Beer–Lambert Law Calculator

What Is Absorbance (A)?

Absorbance is a measure of how much light is absorbed by a solution at a specific wavelength. When light passes through a sample (such as a colored solution in a cuvette), some of the light is absorbed by the molecules in the solution, and the rest is transmitted through. Absorbance quantifies the amount of light absorbed. It is a unitless (dimensionless) quantity, typically ranging from 0 (no absorption—perfectly transparent) to around 2–3 in routine measurements (though higher values are possible in principle).

Higher absorbance means more light is absorbed and less is transmitted—the solution appears darker or more intensely colored at that wavelength. Lower absorbance means more light passes through—the solution is lighter or more dilute. Absorbance is measured using a spectrophotometer, an instrument that shines light of a specific wavelength through the sample and compares the intensity of transmitted light to a reference (blank).

In textbook problems, absorbance is often the measured or given value, and you use it with the Beer–Lambert law to calculate concentration or other parameters. Understanding absorbance conceptually helps you interpret spectroscopic data and predict how changes in sample composition affect light absorption.

What Is the Beer–Lambert Law?

The Beer–Lambert law states that absorbance (A) is directly proportional to three factors:

• Concentration (c): The amount of absorbing substance per unit volume (e.g., mol/L).
• Path length (b or l): The distance light travels through the sample (e.g., cm).
• Molar absorptivity (ε): An intrinsic property of the absorbing species at a given wavelength, indicating how strongly it absorbs light (e.g., L·mol⁻¹·cm⁻¹).

The mathematical form of the law is:

Where:

• A: Absorbance (unitless)
• ε: Molar absorptivity or extinction coefficient (L·mol⁻¹·cm⁻¹, or sometimes M⁻¹·cm⁻¹)
• b: Path length (cm, the standard cuvette size is 1 cm)
• c: Concentration (mol/L or M)

This equation tells us that absorbance increases linearly with concentration (double the concentration, double the absorbance) and with path length (double the path length, double the absorbance). The molar absorptivity ε is a constant for a given substance at a given wavelength and solvent, and higher ε means the substance is a stronger absorber.

Molar Absorptivity (ε) and Extinction Coefficient

Molar absorptivity (also called the extinction coefficient) is a measure of how efficiently a substance absorbs light at a particular wavelength. It's an intrinsic property that depends on the molecular structure of the absorbing species and the wavelength of light used. Common units are L·mol⁻¹·cm⁻¹ (or M⁻¹·cm⁻¹, which is equivalent).

Higher ε (large extinction coefficient): The substance absorbs light very strongly. Even at low concentrations, you'll see significant absorbance. Example: Many dyes and chromophores have ε values in the range of 10⁴ to 10⁵ L·mol⁻¹·cm⁻¹.

Lower ε: Weaker absorption. You'd need higher concentrations or longer path lengths to see measurable absorbance.

In homework problems, ε is usually provided (for example, "the molar absorptivity of compound X at 450 nm is 1.2 × 10⁴ L·mol⁻¹·cm⁻¹"), or you might be asked to calculate it from experimental data (if A, b, and c are all known).

Path Length (b) and Cuvette Size

Path length (often denoted b or l) is the distance light travels through the sample. In most introductory problems and standard spectrophotometry, a 1 cm cuvette is used, so b = 1 cm. This simplifies calculations because the path length term drops out numerically (multiplying by 1 doesn't change the value), though it's still conceptually present.

Some problems or advanced applications use different path lengths (e.g., 0.1 cm, 5 cm, or 10 cm cuvettes). If path length changes, absorbance scales proportionally: doubling the path length doubles absorbance, assuming concentration stays the same. This is why knowing b is essential for accurate Beer–Lambert calculations.

Always check the problem statement or experimental setup to confirm the path length. If not specified, 1 cm is a standard assumption in most textbook chemistry problems.

Transmittance (T) and Percent Transmittance (%T)

Transmittance (T) is the fraction of light that passes through the sample compared to a reference (blank). It's defined as:

Where I is the intensity of light after passing through the sample, and I₀ is the initial intensity (or intensity through a blank). T ranges from 0 (no light transmitted, complete absorption) to 1 (all light transmitted, no absorption).

Percent transmittance (%T) is simply T expressed as a percentage:

So %T ranges from 0% (no transmission) to 100% (full transmission).

Relationship between Absorbance and Transmittance: Absorbance and transmittance are related logarithmically:

This means that as transmittance decreases (more light absorbed), absorbance increases. For example, if T = 0.5 (50% transmittance), then A = −log₁₀(0.5) ≈ 0.301. If T = 0.1 (10%), A = 1.0. Understanding this relationship is crucial when problems provide %T instead of A, or when you need to convert between the two.

