Ep 20 — Introduction to Quantitative Analysis: Lambert-Beer Law

Series: Encyclopedia of Infrared Spectroscopy: From Principles to Practice
Chapter: Part 2 · Beginner Level — Entering the Laboratory
Target Audience: High school/undergraduate students, graduate students, laboratory technicians new to the field
Prerequisites: Ep 04 (How to Read an IR Spectrum), Ep 16 (Sample Preparation), Ep 17 (Instrument Operation), Ep 18 (Spectral Processing), Ep 19 (Qualitative Analysis)
Reading Time: Approximately 40 minutes

Introduction: From "What" to "How Much"

In 1760, German mathematician Johann Heinrich Lambert wrote a seemingly simple proposition in an optical treatise: the intensity attenuation of light passing through a medium is exponentially related to the medium thickness [1][2].

In 1852, German physicist August Beer extended this law to solutions: attenuation is also proportional to concentration [1][2]. This became known as the "Lambert-Beer Law":

$$A = \varepsilon \cdot b \cdot c$$

This short formula establishes the quantitative relationship between absorbance $A$, path length $b$, and concentration $c$, forming the cornerstone of all modern spectroscopic quantitative analysis—UV-Vis, infrared, and fluorescence all rely on it [1][3].

For infrared spectroscopy, the value of the Lambert-Beer law lies in: as long as the absorbance of a characteristic peak of a functional group is known, the concentration can be deduced. This means we can use infrared spectroscopy for quantitative analysis of mixture components, monitoring reaction progress, and quality control of products—without separation [1][4].

In this episode, we will systematically cover: derivation and applicability of the Lambert-Beer law, peak height vs peak area methods, calibration curve method, multicomponent quantification, precautions, comparison with UV-Vis/chromatography, and a complete demonstration of quantitative analysis workflow with an example.

💡 Position of this episode: Ep 19 covers qualitative analysis ("what"), while this episode covers quantitative analysis ("how much"). Together, they form the core capability of infrared spectroscopy analysis. This episode also concludes Part 2 "Beginner Level".

1. Lambert-Beer Law

1.1 Derivation of the Law

Consider a monochromatic light beam of intensity $I_0$ passing through a solution of thickness $b$ and concentration $c$, resulting in intensity $I$ [1][3]:

        I₀ → ────────────────────→ I
              ┌────────────────┐
              │   Solution (path b) │
              │   Concentration c   │
              └────────────────┘

Step 1: Lambert's Law (Lambert, 1760) [1][2]

When light passes through a thin layer $db$, the attenuation is:

$$\frac{dI}{I} = -\alpha \, db$$

Integrating gives:

$$\ln\frac{I_0}{I} = \alpha \cdot b$$

Converting to base-10 logarithm:

$$\log\frac{I_0}{I} = \frac{\alpha}{2.303} \cdot b = K \cdot b$$

where $K$ is a constant related to the medium properties.

Step 2: Beer's Law (Beer, 1852) [1][2]

$K$ is proportional to concentration $c$:

$$K = \varepsilon \cdot c$$

Substituting yields the Lambert-Beer law:

$$\boxed{A = \log\frac{I_0}{I} = \varepsilon \cdot b \cdot c}$$

where [1][3]:

• $A$: absorbance (dimensionless, often expressed as Abs)
• $\varepsilon$: molar absorptivity (L·mol⁻¹·cm⁻¹)
• $b$: path length (cm)
• $c$: concentration (mol/L)

1.2 Transmittance and Absorbance

Transmittance [1][3]:

$$T = \frac{I}{I_0}$$

Percent transmittance: %T = $T \times 100\%$

Absorbance:

$$A = -\log T = \log\frac{I_0}{I} = -\log\left(\frac{I}{I_0}\right)$$

Relationship between T and A:

Table 1: Conversion between transmittance and absorbance (compiled from [1][3])

Optimal absorbance range [1][3][4]:

• 0.3–1.0 (most ideal)
• 0.1–1.5 (acceptable)
• < 0.1: high noise

• 1.5: possible deviation from linearity

1.3 Molar Absorptivity $\varepsilon$

Physical meaning [1][3]: Absorbance per unit concentration (1 mol/L) and per unit path length (1 cm).

