The lowest-frequency pure rotational line for^12 C^16 O occurs at 115, 271 MHz.

Question

The lowest-frequency pure rotational line for^12 C^16 O occurs at 115, 271 MHz. Assume that the bond distances and the force constants are the same for isotopes of the same molecule. What is the C-O bond distance in this molecule? What is the lowest-frequency rotational line for^13 C^16 O? For^12 C^18 O? If the fundamental vibration of^12 C^16 O occurs at 2143 cm^-1, at what wavenumber should the fundamental vibration of^12 C^18 O appear? Which technique is more sensitive to isotopic shifts: infrared or microwave spectroscopy? Explain your answer.

Explanation / Answer

1. Lowest frequency pure rotational line for

12C16O = 115,271 MHz = 1.15271 x 10^11 Hz

a. dE = hv = 6.626 x 10^-34 x 1.15271 x 10^11 = 7.64 x 10^-23 J

rotational constant Be,

Be = dE/hc = 7.64 x 10^-23/6.626 x 10^-34 x 3 x 10^10

      = 3.84 cm-1 = 384 m-1

moment of inertia (I),

I = h/8pi^2cBe

  = 6.626 x 10^-34 x (3.14)^2 x 3 x 10^8 x 384

= 7.29 x 10^-47 kg.m^2

reduced mass of 12X16O (u) = m1m2/(m1+m2) x avogadro's number

                                           = (12 x 16)/(12+16) x 6.023 x 10^26

                                           = 1.77 x 10^-26 kg

C-O bond distance = sq.rt.(I/u) = 6.416 x 10^-11 m

b. For 13C16O

reduced mass (u) = (13 x 16)/(13+16) x 6.023 x 10^26 = 1.19 x 10^-26 kg

moment of inertia (I) = ur^2 = 1.19 x 10^-26 x (6.416 x 10^-11)^2 = 4.90 x 10^-47 kg.m^2

rotational constant (Be) = 6.626 x 10^-34/8 x (3.14)^2 x 3 x 10^8 x 4.90 x 10^-47 x 100 = 5.712 cm-1

Lowest energy frequency = cBe = 3 x 10^8 x 571.2 = 1.714 x 10^11 Hz

For 12C18O

reduced mass (u) = (12 x 18)/(12+18) x 6.023 x 10^26 = 1.195 x 10^-26 kg

moment of inertia (I) = ur^2 = 1.195 x 10^-26 x (6.416 x 10^-11)^2 = 4.921 x 10^-47 kg.m^2

rotational constant (Be) = 6.626 x 10^-34/8 x (3.14)^2 x 3 x 10^8 x 4.921 x 10^-47 x 100 = 5.69 cm-1

Lowest energy frequency = cBe = 3 x 10^8 x 569 = 1.71 x 10^11 Hz

Similarly for 12C18O molecule can be done.

c. fundamental vibration (w) of 12C16O occurs at 2143 cm-1

v = cw

v = cw = 1/2pi x sq.rt.(k/u)

feed the values from above for 12C16O and 12C18O

2143/v = sq.rt.(1.195 x 10^-26/1.77 x 10^-26)

wavenumber for 12C18O = v = 2608 cm-1

d. Rotational spectroscopy shows a greater shift which is easily identifiable for isotopic substitution and thus is more suitable for this determination.

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