Difficulty: Intermediate | Prerequisites: CHM 113 (light, energy, wavelength, frequency basics from Week 4)
Big Picture
Spectrophotometry sits at the intersection of light physics and chemical analysis. It gives you a way to measure how much of a substance is dissolved in a solution, without doing any wet chemistry beyond preparing the sample. The underlying principle, Beer's Law, connects the amount of light a solution absorbs to the concentration of the absorbing molecule. You will use this technique repeatedly in chemistry, biology, and physics labs, so it is worth understanding properly rather than just memorising the formula.
When light passes through a solution, the molecules in that solution absorb some of it. Beer's Law (A = εlc) tells you that the amount of light absorbed is directly proportional to the concentration of the solution and the distance the light travels through it. By measuring absorbance with a spectrophotometer and plotting a standard curve, you can work backwards to find the concentration of an unknown solution.
Wavelength (λ, lambda)
The crest-to-crest distance between successive waves, measured in metres (or nanometres for visible/UV light). In simple terms, it is the physical length of one complete wave cycle.
Frequency (ν, nu)
The number of complete oscillations a wave makes per second, measured in Hertz (Hz). Think of it as how many times the wave "pulses" each second.
Speed of light (c)
The velocity at which light travels in a given medium. In a vacuum, c = 2.998 × 10⁸ m/s. For most CHM 114 purposes, c = 3.00 × 10⁸ m/s in air. In simple terms, this is a universal constant you will plug into wave equations.
Photon
A single particle of light carrying a discrete packet (quantum) of energy. Think of it as the smallest possible "unit" of light.
Planck's constant (h)
A fundamental constant relating a photon's energy to its frequency: h = 6.626 × 10⁻³⁴ J·s. It is the proportionality factor in E = hν.
Quanta
Discrete packets of energy. Molecules absorb or emit energy only in these fixed amounts, not continuously. In simple terms, energy comes in specific "portions," not a smooth stream.
Spectrum (absorption spectrum)
A graph of how much light a substance absorbs at each wavelength (or frequency). Every molecule has a unique spectrum, like a fingerprint. This is how you identify which wavelength to use for a given molecule.
Transmittance (T)
The fraction of incident light that passes through a sample without being absorbed: T = I_E / I_O. Think of it as the percentage of light that makes it out the other side.
Absorbance (A)
A logarithmic measure of how much light a sample absorbs: A = –log₁₀(T). In simple terms, higher absorbance means less light gets through.
Beer's Law (Lambert-Beer Law, Beer-Lambert Law)
The relationship A = abc (or A = εlc), stating that absorbance is proportional to concentration, path length, and a molecule-specific constant. This is the central equation for spectrophotometric analysis.
Molar absorption coefficient (ε, epsilon)
Also called the molar absorptivity or extinction coefficient. A constant specific to a molecule at a given wavelength, with units of M⁻¹ cm⁻¹. It tells you how strongly that molecule absorbs light. Think of it as the molecule's "absorption strength rating."
Path length (b or l)
The distance the light travels through the sample, usually the width of the cuvette, typically 1 cm. In simple terms, a longer path means more molecules in the way, so more absorption.
Cuvette
The small, transparent container that holds the sample inside the spectrophotometer. It must be clean and scratch-free; defects introduce error.
Standard curve (calibration curve)
A graph plotting absorbance against concentration for several known (standard) solutions. You read an unknown sample's concentration off this line. Think of it as a cheat sheet you build so you can look up concentrations from absorbance readings.
Standard solutions
Solutions of precisely known concentration used to build the standard curve.
Light behaves as both a wave and a particle (photon). CHM 113 covered this in Week 4.
Wavelength (λ) and frequency (ν) are related to the speed of light by: Speed = frequency × wavelength, or c = λν.
Because c is constant in a given medium, a shorter wavelength means a higher frequency, and vice versa.
Each photon carries energy proportional to its frequency: E = hν.
Higher frequency (shorter wavelength) = more energy per photon.
Lower frequency (longer wavelength) = less energy per photon.