IR vs UV-Vis [1][4][5]:

Table 2: Comparison of molar absorptivity between UV-Vis and IR (compiled from [1][5])

Key insight [1][5]: IR $\varepsilon$ is 10–1000 times smaller than UV-Vis; thus IR quantification requires longer path lengths or higher concentrations.

1.4 Applicability Conditions

The Lambert-Beer law is not universal; the following conditions must be met [1][3][4]:

① Monochromatic light

• FTIR obtains "virtual monochromatic light" via Fourier transform of interferogram; theoretically satisfies the condition.
• However, insufficient resolution may still cause deviation.

② Dilute solutions

• At high concentrations (> 0.01 mol/L), intermolecular interactions (hydrogen bonding, aggregation) alter $\varepsilon$.
• IR often uses pure liquids or solids with high concentrations; special care is needed.

③ No scattering

• Scattered light does not participate in absorption but increases background.
• KBr pellets must be ground thoroughly (< 2 μm).

④ No fluorescence or photochemical reactions

• IR has low energy; fluorescence is generally absent.
• However, photolabile samples require attention.

⑤ Homogeneous medium

• Sample distribution must be uniform.
• Pellets must be uniform.

⑥ Fixed path length

• Transmission liquid cells have precise path lengths.
• For KBr pellets, "path length" is determined by thickness; internal standard must be used to compensate.

1.5 Causes of Deviation from Lambert-Beer Law

Chemical deviations [1][3][4]:

• Concentration changes lead to changes in molecular state (e.g., hydrogen bonding, dissociation).
• Different acid-base forms at different pH.
• Different solvents cause changes in $\varepsilon$.

Instrumental deviations [1][3][4]:

• Stray light.
• Nonlinear detector response.
• Insufficient resolution.
• Inaccurate path length.

Typical manifestation: The calibration curve bends (upward or downward) at high concentrations.

2. Peak Height vs Peak Area Methods

2.1 Peak Height Method

Method [1][3][4]: Measure the perpendicular distance from the peak top to the baseline.

Absorbance
   ↑
1.0│            ▓▓
0.8│           ▓██▓          ← Peak height = 0.8
0.6│          ▓████▓          
0.4│        ▓▓██████▓▓        
0.2│      ▓▓▓▓████████▓▓      
0.0│────▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓──→ ν
              ↑
            Peak position ν₀

Figure 1: Schematic of peak height method

Advantages [1][3]:

• Simple calculation.
• Less affected by noise.
• Good reproducibility (insensitive to baseline point selection).

Disadvantages [1][3][4]:

• Affected by peak width (when the peak broadens but area remains constant, peak height decreases).
• Affected by peak position shift.
• Affected by resolution.

2.2 Peak Area Method

Method [1][3][4]: Integrate absorbance between "integration limits" on both sides of the peak.

$$\text{Area} = \int_{\nu_1}^{\nu_2} A(\nu) \, d\nu$$

Absorbance
   ↑
1.0│            ▓▓
0.8│           ▓██▓          
0.6│          ▓████▓          ← Shaded area = peak area
0.4│        ▓▓██████▓▓        
0.2│      ▓▓▓▓████████▓▓      
0.0│────▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓──→ ν
        ↑                  ↑
        ν₁              ν₂
        ←── Integration range ──→

Figure 2: Schematic of peak area method

Advantages [1][3][4]:

• Less affected by peak width
• Less affected by peak shift
• Less affected by resolution
• More suitable for broad IR peaks

Disadvantages [1][3][4]:

• Highly affected by baseline selection
• Affected by overlap with neighboring peaks
• More complex calculation

2.3 Selection Recommendations

Table 3: Peak height vs. peak area selection (comprehensive from [1][3][4])

💡 Professional tip: Usually perform both peak height and peak area methods, compare which has lower RSD, and choose the one with lower RSD [3][4].

