Question

: $\Delta E$value is maximum in :
A: ${E_2} - {E_1} = \Delta E$
B: ${E_3} - {E_2} = \Delta E$
C: ${E_4} - {E_3} = \Delta E$
D: ${E_5} - {E_4} = \Delta E$

Answer 1

Hint:In the question ${E_n}$ represents the energy of the electron in a particular orbit $n$. $\Delta E$ is the change of energy of the electron between two shells or orbits. We know that every orbit has different energy and the electron revolves around the nucleus with a certain kinetic energy similar to that of the planets revolving around the sun.

Complete step by step answer:
As we know that the total energy of an electron is the sum of its potential energy and it’s kinetic energy. The total energy of the electron is given by the formula :
$E = - \dfrac{{13.6 \times {Z^2}}}{{{n^2}}}$. Here $Z = $ atomic number and $n = $ orbit number of the shell.
The change in energy between two consecutive shells can be written as $\Delta E = \left| { - 13.6{Z^2}\left[ {\dfrac{1}{{{{(n + 1)}^2}}} - \dfrac{1}{{{n^2}}}} \right]} \right|$. The change in energy will be larger for a smaller value of $n$ possible because it will make the energy difference larger . If we put $n = 1$, the value of $\Delta E$ will become maximum, which is ${E_2} - {E_1} = \Delta E$.
So from the above explanation and calculation it is clear to us that the value of ${E_2} - {E_1} = \Delta E$ is maximum.

So the correct option of the given question is : A: ${E_2} - {E_1} = \Delta E$

Additional information:
 The energy can also be written by the formula $E = \dfrac{{hc}}{\lambda }$. Here $h = $ planck's constant, $c = speed$ and $\lambda = wavelength$ . We can observe that energy is inversely proportional to the wavelength . This relation can be very useful while solving numerical problems related to wavelength and energy. In the formula$E = - \dfrac{{13.6 \times {Z^2}}}{{{n^2}}}$ , the negative sign indicates that some energy is to be given to the electron if it has to overcome the attractive force of the nucleus to escape the atom.

Note:
Always remember that the energy of an electron in an orbit is given by the formula : $E = - \dfrac{{13.6 \times {Z^2}}}{{{n^2}}}$. The energy difference between two consecutive shells is $\Delta E = \left| { - 13.6{Z^2}\left[ {\dfrac{1}{{{{(n + 1)}^2}}} - \dfrac{1}{{{n^2}}}} \right]} \right|$ . Always make sure to avoid calculation errors.