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Energy Levels

A hydrogen at...

Question

A hydrogen atom and $L i^{+ +}$ ion are both in the second excited state. If $I_{H}$ and $I_{L i}$ are their respective electronic angular momenta, and $E_{H}$ and $E_{L}$ i are their respective energies, then

A

I_{H} > I_{L i}

and

| E_{H} | > | E_{L i} |

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B

I_{H} = I_{L i}

and

| E_{H} | < | E_{L i} |

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C

I_{H} = I_{L i}

and

| E_{H} | > | E_{L i} |

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D

I_{H} < I_{L i}

and

| E_{H} | < | E_{L i} |

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Solution

The correct option is B $I_{H} = I_{L i}$ and $| E_{H} | < | E_{L i} |$
Given $Z_{H} = 1, Z_{L i} = 3$ and $n_{H} = n_{L i} = 3$
1. Angular momentum is independent of Z and proportional to $n$ as $m v r = \frac{n h}{2 π}$ , therefore $I_{H} = I_{L i}$

2. $E = - 13.6 \frac{Z^{2}}{n^{2}}$
Energy is directly proportional to square of atomic number, hence $E_{H} < E_{L i}$

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