17.3 Spectrophotometry, Beer's Law & Chromatography

17.3 Spectrophotometry, Beer's Law & Chromatography

Quick Summary: Instrumental chemical analysis utilizes physical phenomena to identify and quantify chemical species. Ultraviolet-Visible (UV-Vis) spectrophotometry measures the attenuation of light passing through an absorbing solution. Transmittance ($T = I / I_0$) is converted into absorbance ($A = -\log_{10} T$), which relates linearly to molar concentration ($c$) via the Beer-Lambert Law ($A = \varepsilon b c$). Optimal measurements occur at the absorption peak ($\lambda_{\text{max}}$) to maximize sensitivity and minimize polychromatic deviation. Chromatography separates complex chemical mixtures via differential partitioning between a stationary phase and a moving mobile phase, quantified in thin-layer chromatography (TLC) by retention factors ($R_f$).

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1. Principles of UV-Vis Spectrophotometry: Transmittance & Absorbance

When UV (200–400 nm) or visible (400–700 nm) light strikes a molecule, photons matching electronic gaps promote electrons from bonding ($\sigma, \pi$) or nonbonding ($n$) orbitals to antibonding ($\pi^*, \sigma^*$) orbitals.

Transmittance & Absorbance

A monochromatic light beam of incident radiant power $I_0$ passes through a cuvette of path length $b$, exiting with transmitted power $I$. Transmittance ($T$) and percent transmittance (%T) are: $T = \frac{I}{I_0}$ $\%T = \left( \frac{I}{I_0} \right) \times 100\%$ Because light attenuation decays exponentially across successive solution layers, transmittance does not scale linearly with concentration. To obtain direct linearity with concentration, Absorbance ($A$, dimensionless) is defined logarithmically: $A = -\log_{10} T = \log_{10}\left(\frac{I_0}{I}\right) = 2.000 - \log_{10}(\%T)$

• $\%T = 100\% \rightarrow A = 0.000$
• $\%T = 10\% \rightarrow A = 1.000$
• $\%T = 1.0\% \rightarrow A = 2.000$
• $\%T = 0.1\% \rightarrow A = 3.000$ Reliable spectrophotometric measurements fall between $0.1$ and $1.0$ (80% to 10% transmittance), where detector photometric noise is minimized.

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2. The Beer-Lambert Law & Calibration Curves

The fundamental equation of absorption spectrophotometry is the Beer-Lambert Law: $A = \varepsilon \times b \times c$

• $A$ (Absorbance): Dimensionless quantity.
• $\varepsilon$ (Molar Absorptivity): In $\text{L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$ (or $\text{M}^{-1}\cdot\text{cm}^{-1}$), representing intrinsic photon absorption probability at a specific wavelength, solvent, and temperature.
• $b$ (Path Length): Internal width of the cuvette, standardly $1.00\text{ cm}$.
• $c$ (Molar Concentration): Analyte concentration in $\text{mol}\cdot\text{L}^{-1}$ (M).

Analytical Wavelength Selection ($\lambda_{\text{max}}$)

Measurements are conducted at the absorption spectrum peak ($\lambda_{\text{max}}$) for two reasons:

1. Maximum Sensitivity: Molar absorptivity $\varepsilon$ peaks at $\lambda_{\text{max}}$, producing the greatest absorbance change per unit concentration change ($\Delta A / \Delta c$).
2. Adherence to Beer's Law: At the apex, the slope is zero ($dA/d\lambda = 0$). Minor wavelength fluctuations or finite monochromator bandpass produce negligible variation in $\varepsilon$.

Calibration Curves & Cuvette Protocols

A series of standard solutions are measured at $\lambda_{\text{max}}$. Plotting $A$ versus $c$ yields a straight line with slope $\varepsilon b$ passing through the origin. An unknown concentration is determined from $c_{\text{unknown}} = A_{\text{unknown}} / \text{Slope}$.