Molecules absorb or emit energy only in discrete amounts (quanta), not continuously.
A photon is absorbed only when its energy exactly matches the gap between two energy levels in the molecule.
Every molecule has a characteristic absorption spectrum: a plot of how much light is absorbed at each wavelength.
Structurally similar molecules (e.g. NAD⁺ and NADH) can have very different spectra.
This difference is exploited in kinetics experiments to track reactions.
For CHM 114, the focus is on absorption, not emission.
Transmittance (T) is the fraction of light that passes through the sample: T = I_E / I_O.
I_O = intensity of incident (incoming) light.
I_E = intensity of exiting (transmitted) light.
Three factors affect transmittance:
Concentration of the solution. More molecules = more photons absorbed = lower transmittance.
Path length (b). A longer path through the sample means light encounters more molecules.
Molecular properties. Different molecules absorb different wavelengths at different efficiencies (captured by the constant a or ε).
Pierre Bouguer (1729) discovered that transmittance relates logarithmically, not linearly, to concentration and path length.
The reason: as light travels through the sample, each successive layer absorbs some of what remains. The light intensity drops in each layer, so the fraction absorbed per layer is always changing.
This is the same mathematical pattern as exponential decay.
Johann Heinrich Lambert (1760) rediscovered the exponential relationship.
August Beer (1852) defined absorbance and showed it is linearly related to concentration.
Absorbance is defined as: A = –log₁₀(T).
When no light is absorbed: T = 1.00, A = 0.
When 90% of light is absorbed: T = 0.1, A = 1.
The Lambert-Beer Law: A = abc (general form) or A = εlc (when using molar units).
a (or ε) = absorption constant specific to the molecule and wavelength.
b (or l) = path length in cm.
c = concentration of the solution.
Absorbance is unitless, so the units of a must cancel the units of b and c.
Method 1: Known molar absorption coefficient (ε)
If ε is listed in the literature for your molecule at the wavelength you are using, plug it in along with the path length and measured absorbance, then solve for c.
Example: cuvette = 1 cm, ε = 3600 M⁻¹cm⁻¹, A = 0.650 → c = 0.650 / (3600 × 1) = 1.80 × 10⁻⁴ M.
Limitation: this assumes your instrument is calibrated the same way as the one used to determine ε, and that you have made no systematic errors.
Method 2: Standard curve (calibration curve)
Prepare several solutions of known concentration (standards) and measure the absorbance of each.
Plot absorbance (y-axis) vs. concentration (x-axis). The result should be a straight line through the origin.
The slope of this line equals a (or ε × b).
To find the concentration of an unknown:
Quick method: read its absorbance off the y-axis, trace horizontally to the line, then down to the x-axis.
Precise method: use the line equation (y = mx + 0), substitute A for y, solve for x.
Standard curves are preferred in practice because they automatically account for instrument-specific errors and systematic mistakes.
Beer's Law holds only for dilute solutions. At high concentrations the linear relationship breaks down.
Common lab errors: dirty or scratched cuvettes, poorly mixed solutions, incorrect pipetting, wrong wavelength setting.
c = \lambda \nuSpeed of light equals wavelength times frequency. c = 3.00 × 10⁸ m/s in air.
E_{\text{photon}} = h\nuEnergy of a single photon equals Planck's constant times frequency. h = 6.626 × 10⁻³⁴ J·s. Note: this gives the energy of one photon, not one mole of photons.
A = -\log_{10}(T)Absorbance is the negative base-10 logarithm of transmittance. T = I_E / I_O.
A = abcGeneral form of Beer's Law. a = absorption constant, b = path length (cm), c = concentration.
A = \varepsilon l cMolar form of Beer's Law. ε = molar absorption coefficient (M⁻¹cm⁻¹), l = path length (cm), c = concentration in molarity (M). Absorbance (A) is unitless.
Spectrophotometry is the standard method for measuring concentrations in clinical labs (blood glucose, haemoglobin), environmental monitoring (water pollutant levels), and biochemistry research (protein and DNA quantification). The NAD⁺/NADH example from the lecture is a case in point: because these two molecules absorb at different wavelengths, you can track enzyme-catalysed reactions in real time by watching the absorbance change at 340 nm.