Mode 1: Calculate Absorbance (A) from ε, b, and c

This is the most straightforward use of the Beer–Lambert law: you know the molar absorptivity, path length, and concentration, and you want to predict the absorbance.

1. Enter molar absorptivity (ε): Input the value in the units shown (typically L·mol⁻¹·cm⁻¹). This is usually provided in the problem statement.
2. Enter path length (b): Input the cuvette or cell path length in cm (commonly 1 cm).
3. Enter concentration (c): Input the concentration in mol/L (M), or convert from other units (mM, µM) before entering.
4. Click Calculate: The tool applies A = ε × b × c and displays the absorbance.
5. Interpret: The result is a unitless absorbance value. Compare it to expected ranges (typically 0–2 for reliable measurements).

Mode 2: Calculate Concentration (c) from A, ε, and b

This is perhaps the most common scenario in lab reports and homework: you've measured absorbance and need to find the concentration of the unknown solution.

1. Enter absorbance (A): The measured or given absorbance value (unitless).
2. Enter molar absorptivity (ε): The known extinction coefficient for the substance.
3. Enter path length (b): Typically 1 cm.
4. Click Calculate: The calculator rearranges the Beer–Lambert law to c = A / (ε × b) and computes concentration.
5. Result: Concentration is displayed in mol/L (M). Convert to other units (mM, µM, mg/mL, etc.) if needed for your problem.

Mode 3: Calculate Molar Absorptivity (ε) from A, b, and c

If you know absorbance, path length, and concentration (from a standard or known sample), you can determine the molar absorptivity of a compound.

1. Enter absorbance (A): The measured value.
2. Enter path length (b): Typically 1 cm.
3. Enter concentration (c): The known concentration of the standard.
4. Click Calculate: The tool solves ε = A / (b × c).
5. Result: Molar absorptivity in L·mol⁻¹·cm⁻¹. This value is specific to the wavelength and solvent used.

Mode 4: Convert Between Transmittance and Absorbance

Some problems provide transmittance (T or %T) instead of absorbance. The calculator can convert between these quantities.

1. If given %T: Enter the percent transmittance value (e.g., 50 for 50% transmittance).
2. Convert to T: T = %T / 100 (e.g., 50% → 0.50).
3. Calculate A: Use A = −log₁₀(T) or A = 2 − log₁₀(%T).
4. Result: Absorbance value, which can then be used with ε and b to find concentration.

Example: If %T = 25%, then T = 0.25, and A = −log₁₀(0.25) ≈ 0.602.

General Tips for Using the Calculator

• Keep units consistent: Ensure ε, b, and c are in compatible units (e.g., L·mol⁻¹·cm⁻¹, cm, and mol/L).
• Check your path length: Don't assume b = 1 cm unless stated. Some problems use different cuvette sizes.
• Understand that A is unitless: Absorbance has no units, but it's not arbitrary—it's based on logarithmic light intensity ratios.
• Use the calculator to verify manual work: In exams, you'll need to calculate by hand, so practice the algebra first and use the tool to check.
• Know when Beer–Lambert applies: The law assumes dilute solutions, monochromatic light, and no chemical reactions or scattering. At very high concentrations or in turbid samples, deviations can occur (covered conceptually in advanced topics).

1. The Beer–Lambert Law (Core Formula)

Variables:

• A: Absorbance (unitless)
• ε: Molar absorptivity / extinction coefficient (L·mol⁻¹·cm⁻¹ or M⁻¹·cm⁻¹)
• b: Path length (cm)
• c: Concentration (mol/L or M)

2. Rearranging the Beer–Lambert Law

Depending on which variable is unknown, the equation can be rearranged:

3. Transmittance and Absorbance Relationships

Worked Example 1: Find Concentration from A, ε, and b

Problem: A solution has an absorbance of 0.60 at 520 nm. The molar absorptivity of the solute at this wavelength is 1.5 × 10⁴ L·mol⁻¹·cm⁻¹, and the path length is 1.0 cm. What is the concentration?

Solution (step-by-step):

1. Identify the values:
   A = 0.60
   ε = 1.5 × 10⁴ L·mol⁻¹·cm⁻¹
   b = 1.0 cm
2. Rearrange Beer–Lambert law to solve for c:
   c = A / (ε × b)
3. Plug in the values:
   c = 0.60 / (1.5 × 10⁴ × 1.0)
   c = 0.60 / (1.5 × 10⁴)
   c = 4.0 × 10⁻⁵ mol/L = 4.0 × 10⁻⁵ M
4. Convert to more convenient units (optional):
   c = 4.0 × 10⁻⁵ M = 0.040 mM = 40 µM

Answer: The concentration is 4.0 × 10⁻⁵ M (or 40 µM).