III. Calibration Curve Method

The calibration curve method (standard curve method, calibration curve method) is the most commonly used method for quantitative analysis [1][3][4].

3.1 Establishing a Calibration Curve

Procedure [1][3][4]:

1. Prepare 5–7 standard solutions (or KBr pellets) at different concentrations
2. Measure the infrared spectrum of each standard
3. Select characteristic analysis peaks
4. Calculate peak height or peak area for each standard
5. Plot absorbance (or area) vs. concentration
6. Perform linear fitting to obtain the calibration curve

Example Data [3][4]:

Linear Fit [3][4]:

$$A = k \cdot c + a$$

where $k$ is the slope (sensitivity) and $a$ is the intercept (theoretically = 0).

Peak Area A
   ↑
20│                                ●   ← 20 mg/mL
  │                            ●
15│
  │                        ●
10│
  │                    ●
 5│                ●
  │            ●
 0│───●─────────────────────────────→ c (mg/mL)
   0   5   10   15   20

Figure 3: Typical calibration curve (linear range 0–20 mg/mL)

3.2 Evaluation Criteria for Calibration Curves

Linear correlation coefficient $R^2$ [1][3][4]:

• $R^2 > 0.999$: Excellent
• $R^2 = 0.99$–$0.999$: Good
• $R^2 = 0.95$–$0.99$: Acceptable (limited precision)
• $R^2 < 0.95$: Poor, need to redesign

Linear Range [1][3][4]:

• Within this range, $A$ and $c$ have a linear relationship
• High concentration end may deviate (chemical deviation)
• Low concentration end may deviate due to poor SNR

Limit of Detection (LOD) [3][4]:

$$\text{LOD} = \frac{3\sigma}{k}$$

where $\sigma$ is the standard deviation of the blank sample, and $k$ is the slope of the calibration curve.

Limit of Quantification (LOQ) [3][4]:

$$\text{LOQ} = \frac{10\sigma}{k}$$

3.3 Unknown Sample Measurement

1. Measure the spectrum of the unknown sample
2. Calculate the absorbance (or area) of the characteristic peak
3. Substitute into the calibration curve: $c = (A - a) / k$
4. Report the concentration

Example [3][4]:

• Unknown sample peak area = 4.50
• Calibration curve: $A = 0.825 \cdot c + 0.05$
• Concentration: $c = (4.50 - 0.05) / 0.825 = 5.39$ mg/mL

3.4 "Quality" of the Calibration Curve

Judging reliability of calibration curve [1][3][4]:

• $R^2 > 0.999$
• Intercept $a$ close to 0 (theoretically)
• Residuals randomly distributed (no systematic trend)
• Linear range covers unknown sample concentration
• At least 3 replicates for each standard

Common Issues [3][4]:

• High concentration bending: dilute and redo
• Low concentration deviation: increase number of scans, improve SNR
• Large intercept: check blank, subtract background

IV. KBr Pellet Quantification

KBr pellet method is the most commonly used for solid quantification, but has special considerations [1][3][4].

4.1 "Pathlength" Issue in KBr Pellets

KBr pellet thickness is difficult to control precisely due to pressure, sample amount, and grinding [1][3][4].

Solution: Internal Standard Method

Common Internal Standards [1][3][4]:

• KSCN: 2050 cm⁻¹ (C≡N stretch), mostly non-overlapping with organic peaks
• NaN₃: 2140 cm⁻¹ (N₃⁻ asymmetric stretch)
• KNO₃: 1384 cm⁻¹ (NO₃⁻ asymmetric stretch)

Principle of Internal Standard Quantification [3][4]:

1. Add the same amount of internal standard (e.g., 1% KSCN) to each standard and unknown sample
2. Measure the sample characteristic peak area $As$ and internal standard peak area $A{\text{std}}$
3. Calculate the ratio $R = As / A{\text{std}}$
4. Plot $R$ vs. concentration $c$ to obtain calibration curve

Advantages [3][4]:

• Eliminates pellet thickness error
• Eliminates pathlength uncertainty
• Improves reproducibility

