Molecular Spectroscopy Fundamentals | Lucas Raposo Carvalho

Energy Levels and Transitions

The energy of electromagnetic radiation is quantized and relates to frequency through Planck's equation:

Planck's Equation (1) $$E = h\nu = \frac{hc}{\lambda}$$

where:

$E$ is the energy of the photon (J)
$h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s)
$\nu$ is the frequency (Hz)
$c$ is the speed of light ($2.998 \times 10^8$ m/s)
$\lambda$ is the wavelength (m)

The energy difference between two quantum states determines the wavelength of absorbed or emitted radiation:

Energy Transition (2) $$\Delta E = E_2 - E_1 = h\nu = \frac{hc}{\lambda}$$

The Beer-Lambert Law

The Beer-Lambert law describes the attenuation of light passing through a solution. It's fundamental to quantitative spectroscopy:

Beer-Lambert Law (3) $$A = \log_{10}\left(\frac{I_0}{I}\right) = \varepsilon \cdot c \cdot l$$

where:

$A$ is the absorbance (dimensionless)
$I_0$ is the incident light intensity
$I$ is the transmitted light intensity
$\varepsilon$ is the molar absorptivity (L mol^-1 cm^-1)
$c$ is the concentration (mol L^-1)
$l$ is the path length (cm)

Transmittance ($T$) is related to absorbance by:

(4) $$T = \frac{I}{I_0} = 10^{-A}$$

$$A = -\log_{10}(T) = 2 - \log_{10}(\%T)$$

Electronegativity Trends

Pauling electronegativity values show periodic trends across the periodic table. Here's an interactive visualization showing how electronegativity varies with atomic number:

Figure 1: Pauling Electronegativity vs. Atomic Number

Hover over points to see element names and electronegativity values. Note the periodic trends and the gap for noble gases.

Spectral Data Tables

Characteristic IR absorption frequencies for common functional groups:

Table 1: Characteristic IR Absorption Frequencies
Functional Group	Bond Type	Frequency (cm^-1)	Intensity	Notes
Alkane	C–H stretch	2850–2960	strong	sp³ hybridized C–H
Alkene	C=C stretch	1620–1680	medium	Variable intensity
Alkene	=C–H stretch	3020–3100	medium	sp² hybridized C–H
Alkyne	C≡C stretch	2100–2260	weak	Terminal alkynes stronger
Alkyne	≡C–H stretch	3260–3330	strong	Sharp, distinctive peak
Alcohol	O–H stretch	3200–3550	strong	Broad; H-bonded
Carboxylic Acid	O–H stretch	2500–3300	strong	Very broad; dimer
Carbonyl	C=O stretch	1670–1820	strong	Exact position varies
Aldehyde	C=O stretch	1720–1740	strong	—
Ketone	C=O stretch	1705–1725	strong	—
Ester	C=O stretch	1735–1750	strong	Higher than ketones
Amine	N–H stretch	3300–3500	medium	Primary: 2 peaks
Nitrile	C≡N stretch	2210–2260	medium	Sharp peak

Molecular Vibrations

A non-linear molecule with $N$ atoms has $3N - 6$ normal modes of vibration (linear molecules have $3N - 5$ modes). For example, water (H₂O) has:

(5) $$\text{Normal modes} = 3N - 6 = 3(3) - 6 = 3$$

The three vibrational modes of water are: symmetric stretch ($\nu_1$), asymmetric stretch ($\nu_3$), and bending ($\nu_2$).

Let's visualize a simulated IR spectrum showing these characteristic peaks:

Figure 2: Simulated IR Spectrum of a Carbonyl Compound

Simulated IR spectrum showing characteristic C=O stretch around 1720 cm^-1 and C–H stretches around 2900 cm^-1.

Comparison of Spectroscopic Techniques

Table 2: Comparison of Common Spectroscopic Methods
Technique	Energy Range	Information Provided	Transitions
UV-Visible	200–800 nm	Electronic structure, conjugation	Electronic (π → π, n → π)
Infrared (IR)	2.5–25 μm	Functional groups, bonding	Vibrational
Raman	Visible laser	Symmetric vibrations, complementary to IR	Vibrational (different selection rules)
NMR	Radio frequency	Molecular structure, connectivity	Nuclear spin
Mass Spec	—	Molecular weight, fragmentation	Ionization

Understanding these techniques and their complementary nature is essential for complete structural elucidation of organic compounds.