Calculate CO bond length from J=0 to J=1 transition

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Discussion Overview

The discussion revolves around calculating the bond length of carbon monoxide (CO) based on the J=0 to J=1 transition frequency. It involves theoretical and mathematical reasoning related to quantum mechanics and molecular physics.

Discussion Character

  • Homework-related, Mathematical reasoning, Technical explanation

Main Points Raised

  • The initial calculation for the moment of inertia (I) and bond length (r) presented by the first participant, which resulted in a value significantly larger than the known bond length.
  • A second participant questions the formula used for rotational constants, suggesting alternative expressions found in external sources.
  • A third participant indicates that their attempts with the suggested formulas also yield results that are orders of magnitude too large.
  • A later reply provides a detailed explanation of the rotational energy levels and the relationship between energy, frequency, and bond length, proposing a bond length of approximately 115.3 pm.
  • An additional comment points out a missed factor in the calculations, suggesting a possible source of error in the initial approach.

Areas of Agreement / Disagreement

Participants express differing views on the correct formulas and calculations for determining the bond length, with no consensus reached on the correct approach or final value.

Contextual Notes

There are unresolved mathematical steps and potential dependencies on specific definitions of constants and units that may affect the calculations presented.

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Homework Statement



The J=0 to J=1 transition fro 12C16O carbon monoxide occurs at 1.153x105 MHz. Calculate the value of the bond length in CO.

Homework Equations



2B = h/(4π2I)

μ = m1*m2/(m1+m2)

I = μr2

The Attempt at a Solution



I = 6.61x10-34/ (4*π2*1.153x108) = 1.45x10-43 kg m2

μ = 12.01*16.00/(12.01+16.00) *1.661x10-27 = 1.14x10-26kg

r = √(I/μ) = √(1.45x10-43/1.14x10-26) = 2.52x10-9 m or 2520 pm, which is significantly larger than the observed value of 113 pm.

Does anyone see where I went wrong?

Thanks!
 
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cep said:
2B = h/(4π2I)

My QM is so rusty I am trying to not touch it as it may fell apart, but I just checked in wikipedia and it is either

\bar B = \frac {h} {8\pi^2cI}

or

B = \frac {\hbar} {2I}

Neither fits what you wrote.
 
I've tried both of those-- still orders of magnitude too large.
 
Even though this is an old question I figured I'd post an answer for people who might have the same question and end up here.

The rotational energy for a specific level J, EJ = J(J+1)[((m1 + m2)*h2)/(8m1m2R2) = J(J+1)B

In this case B = ((m1 + m2)*h2)/(8m1m2R2)

The transition from J = 0 → J = 1 then would be [1(1+1)B] - [0(0+1)B] = 2B

1.153x105 MHz is 1.153x1011 Hz. Solving for E from E = hv where h is still Planck's constant and v is the frequency gives 7.64x10-23 Joules which is equal to 2B as shown above.

7.64x10-23 Joules = 2B = 2[((m1 + m2)*h2)/(8m1m2R2)]

From here you just manipulate the equation to solve for R then plug in the values for m1, m2, E, and h. The value for R (not a radius mind you, but a bond length) should be ~115.3 pm.

Oh and if you still aren't getting the right answer, make sure the check your units!
 
You were right only missed a factor of 3 by 1.153x10^11, the rest check your calculator. I know its been a while :cry:
 

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