4.2 KBr Pellet Quantification Procedure

Steps [3][4]:

1. Prepare a series of standards (5–7 concentrations)
   │ Standard + KBr + KSCN internal standard
   │ Grind and mix thoroughly
   │
   ▼
2. Make pellets (3 replicates per standard)
   │
   ▼
3. Measure spectra
   │
   ▼
4. Measure sample peak and internal standard peak
   │
   ▼
5. Calculate ratio R = A_s / A_std
   │
   ▼
6. Plot R vs c calibration curve
   │
   ▼
7. Measure unknown sample
   │ Add same amount of internal standard
   │ Make pellet, measure, calculate ratio
   │
   ▼
8. Read concentration from calibration curve

Figure 4: KBr pellet quantification procedure (reference [3][4])

4.3 Precision of KBr Pellet Quantification

Typical RSD [3][4]:

• Without internal standard: 5–10%
• With internal standard: 1–3%
• Automatic pellet press: < 1%

V. Liquid Cell Quantification

Liquid cell quantification is the gold standard for liquid samples, with precisely controllable pathlength [1][3][4].

5.1 Measurement of Liquid Cell Pathlength

Interference Fringe Method [1][3][4]:

Transmission measurement of empty liquid cell yields interference fringes:

Transmittance
   ↑
100│  ▓▓▓▓  ▓▓▓▓  ▓▓▓▓  ▓▓▓▓
   │      ▓▓      ▓▓      ▓▓
 0│────────────────────────────→ ν

Optical path formula [1][3]:

$$b = \frac{1}{2 \cdot n \cdot \left(\frac{1}{\nu_2 - \nu_1}\right)} = \frac{\nu_2 - \nu_1}{2n \cdot N}$$

where $n$ is the refractive index of the medium in the cell (air $n=1$), and $N$ is the number of fringes between $\nu_1$ and $\nu_2$.

Example [3]:

• 10 fringes between 1000–2000 cm⁻¹
• $b = (2000-1000)/(2 \times 1 \times 10) = 50\,\mu\text{m} = 0.05\,\text{mm}$

5.2 Quantitative Procedure for Transmission Cells

Steps [1][3][4]:

1. Select a cell with appropriate pathlength (avoid flat-top)
2. Measure actual pathlength (interference fringe method)
3. Prepare a series of standards
4. Measure spectra of each standard
5. Select characteristic peaks, calculate absorbance
6. Construct $A$ vs $c$ calibration curve
7. Measure unknown samples
8. Determine concentration from calibration curve

5.3 Precision of Quantitative Transmission Cell Measurements

Typical RSD [3][4]:

• Fixed cell: 0.5–1%
• Demountable cell: 1–3%
• High-end cell: < 0.5%

6. ATR Quantification

Although ATR pathlength varies with wavelength, quantification is still possible under fixed conditions [3][4][7].

6.1 Feasibility of ATR Quantification

Prerequisites [3][4][7]:

• Fixed crystal material, incident angle
• Fixed sample amount, pressure
• Fixed temperature
• Use peak height ratios or peak area ratios, not absolute absorbance

6.2 ATR Quantitative Procedure

Steps [3][7]:

1. Prepare a series of standards
2. ATR measurement (fixed pressure, crystal)
3. Measure absorbance of characteristic peaks
4. Construct $A$ vs $c$ calibration curve
5. Measure unknown samples

6.3 Precision of ATR Quantification

Typical RSD [3][4][7]:

• Manual pressure ATR: 2–5%
• Pneumatic pressure ATR: 1–2%
• Comparable to transmission cell quantification

⚠️ ATR Quantification Note: Background must be re-collected before each measurement, as crystal temperature and pressure can affect the background [3][7].

7. Multicomponent Quantification

7.1 Two-Component Quantification

Principle [1][3][4]: Both components absorb at two characteristic peaks, but with different ratios.

