The delocalized electrons in a long-chain polyene exist in quantum states that can be reasonably approximated by the particle in a one-dimensional box model. The length L of the box is the end-to-end distance. Suppose one such polymer has a length L = 4.00 nm. How much energy in Joules does an electron emit when it falls from the level with n= 3 to the lowest energy level?
See if this will help.
https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.5%3A_The_Energy_of_a_Particle_in_a_Box_is_Quantized
To calculate the energy emitted when an electron falls from the level with n = 3 to the lowest energy level in a one-dimensional box model, we can use the formula for the energy levels of a particle in a one-dimensional box:
En = (n^2 * h^2) / (8 * m * L^2),
where En is the energy level, n is the quantum number, h is the Planck's constant, m is the mass of the electron, and L is the length of the box.
First, we need to determine the energy of the level with n = 3 using the formula. Let's plug in the given values:
En = (3^2 * h^2) / (8 * m * L^2),
Next, we need to find the energy of the lowest energy level, which corresponds to n = 1. Plugging in the values:
E1 = (1^2 * h^2) / (8 * m * L^2),
To find the energy emitted when the electron falls from n = 3 to n = 1, we subtract the energy of the lower level from the higher level:
ΔE = E3 - E1,
Now we can calculate the energy emitted. However, we need to determine the value of Planck's constant (h) and the mass of the electron (m) from reliable sources or given data in order to get an accurate result.