Question

III An infrared absorbance spectrum C"O is shown below. Based on this information, Iculate the following: a) the fundamental vibrational frequency (in Hz) of this molecule; b) the period of the vibration; c) the force constant; d) the zero-point energy of this molecule in kJ/mole; e) the approximate value of the rotational constant. Note that "zero point" energy means the lowest vibrational energy the molecule can have. The isotopic masses of "C and "O are 12.000000 and 15.994915 amu, respectively. (20 points) CM 시-JnUT 시: TS R branch P branch 8-쵸 2165.9 cA 2169.8 cm

Answer 1

Answer a)

fundamental frequency = 2169.8 cm^-1

1 cm^-1 = 3.0 x10¹⁰ Hz

so, fundamental frequency = 2169.8 cm^-1 = 2169.8 x 3.0 x 10¹⁰ Hz = 6.50 x 10¹³ Hz

Answer b)

period of vibration = 1 / frequency of vibration = (1 / 6.50 x 10¹³) s = 1.54 x 10^-14 s

Answer c)

$\nu$ = (1/2$\pi$) (k/ $\mu$ )^0.5     ...(1)

where k = force constant         $\mu$ = reduced mass

$\mu$ = m₁m₂ / (m₁ + m₂)

where m₁ = 12.000 a.m.u. = 12.000 x 1.67 x 10^-27 kg,     m₂ = 15.994915 a.m.u. = 15.994915 x 1.67 x 10^-27 kg

so it gives $\mu$ = 1.145 x 10^-26 kg

putting in (1) gives,

k = 1907.9 N m^-1

Answer d)

zero point energy = h $\nu$/ 2 = 6.626 x 10^-34 x 6.5 x 10¹³ J = 4.3 x 10^-20 J per molecule

now multiply by N_A (avagadro number gives) E = 4.3 x 10^-20 J x 6.022 x 10²³ J/mole = 2.59 x 10⁴ J/mole = 25.9 kJ/mole

Answer e)

approximate value of rotational constant = (2173.7 cm^-1 - 2165.9 cm^-1) / 4 = 1.95 cm^-1

Note that, the difference between first P and first R line is 4B. where, B = rotational constant