Let components A and B have concentrations $c_A, c_B$, and absorbances at wavenumbers $\nu_1, \nu_2$:

$$\begin{cases} A_{\nu_1} = \varepsilon_{A1} \cdot b \cdot c_A + \varepsilon_{B1} \cdot b \cdot c_B \\ A_{\nu_2} = \varepsilon_{A2} \cdot b \cdot c_A + \varepsilon_{B2} \cdot b \cdot c_B \end{cases}$$

Solving:

$$\begin{pmatrix} c_A \\ c_B \end{pmatrix} = \frac{1}{b} \begin{pmatrix} \varepsilon_{A1} & \varepsilon_{B1} \\ \varepsilon_{A2} & \varepsilon_{B2} \end{pmatrix}^{-1} \begin{pmatrix} A_{\nu_1} \\ A_{\nu_2} \end{pmatrix}$$

Example [3][4]: Determination of ethyl acetate + acetic acid mixture

• Ethyl acetate C=O: 1740 cm⁻¹
• Acetic acid C=O: 1710 cm⁻¹
• Measure absorbance at 1740 and 1710
• Solve simultaneous equations to obtain $c_A, c_B$

7.2 Multicomponent (≥ 3) Quantification

Matrix Method [1][3][4]:

$$\mathbf{A} = \mathbf{E} \cdot \mathbf{C} \cdot b$$

where:

• $\mathbf{A}$: absorbance vector (at multiple wavenumbers)
• $\mathbf{E}$: absorptivity matrix
• $\mathbf{C}$: concentration vector

Inversion:

$$\mathbf{C} = \frac{1}{b} \mathbf{E}^{-1} \mathbf{A}$$

Conditions [3][4]:

• Matrix non-singular (determinant ≠ 0)
• Spectra of components are different
• Number of measurement wavenumbers ≥ number of components

7.3 PLS (Partial Least Squares)

Principle [3][4][8]: Build linear model using full spectral data to avoid subjective peak selection.

Procedure [3][8]:

1. Prepare a large set of known samples (≥ 30, covering concentration range)
2. Measure full spectra (4000–400 cm⁻¹)
3. Build calibration model using PLS algorithm
4. Validate model (using validation set)
5. Measure unknown samples, predict concentrations

Advantages [3][8]:

• Uses full spectral information, more accurate
• Handles overlapping peaks
• Suitable for complex mixtures

Disadvantages [3][8]:

• Requires many calibration samples
• Requires specialized software (PLS Toolbox, Unscrambler)
• High model maintenance cost

Software [3][8]:

• Thermo TQ Analyst
• Bruker OPUS QUANT
• PLS_Toolbox (MATLAB)
• Python (scikit-learn, pyMCR)

8. Precautions in Quantitative Analysis

8.1 Pathlength Consistency

Transmission Cells [1][3][4]:

• Use the same cell for all samples
• Measure actual pathlength (interference fringe method)
• Keep cell windows clean

KBr Pellets [1][3][4]:

• Use internal standard to eliminate thickness differences
• Use same batch of KBr for standards and unknowns
• Keep press pressure constant

ATR [3][7]:

• Pneumatic pressure is better than manual
• Consistent pressure each time
• Constant crystal temperature

8.2 Baseline Selection

Peak Height Baseline [3][4]:

• Select "non-absorbing" regions on both sides of the peak as baseline points
• Baseline length should be > 2× peak width
• Avoid selecting on the "shoulder" of adjacent peaks

Peak Area Baseline [3][4]:

• Set integration limits at "valleys" on both sides of the peak
• Avoid adjacent peaks
• Use same integration limits for each sample

8.3 Analytical Peak Selection

Ideal Analytical Peaks [1][3][4]:

• Moderate intensity (absorbance 0.3–1.5)
• No overlap with other peaks
• Little affected by intermolecular interactions
• Easily assignable (e.g., C=O, C-H, N-H)

Avoid [1][3][4]:

• Overlapping peaks
• Near strong absorption peaks (flat-top risk)
• Weak absorption peaks (poor SNR)
• Regions disturbed by water/CO₂ (3400, 1640, 2350)

🔗 Extension: Characteristics of commonly used analytical peaks (e.g., C=O near 1700) can be found at ftir.fun carbonyl functional group page.

