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,
 $${\displaystyle \frac{m{v}^2}{r}}={\displaystyle \frac{1}{4\pi \epsilon }}{\displaystyle \frac{z{e}^2}{r^2}}--(1)$$ , 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.
$$mvr=n{\displaystyle \frac{h}{2\pi }}--(2)$$ , 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 ($${\displaystyle \frac{1}{2}}m{v}^2$$) and potential energy ($${\displaystyle \frac{1}{4\pi \epsilon }}{\displaystyle \frac{z{e}^2}{r}}$$). Now calculating the total energy of an electron, we get as $$E_n={\displaystyle \frac{z^2}{n^2}}{E}_1$$, 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)
$$E_3={\displaystyle \frac{1}{3^2}}{E}_1$$=-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.