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

1. Oct 26, 2011

### cep

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

2B = h/(4π2I)

μ = m1*m2/(m1+m2)

I = μr2

3. 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!

2. Oct 26, 2011

### Staff: Mentor

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.

3. Oct 26, 2011

### cep

I've tried both of those-- still orders of magnitude too large.

4. Mar 19, 2012

### B. Richter

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!

5. Nov 29, 2012