When light passes through a medium, the intensity of light that successfully makes it through depends on the chemical composition of the medium, how concentrated the medium is, and how far the light travels through it.

Light passing through a medium can be thought of as having a small chance of being absorbed for each small step it takes through the medium. This is described by a differential equation:

\(\frac{dI}{dx} = -kcI\)

where \(I\) is concentration, \(x\) is distance traveled, \(c\) is concentration, and \(k\) is a proportionality constant.

Solving this differential equation yields the following equation:

\(-\log_{10}({\frac{I}{I_o})} = \varepsilon c l\)

The logarithmic ratio of the transmitted light intensity \(I\) to the incident light intensity \(I_o\) is a quantity known as the absorbance.

\(A = -\log_{10}({\frac{I}{I_o})}\)

This is the Beer-Lambert law, which states that absorbance \(A\) is proportional to both concentration \(c\) and path length \(l\) by the molar absorptivity constant \(\varepsilon\) specific to the medium and the wavelength.

\(A = \varepsilon l c\)

A standard curve is a plot of measured absorbances for samples of known concentration. Here, the plot was obtained from serial dilutions of red food dye.

Standard curves are used to estimate unknown concentrations from known ones. According to the Beer-Lambert Law, the relationship between absorbance and concentration should be linear, so simple linear regression can be used:

\(A = (\varepsilon l) c \hspace{0.5cm} \Rightarrow \hspace{0.5cm} A = \beta_0 + \beta_1 c + \varepsilon; \hspace{1cm} \varepsilon \sim \mathcal{N}(0;\sigma^2)\)

For measurements taken at a constant path length \(l\) and in the same medium, with a fixed molar absorptivity constant \(\varepsilon\), absorbance \(A\) should be directly proportional to concentration \(c\) by some parameter \(\beta_1\) with error \(\varepsilon\). A constant \(\beta_0\) accounts for any baseline reading.

The red food dye standard curve can be fitted with a linear regression model.

If we have a sample of unknown concentration in the same medium, we can estimate its concentration using the standard curve.

# Measured absorbances of samples of unknown concentration
unknowns = data.frame(Absorbance = c(0.8, 1.2, 0.1))

# Linear regression model
model1 <- lm(data = absorbance1, Absorbance ~ Concentration)

# Extract coefficients
beta0 <- coef(model1)[1]
beta1 <- coef(model1)[2]

# Invert to get concentration in terms of absorbance
unknowns$Concentration <- (unknowns$Absorbance - beta0) / beta1

From the linear model, we obtain the following concentrations:

unknowns

##   Absorbance Concentration
## 1        0.8     48.152245
## 2        1.2     74.416642
## 3        0.1      2.189549

These concentrations can be plotted against the standard curve.

Is it really necessary to make a new standard curve whenever we want to test a sample? How well does the molar absorptivity constant hold up across samples of the same medium? To test this, we can plot a different batch of red food dye whose absorbances were measured at different times.

We can plot both datasets together to see how well the linear models align.

The models do not align very well; regression from one batch would almost certainly provide different predictions than the other, even when both were measured at 520 nm. External factors make it necessary to reconstruct a new standard curve whenever a new medium is used.

The Beer-Lambert law provides a valuable way to determine the concentration of an unknown sample using linear regression, and it has applications ranging from materials science to cellular biology!

\(A = \varepsilon l c\)