Physical Chemistry: Extracting information from spectroscopic data

[quantumech]

I need help with extracting information from spectroscopic data. Here is the type of question I am trying to understand:

We have an energy level expression for the particle-in-a-box, which is

E_n= n²*h² n=1, 2, 3
. 8mL²

h = Planck’s constant, m = mass of the particle in kg, and L = the length of the
box in m. If an electron is trapped in a one-dimensional space of length L, and shows an
absorption at 523 nm due to the transition from ψ2 --> ψ3, what is the length of the space L in nanometers?Even if I just get a general idea of how to do this so I can compare my answer to it I will really appreciate it.

Thanks

 

[jbsteele]

!The solution to this problem can be found using the equation for the energy of the particle in the box and the energy of the transition between the ψ2 and ψ3 states. The energy of the transition can be found using the equation:E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the absorption. Substituting this into the energy level expression, we get:n2*h2 = hc/λRearranging this equation, we get:L = (hc/n2*h2)*λSubstituting in the values from the question, we get:L = (6.626 x 10-34 x 3.00 x 108)/(9 x (6.626 x 10-34)2 x 5.23 x 10-7)L = 1.01 x 10-9 m = 10.1 nm

 

[baba89]

First, it is important to understand the concept of spectroscopy and how it relates to physical chemistry. Spectroscopy is a technique used to study the interaction between matter and electromagnetic radiation. It provides information about the energy levels and transitions of a system, which can be used to extract valuable information about the system's properties.

In this case, we are dealing with the particle-in-a-box model, which is a simplified representation of a particle confined in a one-dimensional space. The energy levels of this system are given by the expression En = n^2*h^2/(8mL^2), where n is the energy level, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

Now, to extract information from spectroscopic data, we can use the fact that the energy of a photon is directly related to its wavelength by the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. This means that the energy difference between two energy levels can be calculated by taking the difference of their corresponding wavelengths.

In this case, we are given that the transition from ψ2 to ψ3 results in an absorption at 523 nm. Using the above equation, we can calculate the energy difference between these two levels as follows:

ΔE = hc/λ = (6.626 x 10^-34 J*s)(3.00 x 10^8 m/s)/ (523 x 10^-9 m) = 3.80 x 10^-19 J

Now, we can equate this energy difference to the energy expression for the particle-in-a-box model and solve for L:

ΔE = n^2*h^2/(8mL^2)

3.80 x 10^-19 J = (3^2)(6.626 x 10^-34 J*s)^2/(8(9.11 x 10^-31 kg)L^2)

Solving for L, we get L = 1.39 x 10^-9 m or 1.39 nm.

Therefore, the length of the space L is 1.39 nm.

In summary, to extract information from spectroscopic data, we need to use the relationship between energy and wavelength, as well as the energy expression for the system in question. By equating the two and solving for the unknown variable,