A hydrogen atom (Z = 1) and a singly-ionised helium atom (He , Z = 2) are both in…

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

Problem

A hydrogen atom (Z = 1) and a singly-ionised helium atom (He

^{+}

, Z = 2) are both in their fourth excited state, i.e. the principal quantum number is

n = 5

for the electron in each species. Answer the following, showing all algebraic steps clearly. (i) Derive an expression for the ratio of the orbital radii

\dfrac{r_{\text{He}^{+}}}{r_{\text{H}}}

for the two atoms in terms of the atomic numbers and the common principal quantum number. (ii) Hence calculate the numerical value of

\dfrac{r_{\text{He}^{+}}}{r_{\text{H}}}

for

n = 5

. (iii) Derive a similar expression for the ratio of the electron speeds

\dfrac{v_{\text{He}^{+}}}{v_{\text{H}}}

and determine its numerical value for

n = 5

. (iv) Obtain the ratio of the total energies

\dfrac{E_{\text{He}^{+}}}{E_{\text{H}}}

of the electrons in these states and compute its value.

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