Given below is a conjugated system of 11 carbon atoms

Assume the average C-C bond length to be 1.5Å and treat the system as a 1-dimensional box. The frequency of radiation required to cause a transition from the ground state of the system to the first excited state (take \(\frac{h^2}{8m}\) = k) is

Concept:-

A 1-dimensional (1D) box, also known as a particle in a one-dimensional box or a particle in a box, is a simple quantum mechanical model that describes the behavior of a particle confined to move within a one-dimensional region, typically between two walls or barriers. This model is often used in introductory quantum mechanics courses to illustrate the quantization of energy levels and wavefunctions in a confined system.

Given:

\({h^2 \over 8m} = k\)

The average bond length of C-C = 1.5 Å

Explanation:-

Bond length for the whole conjugated system having 10 bonds are

= 1.5 x10 

=15 Å

We know that, 

\(∆E=hν={(∆n)^2}{h^2 \over 8ma^2}\)

\(∆E={(∆n)^2}{k \over 15^2}\)

In the given conjugated system 10π electrons are arranged in the following manner -

The 10 π electrons make the ground state and the first transition state occurs at minimum transition level at n= 6.

\(∆E={(6^2 -5^2)}{k \over 225}\)

\(∆E={11k \over 225}\)

• Now, the frequency required to cause a transition from the ground state of the system to the first excited state is,

\(∆E=hν={11k \over 225}\)

\(ν={11k \over 225 h}\)

Conclusion:-

Therefore, the frequency of radiation required to cause a transition from the ground state to the first excited state of the system is \(\frac{11\,k}{225\,h}\).