chemistry

Vibrational spectroscopy of linear molecules

Vibrational spectroscopy of linear molecules is a chemistry topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Vibrational spectroscopy of linear molecules rather than just read about it. In short: To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum. Degrees of freedom The location of a molecule in a 3-dimensional space can be described by the total number of coordinates.

Key takeaways

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Reference excerpt

To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum.

Degrees of freedom The location of a molecule in a 3-dimensional space can be described by the total number of coordinates. Each atom is assigned a set of x, y, and z coordinates and can move in all three directions. Degrees of freedom is the total number of variables used to define the motion of a molecule completely. For N atoms in a molecule moving in 3-D space, there are 3N total motions because each atom has 3N degrees of freedom.

Vibrational modes N atoms in a molecule have 3N degrees of freedom which constitute translations, rotations, and vibrations. For non-linear molecules, there are 3 degrees of freedom for translational (motion along the x, y, and z directions) and 3 degrees of freedom for rotational motion (rotations in Rx, Ry, and Rz directions) for each atom. Linear molecules are defined as possessing bond angles of 180°, so there are 3 degrees of freedom for translational motion but only 2 degrees of freedom for rotational motion because the rotation about its molecular axis leaves the molecule unchanged. When subtracting the translational and rotational degrees of freedom, the degrees of vibrational modes is determined. Number of degrees of vibrational freedom for nonlinear molecules: 3N-6 Number of degrees of vibrational freedom for linear molecules: 3N-5

Symmetry of vibrational modes All 3N degrees of freedom have symmetry relationships consistent with the irreducible representations of the molecule's point group. A linear molecule is characterized as possessing a bond angle of 180° with either a C∞v or D∞h symmetry point group. Each point group has a character table that represents all of the possible symmetry of that molecule. Specifically for linear molecules, the two character tables are shown below:

However, these two character tables have infinite number of irreducible representations, so it is necessary to lower the symmetry to a subgroup that has related representations whose characters are the same for the shared operations in the two groups. A property that transforms as one representation in a group will transform as its correlated representation in a subgroup. Therefore, C∞v will be correlated to C2v and D∞h to D2h. The correlation table for each is shown below:

Once the point group of the linear molecule is determined and the correlated symmetry is identified, all symmetry element operations associated to that correlated symmetry's point group are performed for each atom to deduce the reducible representation of the 3N Cartsian displacement vectors. From the right side of the character table, the non-vibrational degrees of freedom, rotational (Rx and Ry) and translational (x, y, and z), are subtracted: Γvib = Γ3N - Γrot - Γtrans. This yields the Γvib, which is used to find the correct normal modes from the original symmetry, which is either C∞v or D∞h, using the correlation table above. Then, each vibrational mode can be identified as either IR or Raman active.

Vibrational spectroscopy A vibration will be active in the IR if there is a change in the dipole moment of the molecule and if it has the same symmetry as one of the x, y, z coordinates. To determine which modes are IR active, the irreducible representation corresponding to x, y, and z are checked with the reducible representation of Γvib. An IR mode is active if the same irreducible representation is present in both. Furthermore, a vibration will be Raman active if there is a change in the polarizability of the molecule and if it has the same symmetry as one of the direct products of the x, y, z coordinates. To determine which modes are Raman active, the irreducible representation corresponding to xy, xz, yz, x2, y2, and z2 are checked with the reducible representation of Γvib. A Raman mode is active if the same irreducible representation is present in both.

Example

Carbon Dioxide, CO2 1. Assign point group: D∞h 2. Determine group-subgroup point group: D2h 3. Find the number of normal (vibrational) modes or degrees of freedom using the equation: 3n - 5 = 3(3) - 5 = 4 4. Derive reducible representation Γ3N:

5. Decompose the reducible representation into irreducible components: Γ3N = Ag + B2g + B3g + 2B1u + 2B2u + 2B3u 6. Solve for the irreducible representation corresponding to the normal modes with the subgroup character table: Γ3N = Ag + B2g + B3g + 2B1u + 2B2u + 2B3u Γrot = B2g + B3g Γtrans = B1u + B2u + B3u Γvib = Γ3N - Γrot - Γtrans Γvib = Ag + B1u + B2u + B3u 7. Use the correlation table to find the normal modes for the original point group: v1 = Ag = Σ+g v2 = B1u = Σ+u v3 = B2u = Πu v4 = B3u = Πu 8. Label whether the modes are either IR active or Raman active: v1 = Raman active v2 = IR active v3 = IR active v4 = IR active

References

Illustrations

Vibrational spectroscopy of linear molecules: Carbon dioxide molecule on a Cartesian coordinate — Carbon dioxide molecule on a Cartesian coordinate

Worked examples

Example 1 — a first encounter with Vibrational spectroscopy of linear molecules

Start with the simplest possible case. Write down what Vibrational spectroscopy of linear molecules claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In chemistry, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Vibrational spectroscopy of linear molecules before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Vibrational spectroscopy of linear molecules ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Vibrational spectroscopy of linear molecules

In research: Vibrational spectroscopy of linear molecules appears in chemistry research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Vibrational spectroscopy of linear molecules in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Vibrational spectroscopy of linear molecules is common in secondary-school and first-year university syllabi. It links to neighbouring topics Vibrational spectroscopy, so understanding it makes those chapters shorter.
In everyday life: Look for Vibrational spectroscopy of linear molecules outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Vibrational spectroscopy of linear molecules in simple terms?

Why does Vibrational spectroscopy of linear molecules matter?

Because it connects several chemistry ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Vibrational spectroscopy of linear molecules?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

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