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The energy level of a hydrogen atom in its ground state is $-13.6\ \mathrm{eV}$. How is this value calculated and how can we calculate the same for different orbits?

Adarsh
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From the Bohr theory: $$\large E=-\frac{Z^2e^4m}{8\epsilon_0^2n^2h^2}=-13.6\times\frac{Z^2}{n^2}\ \mathrm{eV}$$ where:

$$\begin{array}{c|c|} & Z&\text{atomic number}&-\\ & e&\text{charge of electron}&1.60 \times 10^{-19}\ \mathrm{C}\\ & m&\text{mass of electron}&9.10 \times 10^{-31}\ \mathrm{kg}\\ & \epsilon_0&\text{electric permittivity}& 8.85 \times 10^{−12}\ \mathrm{F/m}\\ & n&\text{principal quantum number}&-\\ & h&\text{Planck's constant}&6.62 \times 10^{-34}\ \mathrm{m^2\ kg/s}\\ \end{array}$$

You can put values of $Z$ and $n$ for any shell of a hydrogenic species like $\ce{H}$, $\ce{He+}$, $\ce{Li^2+}$, $\ce{Be^3+}$, $\ce{B^4+}$…

RE60K
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    The equation gives the answer in Joules. To convert to electron volts use $\pu{ 1 eV = 1.602~10^{-19} J}$. To show that the equation is in Joules you need to know that the Farad has units of Coulomb/Volt $\pu{F=C/V}$ and $\pu{V=J/C}$ and $\pu{J= kgm^2s^{-2}}$. The $13.6 $ is the Rydberg constant in eV. – porphyrin Oct 12 '16 at 15:04