Question

Physcial Chemistry:Ch 10 Introduction to Quantum Mechanics

Carotenes are molecules with alternating C–C and C=C bonds in which the electrons are delocalized across the entire alternating bond system. As such, the electrons can be approximated as being a particle-in-a-box. Lycopene is a carotene found in tomatoes and watermelon. Assuming that the alternating carbon-carbon bond system is 2.64 nm wide:

(a) What is the energy of the n = 11 level?

(b) What is the energy of the n = 12 level?

(c) What is the ∆E between n = 11 and n = 12?

(d) If, by the Bohr frequency condition, ∆E = hν, what frequency and wavelength of light corresponds to the transition from n = 11 to n = 12?

Answer:

(a) 1.05×10⁻¹⁸ J (b) 1.24×10⁻¹⁸ J (c) 1.9×10⁻¹⁹ J (d) ν = 2.87×10¹⁴ s ⁻¹ , λ = 1.05 µm

Answer 1

E = n²h²/8ml²

Given that l = 2.64 nm = 2.64 x 10^-9 m

m = mass of electron = 9.1 x 10^-31 kg

a) energy of the n = 11 level

E = n²h²/8ml²

   = (11)² (6.626 x 10^-34 J.s)² / 8 (9.1 x 10^-31 kg)(2.64 x 10^-9 m)²

   = 1.05×10⁻¹⁸ J

b)  energy of the n = 12 level

E = n²h²/8ml²

      = (12)² (6.626 x 10^-34 J.s)² / 8 (9.1 x 10^-31 kg)(2.64 x 10^-9 m)²

     = 1.24×10⁻¹⁸ J

c)  ∆E between n = 11 and n = 12

  ∆E = 1.24×10⁻¹⁸ J - 1.05×10⁻¹⁸ J = 1.9×10⁻¹⁹ J

d) frequency:

  ∆E = hν

v =   ∆E / h =  1.9×10⁻¹⁹ J / 6.626 x 10^-34 J.s = 2.87×10¹⁴ s −1

wavelength:

  ∆E = hc/ λ

λ = hc/ ∆E = (6.626 x 10^-34 J.s ) x( 3 x10⁸ m/s ) / 1.9×10⁻¹⁹ J =  1.05 µm