8.4 Consistent Treatment of Standards and Samples

Key Principles [1][3][4]:

• Same solvent, same matrix
• Same preparation method
• Same measurement conditions
• Same data processing procedure

Examples [3][4]:

• Standards prepared as KBr pellets → unknowns must be KBr pellets
• Standards measured in CaF₂ cell → unknowns must be measured in CaF₂ cell

8.5 Replicate Experiments

Minimum Number of Replicates [3][4]:

• General analysis: 3
• Quantitative analysis: ≥ 3 (RSD < 2%)
• Pharmacopoeia quantification: 6

RSD Requirements [3][4][6]:

• Pharmacopoeia quantification: RSD < 2%
• Industrial analysis: RSD < 5%
• Research analysis: RSD < 1%

9. Example: Determination of a Component in a Mixture

Let us demonstrate the infrared quantitative analysis procedure with a complete example.

9.1 Task

Objective: Determine acetone content in industrial acetone-ethanol mixture (known to be about 20–50%)

Samples: 3 unknown mixtures (labeled M1, M2, M3)

9.2 Method Selection

Method: Transmission cell method

• Liquid sample → transmission cell (see Ep 16)
• Need precise quantification → transmission is better than ATR
• Suitable standards available → calibration curve method

Pathlength Selection [3][4]:

• Acetone C=O (1715 cm⁻¹), $\varepsilon \approx 500$ L·mol⁻¹·cm⁻¹
• Concentration range 20–50% ≈ 3–8 mol/L
• Recommended path length: $b = 1/(\varepsilon \cdot c) = 1/(500 \times 5) = 0.0004$ cm = 4 μm

Practical choice: 6 μm path length CaF₂ liquid cell (to avoid too short, easier operation)

9.3 Standard Preparation

Prepare 5 standard solutions [3][4]:

• S1: 10% acetone + 90% ethanol
• S2: 20% acetone + 80% ethanol
• S3: 30% acetone + 70% ethanol
• S4: 40% acetone + 60% ethanol
• S5: 50% acetone + 50% ethanol

9.4 Spectrum Acquisition

Instrument: Bruker Tensor II
Accessory: CaF₂ liquid cell, 6 μm path length
Parameters: 4 cm⁻¹ resolution, 32 scans, 4000–400 cm⁻¹

Measurement [3][4]:

1. Use pure ethanol as background (to eliminate ethanol effect)
2. Measure spectra S1–S5
3. Measure spectra M1–M3

9.5 Data Processing

Analysis peak: 1715 cm⁻¹ (acetone C=O stretch) [1][5]

🔗 Extension: Characteristic peaks of acetone and other carbonyl compounds can be found at ftir.fun carbonyl functional group page.

Peak area calculation [3][4]:

• Integration range: 1740–1690 cm⁻¹
• Baseline: connect points at 1740 and 1690
• Software (OMNIC/OPUS) calculates automatically

Results [3][4]:

9.6 Calibration Curve

Linear fit [3][4]:

$$A = 0.2065 \cdot c + 0.005$$

Linear parameters:

• Slope $k = 0.2065$ (area/%)
• Intercept $a = 0.005$ (close to 0, as expected)
• Correlation coefficient $R^2 = 0.9998$
• Linear range: 10–50%

Peak area
   ↑
12│                                ● S5
  │
10│                            ● S4
  │
 8│                        ●
  │
 6│                    ● S3
  │
 4│                ● S2
  │
 2│            ● S1
  │
 0│─────────────────────────────→ c (%)
   0   10  20  30  40  50

Figure 5: Acetone quantitative calibration curve

9.7 Unknown Sample Determination

Measure spectra of M1, M2, M3 [3][4]:

• M1 peak area: 5.20
• M2 peak area: 7.85
• M3 peak area: 3.50

Substitute into calibration curve [3][4]:

• M1: $c = (5.20 - 0.005) / 0.2065 = 25.2\%$
• M2: $c = (7.85 - 0.005) / 0.2065 = 38.0\%$
• M3: $c = (3.50 - 0.005) / 0.2065 = 16.9\%$

