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Question: The first excitation energy of an electron of a hydrogen like atom is \(40.8eV\). Find the energy ne...

The first excitation energy of an electron of a hydrogen like atom is 40.8eV40.8eV. Find the energy needed to remove the electron to form the ion.

Explanation

Solution

In order to solve this question, first we need to find the atomic numberZZby equating the given 1st1stexcitation energy with the help of the formula mentioned below. Then put this value in the formula to get the answer.

Formula used:
En=−13.6Z2n2eV{E_n} = \dfrac{{ - 13.6{Z^2}}}{{{n^2}}}eV

\\\ \\\ $$where $Z = $atomic number of the atom, $n = $number of levels, Excitation energy, $\Delta E = {E_n} - {E_{n - 1}}$ **Complete step-by-step solution:** For a hydrogen like atom the the energy is given by $${E_n} = \dfrac{{ - 13.6{Z^2}}}{{{n^2}}}eV - - (i)$$ For energy level $$n = 1$$the energy is equal to: $${E_1} = \dfrac{{ - 13.6{Z^2}}}{{{{(1)}^2}}}eV - - (ii)$$ For energy level $n = 2$the energy is equal to: ${E_2} = \dfrac{{ - 13.6{Z^2}}}{{{{\left( 2 \right)}^2}}} - - (iii)$ Subtracting equation$(i)$from $(ii)$we get: $ \Delta E = {E_2} - {E_1} \\\ \Delta E = - 13.6{Z^2}\left( {\dfrac{1}{{{{\left( 2 \right)}^2}}} - \dfrac{1}{{{{\left( 1 \right)}^2}}}} \right) - - (iv) \\\ $ It is provided that $\Delta E = 40.8eV$. Now putting it in equation$(iv)$ we get: $40.8 = - 13.6{Z^2}\left( {\dfrac{1}{{{{\left( 2 \right)}^2}}} - \dfrac{1}{{{{\left( 1 \right)}^2}}}} \right)$ $\therefore Z = 2$ To find the energy needed to remove the electron from ion we have: $Z = 2$,$n = 2$ Then we put it in equation$(i)$we get: $ {E_2} = \dfrac{{ - 13.6{{\left( 2 \right)}^2}}}{{{{\left( 2 \right)}^2}}}eV \\\ {E_2} = - 54.4eV \\\ $ Energy needed to remove the electron to form the ion will be: $ \- \left( { - 54.4} \right)eV \\\ 54.4eV \\\ $ The energy required to remove the electron to form the ion will be: $54.4eV$ **Note:** Some points to keep in mind while solving these types of problems: Remember all the formulae of energies so that you should apply those in the right place. You have to keep a negative sign in the final answer so that we get the energy needed to remove the electron to form the ion.