Worked Example 2: From %T to Concentration

Problem: A spectrophotometer reads 25% transmittance (%T) for a solution. The molar absorptivity is 2.0 × 10³ L·mol⁻¹·cm⁻¹, and the path length is 1.0 cm. Find the concentration.

Solution (step-by-step):

1. Convert %T to T:
   T = %T / 100 = 25 / 100 = 0.25
2. Convert T to absorbance:
   A = −log₁₀(T) = −log₁₀(0.25)
   A ≈ 0.602
3. Solve for concentration using Beer–Lambert law:
   c = A / (ε × b)
   c = 0.602 / (2.0 × 10³ × 1.0)
   c = 0.602 / 2000
   c = 3.01 × 10⁻⁴ mol/L
4. Express in convenient units:
   c ≈ 3.0 × 10⁻⁴ M = 0.30 mM = 300 µM

Answer: The concentration is approximately 3.0 × 10⁻⁴ M (or 0.30 mM).

Interpretation: Notice that 25% transmittance means 75% of the light was absorbed, giving a fairly high absorbance (0.602) and correspondingly measurable concentration. This example shows the complete workflow from %T to concentration.

1. General Chemistry Lab Report: Determining Unknown Concentration

Scenario: In a general chemistry lab, you measure the absorbance of a colored solution (e.g., copper(II) sulfate) at 635 nm and get A = 0.450. The lab manual provides ε = 12.5 L·mol⁻¹·cm⁻¹ for CuSO₄ at this wavelength, and you used a standard 1 cm cuvette. You need to report the molar concentration of Cu²⁺ in your solution.

How the calculator helps: Enter A = 0.450, ε = 12.5, b = 1.0 cm, and solve for c. The calculator immediately gives c = 0.036 M (36 mM), which you can include in your lab report with proper significant figures and units. This saves time and reduces arithmetic errors when processing multiple samples.

2. Analytical Chemistry Exam: Transmittance to Concentration

Scenario: An exam question states: "A solution shows 40% transmittance at 450 nm. The molar absorptivity is 8.5 × 10³ L·mol⁻¹·cm⁻¹. Calculate the concentration." You need to convert %T to A, then use the Beer–Lambert law.

How the calculator helps: Quickly verify your manual calculation. Convert 40% to T = 0.40, find A = −log₁₀(0.40) ≈ 0.398, then solve for c using the calculator. This confirms your hand-calculated answer and ensures you didn't drop a power of ten or make a logarithm error.

3. Biochemistry Problem: DNA Quantification at A260

Scenario: A biochemistry homework asks: "You measure A260 = 0.5 for a DNA solution. Using the standard approximation that A260 = 1 corresponds to 50 µg/mL dsDNA, what is the concentration?" While not strictly Beer–Lambert (since the standard uses mass concentration), the conceptual connection is the same: absorbance relates linearly to concentration.

How the calculator helps: For a true Beer–Lambert calculation, you'd need ε for DNA at 260 nm (which varies by sequence but can be approximated). Use the calculator to explore how varying ε affects the calculated molarity if you convert mass to moles. This deepens understanding of spectroscopic concentration determination.

4. Comparing Two Solutions: Which is More Concentrated?

Scenario: An exam gives absorbance values for two solutions of the same dye (Solution A: A = 0.75, Solution B: A = 0.45). Both measured in 1 cm cuvettes. Which solution has higher concentration?

How the calculator helps: Since A = ε × b × c and ε and b are the same for both, concentration is directly proportional to absorbance. Solution A (higher A) has higher c. Use the calculator to compute actual concentrations if ε is given, reinforcing the linear relationship between A and c.

5. Effect of Path Length: Conceptual Exploration

Scenario: A homework problem asks: "If you switch from a 1 cm cuvette to a 5 cm cuvette, how does absorbance change for the same solution?" You need to understand that A scales linearly with b.

How the calculator helps: Enter the same ε and c, but change b from 1 to 5. The calculator shows A increases by a factor of 5. This numerical verification reinforces the proportionality and helps you check your understanding conceptually.

6. Determining Molar Absorptivity from Standard Data

Scenario: In a lab, you prepare a standard solution with known concentration (e.g., 0.010 M) and measure its absorbance (A = 1.20) using a 1 cm cuvette. The lab manual asks you to calculate ε for this compound at the wavelength used.

