Question

Find the value of Energy of electron in eV in the third Bohr orbit of Hydrogen atom.
 (Rydberg’s Constant(R)= \[1.097\times {{10}^{7}}{{m}^{-1}}\],Planck’s Constant (h)= \[6.63\times {{10}^{-34}}J\cdot s\], Velocity of Light in air (c)= \[3\times {{10}^{8}}m/s\])

Answer 1

Hint:We will first try to know about Bohr’s theory. First, electrons are moving in a circle around the nucleus. His second postulate was angular momentum is quantized.

Complete step by step answer:
In first postulate, when an electron has velocity (v) while circling around nucleus with radius r, it will follow the equation,
 \[\dfrac{m{{v}^{2}}}{r}=\dfrac{1}{4\pi \varepsilon }\dfrac{z{{e}^{2}}}{{{r}^{2}}}--(1)\] , R.H.S is Coulomb force between the electron and nucleus with proton number Z. This is true for any orbit.
According to the second postulate, angular momentum will be quantized.
\[mvr=n\dfrac{h}{2\pi }--(2)\] , where r is radius of \[{{n}^{th}}\] orbital, v is velocity of the electron at that orbital, and h is Planck’s constant.
From (1) and (2), we can calculate the velocity of an electron and radius of an orbital for any orbital. Energy of an electron is sum of kinetic energy (\[\dfrac{1}{2}m{{v}^{2}}\]) and potential energy (\[\dfrac{1}{4\pi \varepsilon }\dfrac{z{{e}^{2}}}{r}\]). Now calculating the total energy of an electron, we get as \[{{E}_{n}}=\dfrac{{{z}^{2}}}{{{n}^{2}}}{{E}_{1}}\], where \[{{E}_{n}}\] is energy of an electron at \[{{n}^{th}}\]orbital, \[{{E}_{1}}\] is energy of an electron at first orbital and Z is number of proton of the atom.
Now, \[{{E}_{1}}\] can also be represented as \[{{E}_{1}}\]=Rhc, i.e multiple of Rydberg’s constant, Planck’s constant and speed of light. The value of \[{{E}_{1}}\]in eV is -13.6 eV. The negative sign here represented that electrons are bound and under attractive force.
So, value of energy of an electron in hydrogen atom at third orbital, (n=3, z=1)
\[{{E}_{3}}=\dfrac{1}{{{3}^{2}}}{{E}_{1}}\]=-1.51 eV.
This is the answer to our question.

Note:Sometimes students are confused about how to know in which state is in the atom so always remember that the value of n determines the state of an electron. Hence,n=1 corresponds to the ground state of the atom and n=2 corresponds to the first excited state of an atom and so on.