Physcial Chemistry:Ch 10 Introduction to Quantum Mechanics Carotenes are molecul
ID: 946362 • Letter: P
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 = h, what frequency and wavelength of light corresponds to the transition from n = 11 to n = 12?
Answer:
(a) 1.05×1018 J (b) 1.24×1018 J (c) 1.9×1019 J (d) = 2.87×1014 s 1 , = 1.05 µm
Explanation / Answer
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
= 1.05×1018 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
= 1.24×1018 J
c) E between n = 11 and n = 12
E = 1.24×1018 J - 1.05×1018 J = 1.9×1019 J
d) frequency:
E = h
v = E / h = 1.9×1019 J / 6.626 x 10-34 J.s = 2.87×1014 s 1
wavelength:
E = hc/
= hc/ E = (6.626 x 10-34 J.s ) x( 3 x108 m/s ) / 1.9×1019 J = 1.05 µm