## Wednesday 6 October 2010

### The Beer-Lambert Law, a straight line and the units of the extinction coefficient

If you are struggling with 'science maths' then have a look at Maths4Biosciences - there is also a course on Beer-Lambert Law and Spectrophotometry.

If you would like to test your skills working with the Beer-Lambert Law then you might like to look at the Spectrophotometry tests at: Maths4Biosciences.com.

Some students struggle to understand the relationship between the Beer-Lambert Law and a straight line, and working out the units of the extinction coefficient (ε).

The Beer-Lambert Law states:

A = ε . c . l

Where:

A = absorbance
ε = extinction coefficient
c = concentration
l = path length (i.e. the distance the light travels through the sample)

So, what connection between this and a straight line, and what are the units of the extinction coefficient?

The units of the extinction coefficient

In my opinion, the extinction coefficient has some of the craziest units out there.
Absorbance (A) has no units so the units of the extinction coefficient (ε) are determined by how the concentration (c) and path length (l) are being measured. That is, the units of the extinction coefficient must cancel out the units of the concentration and path length so that the absorbance can have no units!

A worked example.

A = ε . c . l

A = absorbance, units - so put in 1
ε = extinction coefficient, units - unknown
c = concentration, units - milli-Molar, mM
l = path length (i.e. the distance the light travels through the sample), units - cm

So...

A = ε . c . l

And, with units:

[1] = ε . [mM] . [cm]
Rearranging...
[1] / ε = [mM] . [cm]

ε = 1 / ([mM] . [cm])

or

ε = [mM]-1 . [cm]-1

So, the units are mM-1 . cm-1

This can be checked by putting it all back together:

A = ε . c . l

A = ([mM]-1 . [cm]-1) . [mM] . [cm]

This gives:

A = [mM]/[mM] . [cm]/[cm]

So, the mM and the cm cancel each other out, leaving no units for absorbance A.

A straight line

The Beer-Lambert Law:

A = ε . c . l

Where:

A = absorbance
ε = extinction coefficient
c = concentration, units
l = path length

The equation for a straight line is:

y = mx + c

Where:

c = the y-intercept

If you plot concentration, against absorbance, then x = concentration and y = absorbance. Plus, from the Beer-Lambert Law, we know that if the concentration is zero, then absorbance must be zero.

A = ε . c . l
A = ε . 0 . l
A = 0
So...

y = mx + c
absorbance = m . concentration + c

From above, if concentration = 0, then absorbance = 0, hence c must be zero
y = mx + c
absorbance = m . concentration + c*
0 = m . 0 + c
c = 0

(* note, this c is the y-intercept and not the concentration)

Therefore...
y = mx + c
absorbance = m . concentration + 0
or
y = mx

Comparing:

y = mx
absorbance = m . concentration

With (and rearranging):

A = ε . c . l

A = (ε . l) . c

y = m . x
It can be seen that if y = absorbance, and x = concentration, then m (the gradient) must equal extinction coefficient (ε) multiplied by the path length, l, or ε . l. As l is typically 1 cm, then the gradient, m, must equal the extinction coefficient (ε).

If you are struggling with 'science maths' then have a look at Maths4Biosciences - there is also a course on Beer-Lambert Law and Spectrophotometry.