How the calculator helps: Enter A = 1.20, b = 1.0 cm, c = 0.010 M, and solve for ε. The calculator gives ε = 120 L·mol⁻¹·cm⁻¹, which you report as the experimentally determined molar absorptivity. This is a common lab exercise in quantitative analysis courses.

7. Dilution Problem: Predicting Absorbance After Dilution

Scenario: You have a solution with A = 1.50 and concentration c₁. You dilute it 1:10 (c₂ = c₁/10). What will the new absorbance be?

How the calculator helps: Calculate the original c from A = 1.50 (if ε and b are known), then divide by 10 for the new concentration, and calculate the new A. The calculator confirms that A₂ = A₁/10 = 0.15, illustrating the linear relationship and helping you solve multi-step dilution problems.

8. Exam Prep: Unit Conversion Practice

Scenario: A practice problem mixes units: ε is given in M⁻¹·cm⁻¹, concentration in mM, and path length in mm. You need to convert everything to consistent units before applying Beer–Lambert.

How the calculator helps: Convert mM to M and mm to cm, then use the calculator to check your final answer. This enforces unit discipline and helps you catch conversion errors before submitting homework or taking an exam.

1. Using Inconsistent Units

Mistake: Using path length in mm when the molar absorptivity is defined for cm, or using concentration in mM when ε expects mol/L, without converting.

Why it matters: If ε = 5000 L·mol⁻¹·cm⁻¹ is meant for b in cm and c in M, but you use b = 10 mm (without converting to 1 cm) or c = 50 mM (without converting to 0.050 M), your answer will be off by a factor of 10 or 1000.

How to avoid: Always write out units explicitly and convert to a consistent set (typically cm for path length, M for concentration, L·mol⁻¹·cm⁻¹ for ε) before plugging into the formula.

2. Forgetting That Absorbance Is Unitless

Mistake: Trying to attach units to absorbance (e.g., writing "A = 0.5 cm" or "A = 0.5 M") or treating it as a concentration.

Why it matters: Absorbance is a logarithmic ratio of light intensities and has no units. Misunderstanding this can lead to confusion when rearranging equations or interpreting results.

How to avoid: Remember: A is unitless, always. It's a pure number derived from log₁₀(I₀/I). Don't confuse it with concentration, which does have units (M, mM, etc.).

3. Confusing %T and T

Mistake: Plugging %T = 25 directly into A = −log₁₀(T), instead of converting to T = 0.25 first.

Why it matters: A = −log₁₀(25) ≈ −1.40 (negative absorbance, nonsensical!), whereas A = −log₁₀(0.25) ≈ 0.602 (correct). This error completely changes the result.

How to avoid: Always convert %T to T as a fraction (T = %T / 100) before calculating absorbance. Double-check that T is between 0 and 1, and A is positive for solutions that absorb light.

4. Using Wrong ε Value

Mistake: Using a molar absorptivity value from a different wavelength, solvent, or compound without noticing.

Why it matters: ε is highly wavelength-dependent. Using ε at 450 nm when your measurement is at 600 nm will give completely wrong concentrations. Similarly, ε in water vs. ethanol can differ significantly.

How to avoid: Always verify that the ε value matches the wavelength and solvent specified in the problem. If the problem gives multiple ε values, make sure you're using the correct one for your conditions.

5. Assuming Beer–Lambert Always Holds Perfectly

Mistake: Applying the law without recognizing that it has limitations (very high concentrations, chemical reactions, scattering, non-monochromatic light).

Why it matters: At very high concentrations (> 0.01 M for many substances), molecular interactions cause deviations from linearity. In turbid or scattering samples, Beer–Lambert doesn't apply directly.

How to avoid: For homework/exam problems, assume the law holds (since problems are designed that way). But be aware that in real labs, you'd check linearity by measuring standards at multiple concentrations. This conceptual awareness helps in advanced courses.

6. Mis-Handling Scientific Notation

Mistake: Dropping powers of ten when ε is given as, say, 1.5 × 10⁴, or miscalculating logarithms.

Why it matters: If ε = 1.5 × 10⁴ and you accidentally use 1.5 (forgetting the 10⁴), your concentration will be off by 10,000-fold. Similarly, logarithm errors change the absorbance-transmittance relationship completely.

How to avoid: Use a calculator carefully for scientific notation and logarithms. Write out intermediate steps to catch errors. The Beer–Lambert calculator can verify your arithmetic.

