Question

A C-D (carbon–deuterium) bond is electronically much like a C-H bond, and it has a similar stiffness, measured by the spring constant, k. The deuterium atom has twice the mass (m) of a hydrogen atom, however.
(a) The infrared absorption frequency is approximately proportional to

\sqrt{\frac{k}{m}}

, when one of the bonded atoms is much heavier than the other, and m is the lighter of the two atoms (H or D in this case). Use this relationship to calculate the IR absorption frequency of a typical C-D bond. Use 3000 cm^–1 as a typical C-H absorption frequency.
(b) A chemist dissolves a sample in deuterochloroform (CDCl₃) and then decides to take the IR spectrum and simply evaporates most of the CDCl₃. What functional group will appear to be present in this IR spectrum as a result of the CDCl₃ impurity?

Accepted Answer

Step 1: Understand the relationship between the IR absorption frequency and the reduced mass of the bond. The frequency (ν) is proportional to the square root of the spring constant (k) divided by the reduced mass (μ). For this problem, the reduced mass is approximately equal to the mass of the lighter atom (m) since the other atom (carbon) is much heavier. The relationship can be expressed as: ν ∝ √(k/m). Step 2: Compare the C-H and C-D bonds. Since the spring constant (k) is the same for both bonds, the ratio of their frequencies will depend on the square root of the ratio of their masses. For the C-H bond, the lighter atom is hydrogen (m_H), and for the C-D bond, the lighter atom is deuterium (m_D). The ratio of the frequencies can be written as: ν(C-D)/ν(C-H) = √(m_H/m_D). Step 3: Substitute the given values. The mass of deuterium (m_D) is twice the mass of hydrogen (m_H), so m_H/m_D = 1/2. Using this ratio, calculate the proportional frequency of the C-D bond relative to the C-H bond. The typical C-H absorption frequency is given as 3000 cm⁻¹. Step 4: For part (b), consider the chemical nature of deuterochloroform (CDCl3). When CDCl3 is evaporated, residual CDCl3 may remain in the sample. CDCl3 contains a C-D bond, which will appear in the IR spectrum. The C-D bond absorbs at a lower frequency than the C-H bond due to the heavier mass of deuterium. This absorption typically appears in the range of 2100–2200 cm⁻¹. Step 5: Conclude that the functional group appearing in the IR spectrum due to the CDCl3 impurity is the C-D bond. This is because the residual deuterochloroform contributes its characteristic C-D stretching absorption in the IR spectrum.

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Key Concepts

Spring Constant (k)

Mass of Atoms (m)

Infrared (IR) Spectroscopy