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Well the reason the equation En= -hR/n^2 only works for the hydrogen atom is because for hydrogen z=1. For other one-electron ions (ex: He^+ or C^5+) with an atomic number z=# we use the equation En= (-z^2)HR/n^2. This is because the greater the value of the nuclear charge, the more tightly bound the electron is to the nucleus.
The Rydberg equation does not take into account electron-electron repulsion, which is why it's okay to use for Hydrogen and H-spectrum (only one electron to worry about). It will not work for atoms with multiple electrons because of electron-electron repulsion.
The empirical formula does not work for atoms other than hydrogen and multi-electron atoms. For multi-electron atoms, we would use the other formula (the one with Zeff) because of electron shielding and repulsion caused by multiple electrons.
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