Students often think transmittance and absorbance are linearly related. They are not. Absorbance is the negative log of transmittance, so the relationship is logarithmic.
Students sometimes assume Beer's Law works at any concentration. It does not. The linear relationship between absorbance and concentration breaks down at high concentrations.
E = hν gives the energy of one photon, not one mole of photons. If an exam question asks for the energy per mole, you must multiply by Avogadro's number.
Students frequently confuse the molar absorption coefficient (ε) with the general absorption constant (a). They are the same concept, but ε specifically requires concentration in molarity and path length in centimetres, giving units of M⁻¹cm⁻¹.
⚠️ You will almost certainly be asked to solve for concentration using A = εlc. Know how to rearrange the equation for any variable.
⚠️ Standard curve problems are common. Be ready to read a concentration off a graph or calculate it from a line equation (y = mx).
⚠️ Expect a question testing whether you understand why absorbance (not transmittance) is linearly proportional to concentration.
⚠️ Unit analysis is heavily tested. If ε is in M⁻¹cm⁻¹, concentration must be in M and path length in cm. Watch for questions that give concentration in mg/mL or path length in mm to see if you convert correctly.
⚠️ Know the three factors that affect transmittance: concentration, path length, and molecular properties.
True or false: Absorbance is directly proportional to transmittance. (False. A = –log₁₀(T), so the relationship is logarithmic.)
Fill in the blank: The slope of a standard curve (absorbance vs. concentration) equals ______. (The absorption constant a, or ε × b when using molar units.)
True or false: Beer's Law is valid at all concentrations. (False. It applies only to dilute solutions.)
Fill in the blank: If T = 0.01, then A = ______. (A = –log(0.01) = 2.)
True or false: Increasing the path length of a cuvette will increase the measured absorbance. (True. A longer path means light encounters more molecules.)
Q: A solution has a transmittance of 0.25 at 540 nm. What is its absorbance?
A: A = –log₁₀(0.25) = 0.602.
Q: A dye has ε = 1200 M⁻¹cm⁻¹ at 490 nm. You measure an absorbance of 0.360 using a 1.00 cm cuvette. What is the concentration of the dye?
A: c = A / (εl) = 0.360 / (1200 × 1.00) = 3.00 × 10⁻⁴ M.
Q: A student prepares a standard curve and obtains the line equation y = 2.5x. An unknown solution gives an absorbance of 0.75. What is the concentration?
A: 0.75 = 2.5x, so x = 0.30 M.
Q: Explain why a standard curve is preferred over using a literature value of ε to determine concentration.
A: A standard curve accounts for instrument-specific calibration differences and systematic errors (e.g. dirty cuvettes, pipetting bias) because both the standards and the unknown are measured on the same instrument under the same conditions.
Q: A student doubles the concentration of a solution. Assuming Beer's Law holds, what happens to the absorbance?
A: The absorbance doubles. Beer's Law states A = εlc, and absorbance is directly proportional to concentration.
Q: Why does Beer's Law use absorbance rather than transmittance for relating light measurements to concentration?
A: Transmittance varies exponentially (logarithmically) with concentration and path length, which makes calculations harder. Absorbance, defined as –log₁₀(T), converts that exponential relationship into a linear one, making it straightforward to graph and calculate.
This material connects directly to chemical kinetics (CHM 114, Week 8): you will use spectrophotometry to monitor reaction rates by tracking absorbance changes over time, exploiting the fact that reactants and products often absorb at different wavelengths (as with NAD⁺ and NADH).
The wave-particle duality and quantised energy levels link back to atomic structure and quantum mechanics from CHM 113. Understanding why molecules absorb only specific wavelengths requires the same energy-level model you used for electron configurations.
Beer's Law and standard curves are foundational tools in analytical chemistry and biochemistry. Any time you need to measure how much of a substance is present in solution, from protein assays to environmental water testing, you are applying these principles.
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