A singly ionized helium atom (He ) has a nucleus of charge orbited by a single…

CHEMISTRY · Multi-part · CBSE · Chemistry · Structure of Atom · Structure of Atom

Problem

A singly ionized helium atom (He

^+

) has a nucleus of charge

+2e

orbited by a single electron that is assumed to move in a circular stationary state characterized by a principal quantum number

n

. The electron’s motion is treated non-relativistically and the nucleus is taken to be infinitely massive compared with the electron. The following fundamental constants may be useful: • elementary charge

e = 1.60\times10^{-19}\,\text{C}

• permittivity of free space

\varepsilon_0 = 8.85\times10^{-12}\,\text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}

• Planck’s constant

h = 6.63\times10^{-34}\,\text{J\,s}

• electron mass

m_e = 9.11\times10^{-31}\,\text{kg}

(a) Derive an expression for the allowed orbital radius

r_n

of the electron in terms of

n

, fundamental constants, and the nuclear charge number

Z

. (b) Using your result from part (a), obtain an expression for the total mechanical energy

E_n

of the electron and show explicitly how

E_n

depends on both

n

and

Z

. (c) Calculate the numerical value of

E_{n=2}

for the He

^+

ion in electron-volts (eV). (d) Discuss the physical significance of (i) the negative sign of

E_n

and (ii) its proportionality to

Z^2

. Relate each feature to the binding of the electron to the nucleus.

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