9.8 Replicates and Precision

M1 triplicate measurements [3][4]:

• 1st: 25.2%
• 2nd: 24.8%
• 3rd: 25.5%
• Mean: 25.2%
• RSD: 1.4% (meets pharmacopoeia requirement < 2%)

9.9 Report

Quantitative Analysis Report
============================

Sample: M1 (acetone-ethanol mixture)
Method: Liquid cell transmission (CaF₂, 6 μm path length)
Analysis peak: 1715 cm⁻¹ (acetone C=O)
Calibration curve: A = 0.2065·c + 0.005
          R² = 0.9998
          Linear range 10–50%

Result:
  Acetone content = 25.2% (w/w)
  RSD = 1.4% (n=3)

Operator: Zhang San
Date: 2026-07-24

10. IR Quantification vs UV-Vis, Chromatography

10.1 Advantages of IR Quantification

① No separation required [1][4]

• Directly measure mixtures
• Non-destructive
• Suitable for process analysis

② Provides chemical information [1][4]

• Not only quantifies but also identifies
• Can detect functional group changes
• Suitable for reaction monitoring

③ Wide applicability [1][4]

• Gas, liquid, solid all possible
• Organic and inorganic both possible
• Aqueous and non-aqueous solutions both possible

④ Speed [1][4]

• Single measurement in 1 minute (ATR)
• Suitable for high throughput

10.2 Disadvantages of IR Quantification

① Low sensitivity [1][4][5]

• $\varepsilon$ is 10–1000 times smaller than UV-Vis
• Detection limit typically > 0.1%
• Not suitable for trace analysis (below ppm level)

② Path length control [1][4]

• Path length in KBr pellets is difficult to control
• ATR path length varies with wavelength
• Affects quantification accuracy

③ Overlap interference [1][4]

• Complex mixture spectra overlap
• Requires chemometric methods

④ Sample preparation [1][4]

• More complex than UV-Vis
• KBr pellet requires skill
• Liquid cell requires cleaning

10.3 Comparison with UV-Vis

Table 4: IR vs UV-Vis quantification comparison (compiled from [1][4][5])

10.4 Comparison with Chromatography

Table 5: IR vs chromatography quantification comparison (compiled from [1][4])

10.5 When to Choose IR Quantification?

Suitable scenarios [1][4]:

• Higher concentration (> 0.1%)
• Need rapid measurement (process control)
• Non-destructive
• Simultaneous need for quantification and identification
• Reaction monitoring, in situ analysis

Unsuitable scenarios [1][4]:

• Trace analysis (< 0.1%)
• Complex mixtures (> 5 components)
• Extremely high precision requirements (< 0.5% RSD)
• Compounds without characteristic peaks

11. Best Practices for Quantitative Analysis

11.1 Method Development Workflow

1. Requirement Analysis
   │ Concentration range, precision requirements, sample type
   │
   ▼
2. Method Selection
   │ Transmission vs ATR; liquid cell vs pellet
   │
   ▼
3. Path Length Selection
   │ Based on ε and concentration
   │
   ▼
4. Analysis Peak Selection
   │ Moderate intensity, no overlap
   │
   ▼
5. Standard Preparation
   │ 5–7 concentrations covering the range
   │
   ▼
6. Calibration Curve
   │ Measurement, linear fitting, evaluation

