Question

Using Rutherford model of the atom, derive the expression for the total energy of the electron in hydrogen atom. What is the significance of total negative energy possessed by the electron?

OR
Using Bohr’s postulates of the atomic model derive the expression for radius of nth electron orbit. Hence obtain the expression for Bohr’s radius.

Solution

According to Rutherford’s model, we have
$mv squared over straight r space equals space fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator z e squared over denominator r squared end fraction rightwards double arrow space mv squared space equals space fraction numerator 1 over denominator 4 pi epsilon subscript o end fraction fraction numerator z e squared over denominator r end fraction Therefore comma space Total space energy space equals space straight P. straight E space plus space straight K. straight E straight T. straight E. space equals space minus fraction numerator 1 over denominator 4 πε subscript straight o end fraction ze squared over straight r space plus space 1 half mv squared space space space space space space space equals space minus 1 half. fraction numerator 1 over denominator 4 πε subscript straight o end fraction space ze squared over straight r space space space space space space space equals space minus space. fraction numerator 1 over denominator 8 πε subscript straight o end fraction ze squared over straight r$
Energy is negative implies that the electron –nucleus is a bound or attractive system.

OR

According to the Bohr’s atomic model, electrons revolve around the nucleus only in those orbits for which the angular momentum is an integral multiple of $fraction numerator straight h over denominator 2 straight pi end fraction$ .
So, as per Bohr’s postulate, we have
$mvr space equals space fraction numerator nh over denominator 2 straight pi end fraction space Therefore comma space mv squared over straight r space equals space fraction numerator 1 over denominator 4 πε subscript straight o end fraction ze squared over straight r squared mvr space equals fraction numerator nh over denominator 2 straight pi end fraction therefore space space straight m squared straight v squared straight r squared space equals space fraction numerator straight n squared straight h squared over denominator 4 straight pi squared end fraction This space implies comma straight r space equals space fraction numerator straight epsilon subscript straight o straight n squared straight h squared over denominator πze squared straight m end fraction$
This is the required expression for Bohr's radius.

For Daily Free Study Material Join wiredfaculty

Wave Optics

Some More Questions From Wave Optics Chapter

In the ground state of electrons are in stable equilibrium, while in electrons always experience a net force. (Thomson's model/Rutherford's model).

A classical atom based on __________ is doomed to collapse. (Thomson's model/Rutherford's model)

An atom has a nearly continuous mass distribution in a ___ but has a highly non-uniform mass distribution in (Thomson's model/Rutherford's model)

The positively charged part of the atom possesses most of the mass in ___________ (Rutherford's model/both the models).

Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction