For a hydrogen-like atom of atomic number

CHEMISTRY · Subjective · CBSE · Chemistry · Structure of Atom · Structure of Atom

Problem

Z

, the electron moves in a stationary circular orbit of radius

r_n

corresponding to the principal quantum number

n

. (i) Starting with the condition that the electrostatic force of attraction between the nucleus and the electron provides the required centripetal force, derive an expression for the radius

r_n

in terms of

n

Z

and the fundamental constants. (ii) Hence obtain a symbolic relation for the speed

v_n

of the electron in the same orbit. (iii) Show that the total mechanical energy

E_n

of the electron varies as

-Z^2/n^2

and state the significance of the negative sign. (iv) Using your results, calculate the numerical value of

r_3

v_3

and

E_3

for the

{\rm He}^+

ion. Express the radius in ångströms, the speed as a fraction of the speed of light

c=3.00\times10^8\,\text{m\,s}^{-1}

, and the energy in electron-volt. (v) Comment on how the size and binding energy of the orbit change when

Z

is doubled while

n

remains the same.

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