[Nisarg Shah]

I seem to have difficulty solving this question -

The effective length of the system in the conjugated linear hydrocarbon hexa-1,3,5-triene, C6H8, is 7.4 Å. By treating the electrons as particles in a one-dimensional box, calculate the wavelength of the lowest energy absorption in the electronic spectrum of hexa-1,3,5-triene.
I believe I only have to apply the condition for resonance where f=2L/n but am not getting anywhere with it...


Can anyone help me? Thanks!

[MrTeo]

I believe I only have to apply the condition for resonance where f=2L/n but am not getting anywhere with it...

Is f the frequency? And L the length of the "box"? Remember that it's a quantum mechanics problem, not a vibrating string.

[Nisarg Shah]

Is f the frequency? And L the length of the "box"? Remember that it's a quantum mechanics problem, not a vibrating string.

Yes you're right. Its λ=2L/n.

[Corribus]

What are the energy levels in a particle in a box model?

[Nisarg Shah]

What are the energy levels in a particle in a box model?

They're given by n²h²/8ml².

[Corribus]

Right, so the question wants to know what the lowest energy transition is. A transition must be between two states that have integer values of n. So, for instance, n = 3 to n = 4 is the transition between the n = 3 state and the n = 4 state,

Based on the formula you provided, what transition is going to have the smallest energy?

Example: for a transition between states with n = 3 to n = 4, the energy is 7 (4² - 3² = 7) in units of h² / 8ml².

Once you figure out what the lowest energy transition will be (from what n to what n + 1), you need to determine what the energy change is going to be for that transition in a box with length stipulated in the original problem.

[Nisarg Shah]

Right, so the question wants to know what the lowest energy transition is. A transition must be between two states that have integer values of n. So, for instance, n = 3 to n = 4 is the transition between the n = 3 state and the n = 4 state,

Based on the formula you provided, what transition is going to have the smallest energy?

Example: for a transition between states with n = 3 to n = 4, the energy is 7 (4² - 3² = 7) in units of h² / 8ml².

Once you figure out what the lowest energy transition will be (from what n to what n + 1), you need to determine what the energy change is going to be for that transition in a box with length stipulated in the original problem.

Okay, so the lowest energy transition for this case will be from n=1 to n=2, so (2²-1²)h²/(8ml²)=hc/λ, right?

[Corribus]

That would be the right approach, yes. Just be careful with units - the non-SI units for wavelength, molecular length and light speed can be a minefield.

[Nisarg Shah]

Thanks for the help. I'll try it out...