Assume the bond distances in ^13 C^16 O,^13 C^17 O, and ^12 C^17 O are the same as in ^12 C^16 O

Assume the bond distances in ^13 C^16 O,^13 C^17 O, and ^12...

Key Concepts

Isotopic Substitution Effects

Isotopic substitution involves replacing one atom in a molecule with another isotope, thereby altering the atomic mass while keeping the bond distance essentially unchanged. This change affects the reduced mass of the molecule, which in turn modifies the moment of inertia and the rotational constant. Consequently, rotational transition frequencies shift when comparing different isotopomers, such as those seen in various forms of CO.

Rotational Constant (B)

The rotational constant, typically denoted by B, is a fundamental parameter in rotational spectroscopy that relates to the spacing between rotational energy levels. It is defined by the expression B = h/(8?²cI), where h is Planck's constant, c is the speed of light, and I is the moment of inertia. Ethically, B is expressed in units of cm?¹ and determines the energy difference between successive rotational states, particularly the value 2B for the first transition from J=0 to J=1.

Rotational Spectroscopy

Rotational spectroscopy involves the study of rotational transitions in molecules, which occur when a molecule absorbs or emits electromagnetic radiation corresponding to changes in its rotational energy levels. For diatomic molecules, these energy levels are quantized and are typically described using the rotational quantum number J, with transitions following the selection rule ?J=±1. Measurements in units of cm?¹ (wavenumbers) are common, linking directly to the energy differences between levels.

Moment of Inertia

The moment of inertia of a diatomic molecule is a measure of its resistance to rotational acceleration and is calculated as the product of its reduced mass and the square of the bond length. Changes in the masses of the constituent atoms, as in different isotopomers, directly affect the moment of inertia, and consequently, the spacing of the rotational energy levels and the corresponding transition frequencies.

Reduced Mass

The reduced mass is a key concept in molecular physics and is defined for a diatomic molecule as ? = (m?m?)/(m? + m?), where m? and m? are the masses of the two atoms. In the context of rotational transitions, the reduced mass is used to compute the moment of inertia of the molecule, thereby influencing the rotational constant and the energies of the quantized rotational levels.

Transcript

00:02 Okay, so this question is about the rotational spectrum of the molecule.

00:07 Let's see the first question.

00:09 The first question asks the bond lens.

00:19 And let's look at the question.

00:21 The question actually gives us the rotational constant b, right, equals to 1 .92.

00:41 And then we need to calculate the bone length.

00:46 So the first thing we need to do is to write down the expression of the rotational constant, which you can check the book equals to b equals to plung constant over 8 pi square times speed of light and times the moment of inner real.

01:13 And for the moment of inner tier, right, the i also equals to the reduced mass of the molecule times the bone lengths.

01:28 And the bone lens is what we need to calculate.

01:32 So now the question, the answer becomes very clear, right? in order to calculate the bone lens, we first need to know the, the moment in it here.

01:49 And in order to know the moment in it here, right, we need to use this formula.

01:56 So let's start with this formula.

01:58 We can make some changes, right, from the rotational constant.

02:05 Let's just write down again, the rotational constant equals to this.

02:13 And we can just do arrangement, and we find the moment of unit here equals to the plus plump constant over 8 pi square times the speed light and times the rotational constant.

02:35 And now we can do the calculation.

02:38 And one thing you need to be careful is a unit because the unit of the rotational constant given use the inverse of the centimeters.

02:53 But if you look as an expression of the moment in it here, they use meters.

03:02 So the first thing you need to do is to convert the rotational constant unit unit to inverse of meters.

03:15 So for this one, you can convert to a conversion.

03:17 It actually equals to 192 inverse meters.

03:23 Now we can just ploy in the numbers.

03:26 So the plot constant here equals to 6 .62 times 10 to minus 34 and over 8 times 3 .14 square times speed light, which is 3 times 10 to 8.

03:48 And the rotational constant here we need to use 192.

03:58 And use your calculator.

04:00 You can forget the answer, which is 1 .457 times 10 to minus 46.

04:14 And unit is kilograms times meter squared.

04:18 Okay, so now we get a moment in it here, right? and in order to calculate the bone length, another parameter we need to know is a reduced mass.

04:34 So now we can then calculate the reduced mass.

04:38 The reduced mass, mu equals to check the book, the reduced mass equals to the multiplication of the two mass.

04:50 Right over the addition of two masses which here is two atoms the first one is carbon another one is oxygen right so let's just do and one is carbon and its mass is 12 times 1 .66 times 10 to minus 27 times the mass of the oxygen which is 15 .99 times 10, 1 .66 times 10 to minus 27 over the addition.

05:30 Let's just write down again for your convenience plus the mass of the oxygen.

05:45 And same, you have to use your calculator and be careful.

05:49 And you will get the answer.

05:51 The reduced mass of the co molecule is 11 .3.

05:57 8 times 10 to minus 27 kilograms.

06:05 And now, right, for this expression, right, we already know the moment of in it here, and we also know the reduced mass...

Please give Ace some feedback