7. Not Checking If A Is in the Linear Range

Mistake: Calculating concentration from A = 3.5 without recognizing that most spectrophotometers are only accurate for A roughly between 0.1 and 2.

Why it matters: At very high absorbance, little light gets through, and detector noise increases, reducing accuracy. At very low A, small errors in measurement have big effects on calculated concentration.

How to avoid: For homework, if A is outside 0–2, the problem may expect you to dilute conceptually or note that the measurement isn't ideal. The calculator will still compute the math, but understanding practical limits is important.

8. Confusing Path Length "l" and "b" Notation

Mistake: Not recognizing that different textbooks use "l" or "b" for path length (they mean the same thing).

Why it matters: Confusion over notation can lead to using the wrong variable or forgetting path length entirely in calculations.

How to avoid: Understand that A = ε × l × c and A = ε × b × c are equivalent (just different symbols for path length). Always identify what the path length is called in your specific problem.

9. Over-Rounding Intermediate Values

Mistake: Rounding A or ε to very few significant figures too early, introducing cumulative errors.

Why it matters: If A = 0.4567 and you round to 0.5 early, then use that in calculations, your final concentration might differ noticeably from the more precise answer.

How to avoid: Carry full precision (or at least 3-4 significant figures) through intermediate steps. Round only your final answer to appropriate significant figures based on measurement precision.

10. Forgetting to Blank Correct

Mistake: In conceptual problems, not accounting for the "blank" or baseline absorbance of the solvent.

Why it matters: Real spectrophotometers measure absorbance relative to a blank (pure solvent). If a problem states "absorbance after blank correction," the value already accounts for this. But if not, you might need to subtract blank absorbance conceptually.

How to avoid: Read problem statements carefully. Most homework assumes blank correction is done, but understanding the concept prepares you for real lab work and more advanced problems.

1. Explore the Linear Relationships Quantitatively

Use the calculator to test how absorbance scales: double c and see A double (if ε and b are constant); double b and see A double. This numerical exploration reinforces the "directly proportional" concept and builds intuition for predicting how changes affect absorbance.

2. Understand the Role of Wavelength Selection

Different wavelengths give different ε values for the same substance. In conceptual problems, recognize that choosing λ where ε is highest gives maximum sensitivity (highest A for a given c). Use this idea to explain why specific wavelengths are chosen in textbook examples.

3. Think About the "Goldilocks" Absorbance Range

Conceptually, A values around 0.2–1.5 are often ideal for accurate measurements (not too low, not too high). If a calculated A is outside this range, consider how you'd dilute or concentrate the sample in a real scenario (though remember, this calculator is for homework math, not experimental design).

4. Practice Reverse Calculations to Build Flexibility

Don't just solve for c every time. Occasionally solve for ε (given A, b, c) or for b (given A, ε, c). This flexibility helps you tackle diverse exam questions and deepens understanding of how all variables interrelate.

5. Connect to Calibration Curves

Understand that plotting A vs. c (at fixed ε and b) gives a straight line through the origin (slope = ε × b). This is the basis of calibration curves in analytical chemistry. The calculator helps you verify individual points on such a curve and reinforces the linear relationship conceptually.

6. Appreciate Logarithmic vs. Linear Scales

Absorbance (A) is linear with concentration, but transmittance (T) is exponential (A = −log T). Understanding this difference helps you interpret why small changes in T at low T values correspond to large changes in A, and why %T scales aren't as intuitive as absorbance for concentration determination.

7. Use Dimensional Analysis as a Check

Track units through Beer–Lambert calculations: [L·mol⁻¹·cm⁻¹] × [cm] × [mol/L] = unitless (correct!). If your units don't cancel properly, you've made an error in setup. This discipline catches mistakes before you even calculate numbers.

8. Recognize When Beer–Lambert Might Not Apply

In advanced problems, be aware of deviations at high concentration (molecular interactions), in turbid samples (scattering), or with polychromatic light. For homework, these are usually ignored, but knowing the limitations prepares you for upper-level analytical chemistry and real lab work.

9. Compare Multiple Substances' ε Values Conceptually

If two dyes have ε values of 1 × 10³ and 1 × 10⁵ L·mol⁻¹·cm⁻¹, the second is 100× more absorbing. Use the calculator to see how much lower concentration is needed for the high-ε dye to give the same absorbance. This builds intuition for sensitivity and detection limits.

10. Use the Calculator as a Learning Tool, Not a Crutch

In exams, you'll work problems by hand. Practice the algebra and arithmetic manually first, then use the calculator to verify your answers. This dual approach—manual practice + verification—builds true mastery and confidence in applying the Beer–Lambert law.