JEE Main & Advanced Physics Atomic Physics Energy

(1) Potential energy : An electron possesses some potential energy because it is found in the field of nucleus potential energy of electron in \[{{n}^{th}}\] orbit of radius \[{{r}_{n}}\] is given by \[U=k.\frac{(Ze)\,(-e)}{{{r}_{n}}}=-\frac{kZ{{e}^{2}}}{{{r}_{n}}}\]

(2) Kinetic energy : Electron posses kinetic energy because of it's motion. Closer orbits have greater kinetic energy than outer ones.

As we know \[\frac{m{{v}^{2}}}{{{r}_{n}}}=\frac{k.\,(Ze)\,(e)}{r_{n}^{2}}\]

\[\Rightarrow \] Kinetic energy \[K=\frac{kZ{{e}^{2}}}{2{{r}_{n}}}=\frac{|U|}{2}\]

(3) Total energy : Total energy (E) is the sum of potential energy and kinetic energy i.e. \[E=K+U\]

\[\Rightarrow \]               \[E=-\frac{kZ{{e}^{2}}}{2{{r}_{n}}}\] also \[{{r}_{n}}=\frac{{{n}^{2}}{{h}^{2}}{{\varepsilon }_{0}}}{\pi mz{{e}^{2}}}\].

Hence \[E=-\,\left( \frac{m{{e}^{4}}}{8\varepsilon _{0}^{2}{{h}^{2}}} \right).\frac{{{z}^{2}}}{{{n}^{2}}}\]\[=-\,\left( \frac{m{{e}^{4}}}{8\varepsilon _{0}^{2}c{{h}^{3}}} \right)\,ch\frac{{{z}^{2}}}{{{n}^{2}}}\]

\[=-R\,ch\frac{{{Z}^{2}}}{{{n}^{2}}}=-13.6\frac{{{Z}^{2}}}{{{n}^{2}}}eV\]

where \[R=\frac{m{{e}^{4}}}{8\varepsilon _{0}^{2}c{{h}^{3}}}\]= Rydberg's constant \[=1.09\times {{10}^{7}}\] per m.

(4) Ionisation energy and potential : The energy required to ionise an atom is called ionisation energy. It is the energy required to make the electron jump from the present orbit to the infinite orbit.

Hence \[{{E}_{ionisation}}={{E}_{\infty }}-{{E}_{n}}=0-\left( -13.6\frac{{{Z}^{2}}}{{{n}^{2}}} \right)\]\[=+\frac{13.6{{Z}^{2}}}{{{n}^{2}}}eV\]

For \[{{H}_{2}}-\]atom in the ground state

\[{{E}_{ionisation}}=\frac{+13.6{{(1)}^{2}}}{{{n}^{2}}}=13.6\,eV\]

The potential through which an electron need to be accelerated so that it acquires energy equal to the ionisation energy is called ionisation potential. \[{{V}_{ionisation}}=\frac{{{E}_{ionisation}}}{e}\]

(5) Excitation energy and potential : When energy is given to an electron from external source, it jumps to higher energy level. This phenomenon is called excitation.

The minimum energy required to excite an atom is called excitation energy of the particular excited state and corresponding potential is called exciting potential. 

\[{{E}_{Excitation}}={{E}_{Final}}-{{E}_{Initial}}\] and \[{{V}_{Excitation}}=\frac{{{E}_{excitation}}}{e}\]

(6) Binding energy (B.E.) : Binding energy of a system is defined as the energy released when it's constituents are brought from infinity to form the system. It may also be defined as the energy needed to separate it's constituents to large distances. If an electron and a proton are initially at rest and brought from large distances to form a hydrogen atom, 13.6 eV energy will be released. The binding energy of a hydrogen atom is therefore 13.6 eV.

(7) Energy level diagram : The diagrammatic description of the energy of the electron in different orbits around the nucleus is called energy level diagram. Energy level diagram of hydrogen/hydrogen like atom

	\[n=\infty \]	Infinite	Infinite	\[0\,eV\]
	\[n=4\]	Fourth	Third	\[-0.85\,eV\]
	\[n=3\]	Third	Second	\[-1.51\,eV\]
	\[n=2\]	Second	First	\[-3.4\,eV\]
	\[n=1\]	First	Ground	\[-13.6\,eV\]
	Principle quantum number	Orbit	Excited state	Energy for \[{{H}_{2}}-\]atom