• Blank Calibration: A cuvette with solvent and reagents (minus analyte) zeros the instrument ($A = 0.000$, $\%T = 100.0\%$), compensating for solvent absorption, reflection, and cuvette attenuation.
• Cuvette Handling: Cuvettes are handled by frosted faces and wiped with optical tissues (Kimwipes). Fused quartz cuvettes are required for UV (<340 nm) because standard glass and plastic absorb UV light.

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3. Deviations from Beer's Law

• High Concentrations ($c > 0.01\text{ M}$): Solute particles reside close together, causing electrostatic interactions that perturb electronic energy levels and alter $\varepsilon$.
• Chemical Equilibria & pH Shifts: If the absorbing species participates in equilibria, non-linearity occurs. For example, unbuffered potassium chromate shifts between yellow chromate ($\text{CrO}_4^{2-}$) and orange dichromate ($\text{Cr}_2\text{O}_7^{2-}$): $2\text{ CrO}_4^{2-}(aq) + 2\text{ H}^+(aq) \rightleftharpoons \text{Cr}_2\text{O}_7^{2-}(aq) + \text{H}_2\text{O}(l)$ Dilution shifts the equilibrium, altering the ratio of two absorbing species with different spectra.
• Instrumental Deviations: Polychromatic light across a wide slit bandpass causes downward curvature at high absorbance. Stray light striking the detector without traversing the sample produces severe negative deviations.

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4. Chromatography Fundamentals & Retention Factor ($R_f$)

Chromatography separates mixture components based on differential partitioning between a stationary phase and a mobile phase.

Thin-Layer Chromatography (TLC)

In TLC, a thin layer of silica gel ($\text{SiO}_2$) with polar silanol groups ($\text{Si–OH}$) serves as a polar stationary phase. A sample spot is placed on a baseline, and an organic eluent (mobile phase) ascends via capillary action.

• Separation Principle: Polar compounds adhere strongly to silica via dipole interactions and hydrogen bonding, migrating slowly. Nonpolar compounds partition into the organic solvent, migrating rapidly.
• Retention Factor ($R_f$): $R_f = \frac{\text{Distance traveled by solute spot from baseline}}{\text{Distance traveled by solvent front from baseline}}$ $R_f$ is dimensionless ($0.00$ to $1.00$) and serves as a reproducible characteristic under standardized conditions.

Column Chromatography & HPLC

• Column Chromatography: Stationary phase is packed in a vertical column; solvent flows downward under gravity to collect separated fractions.
• HPLC: Employs microparticulate silica ($3\text{--}5\ \mu\text{m}$) under high pressure (100–400 bar). In reversed-phase HPLC, the stationary phase is nonpolar ($C_{18}$ bonded silica) and the mobile phase is polar (water-acetonitrile), causing polar compounds to elute first.

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5. Comparative Reference Tables & Worked Quantitative Problem

Spectrophotometric Parameters & Beer-Lambert Variables

Worked Example: Unknown Concentration Determination

A $2.50 \times 10^{-4}\text{ M}$ solution of $\text{KMnO}_4$ in a $1.00\text{-cm}$ cuvette produces $A = 0.585$ at $\lambda_{\text{max}} = 525\text{ nm}$: $\varepsilon = \frac{A}{b \times c} = \frac{0.585}{1.00\text{ cm} \times 2.50 \times 10^{-4}\text{ M}} = 2.34 \times 10^3\text{ L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1}$ An unknown $\text{KMnO}_4$ solution gives $\%T = 35.5\%$ in the same cell: $A_{\text{unknown}} = 2.000 - \log_{10}(35.5) = 2.000 - 1.550 = 0.450$ $c_{\text{unknown}} = \frac{A_{\text{unknown}}}{\varepsilon \times b} = \frac{0.450}{2.34 \times 10^3\text{ L}\cdot\text{mol}^{-1}\cdot\text{cm}^{-1} \times 1.00\text{ cm}} = 1.92 \times 10^{-4}\text{ M}$