1. Validation
   │ Accuracy, Precision, LOD, LOQ
   │
   ▼
2. Sample Measurement
   │ Replicates ≥ 3
   │
   ▼
3. Reporting
   │ Include complete method description
   ```

Figure 6: Quantitative analysis method development workflow (compiled from [1][3][4])

11.2 Validation Parameters

Pharmacopoeia method validation [6]:

• Accuracy: Spike recovery 98–102%
• Precision: RSD < 2% (n=6)
• Linearity: $R^2 > 0.999$
• Range: 80–120% of target concentration
• Detection Limit (LOD): Signal-to-noise ratio 3:1
• Quantitation Limit (LOQ): Signal-to-noise ratio 10:1
• Robustness: Minor parameter changes do not affect results

11.3 Error Source Analysis

Table 6: Sources of quantitative analysis errors (compiled from [1][3][4])

Summary of This Episode

Questions for Thought

1. Under which conditions does the Beer-Lambert law $A = \varepsilon bc$ fail? List at least 3 common causes and explain how to detect and avoid them.

2. You need to determine the content (about 50%) of an active pharmaceutical ingredient in a KBr pellet. Design a complete quantitation scheme, including internal standard selection, standard preparation, calibration curve construction, unknown sample measurement, and expected precision.

3. Use IR quantitation to measure the acetone content in an acetone-ethanol mixture. Given that acetone has $\varepsilon_{1715} = 500$ L·mol⁻¹·cm⁻¹ and an expected concentration of 30% (about 4 mol/L), calculate the recommended pathlength. What would happen if a 0.1 mm cell is used?

4. RSD for IR quantitation is typically 1–5%, while HPLC can achieve 0.5%. Why is IR quantitation less precise than chromatography? Which error sources are unique to IR?

5. You need to in-situ monitor an esterification reaction (acid + alcohol → ester + water) and track ester formation. Design an IR quantitation scheme, explaining whether to choose ATR or transmission. Which peaks are used to track the reaction? How to establish a calibration curve?

References

[1] Griffiths P R, de Haseth J A. Fourier Transform Infrared Spectrometry. 2nd ed. Wiley, 2007. Chapter 11–13. ISBN: 978-0-471-19404-0.

[2] Beer A. "Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten." Annalen der Physik und Chemie, 1852, 162(5): 78–88. DOI:10.1002/andp.18521620505.

[3] Smith B C. Quantitative Spectroscopy: Theory and Practice. Elsevier, 2002. ISBN: 978-0444511246.

[4] Smith B C. Fundamentals of Fourier Transform Infrared Spectroscopy. 2nd ed. CRC Press, 2011. Chapter 8–10. ISBN: 978-1420069297.

[5] Socrates G. Infrared and Raman Characteristic Group Frequencies. 3rd ed. Wiley, 2001. ISBN: 978-0470093078.

[6] Chinese Pharmacopoeia Commission. Pharmacopoeia of the People's Republic of China 2020 Edition, Part IV, General Rule 0402, Infrared Spectrophotometry. China Medical Science and Technology Press.

[7] Kazarian S G, Ewing A V. "Applications of FTIR Spectroscopic Imaging to Tablet Dissolution and Drug Release." Expert Opinion on Drug Delivery, 2013, 10(9): 1207–1221. DOI:10.1517/17425247.2013.802832.

[8] Wold S, Sjöström M, Eriksson L. "PLS-Regression: A Basic Tool of Chemometrics." Chemometrics and Intelligent Laboratory Systems, 2001, 58(2): 109–130. DOI:10.1016/S0169-7439(01)00155-1.

[9] ASTM E168-16. "Standard Practices for General Techniques of Infrared Quantitative Analysis." ASTM International, 2016.

[10] USP General Chapter <851>. "Spectrophotometry and Light-Scattering." United States Pharmacopeia.

[11] ICH Q2(R1). "Validation of Analytical Procedures: Text and Methodology." International Conference on Harmonisation, 2005.

[12] ftir.fun. "Infrared Spectral Functional Group Database: Carbonyl."
https://ftir.fun/ir/group/car…

Preview of Next Episode: Ep 21 — Entering the intermediate-level section, focusing on real-world applications of infrared spectroscopy in various industries.
This episode concludes the beginner-level section (Part 2). From Ep 1 to Ep 20, we have completed a full journey covering infrared light principles, molecular vibrations, functional group identification, instrument principles, sampling techniques, sample preparation methods, operating procedures, spectral processing, qualitative analysis, and quantitative analysis. Starting from the next episode, we will enter the "intermediate level," combining real cases from pharmaceuticals, food, polymers, environment, forensics, etc., to deepen practical skills in infrared spectroscopy.

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