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A single electron atom has nuclear charge $+Ze$ where $Z$ is atomic number and e is electronic charge requires $\pu{16.52ev}$ to excite the electron from the second bohr orbit to third bohr orbit. Find the atomic number of the element?

A) 1 B) 2 C) 3 D) 4

I wish to ask if we can use energy formula $E= -(13.6)\frac{Z^2}{n^2}$ for this problem.

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Using the formula you stated, Energy difference between the orbits can be evaluated as $$∆E=-13.6\ \mathrm{eV}\left(\frac{z^2}{3^2}\right)-\left(-13.6\ \mathrm{eV}\left(\frac{z^2}{2^2}\right)\right)$$ Equate this expression to the energy you have given, i.e. $16.52\ \mathrm{eV}$

Solve this for $z$ which comes out to be $2.96$ that may be approximated to $3$.

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The charge must have been $+(Z-1)e$, since the species must be monoelectronic to apply Bohr's model.

Coming to your question, the formula you have mentioned in your question is used for finding the energy of an electron in a particular shell of a monoelectronic species. What you are supposed to do is use it to subtract the energies of the given shells and equate it to the energy (16.52 eV) given in the question. This should give the answer as $Z = 3$.

PS: You could have found the answer by little self study.

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