$ $Difference in energy = 23.6 - 11.8 = 11.8 eV$

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Solution

Ans. (11.80) E1=hν1=13.6+10=23.6 eV Energy of positronium in ground state =-13.6μm(Zn)2eV=-13.6×12 eV=-6.8 eV E2=hν2=(10-5)+6.8=11.8 eV ∴ Difference in energy=23.6-11.8=11.8 eV

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Ans.
(11.80)

$E_{1} = h ν_{1} = 13.6 + 10 = 23.6 eV$

Energy of positronium in ground state

$= - 13.6 \frac{μ}{m} {(\frac{Z}{n})}^{2} eV = - 13.6 \times \frac{1}{2} eV = - 6.8 eV$

$E_{2} = h ν_{2} = (10 - 5) + 6.8 = 11.8 eV$

$∴$ $Difference in energy = 23.6 - 11.8 = 11.8 eV$