MathBench- Australia
Straight Lines
December 2015
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Measurement:
Straight Lines/ Standard Curves
URL: http://mathbench.umd.edu/modules-au/measurement_beerslaw/page01.htm
Learning Outcomes
After completing this module you should be able to:
 Describe the linear relationships between two parameters
 Construct and use a standard curve for determining the concentration of a unknown solution using
spectroscopy
 Use Beer’s Law and the gradient of a standard curve to determine the concentration of an unknown
solution using spectroscopy
Why do we use line graphs in biology?
Line graphs are one of the most common ways that trends or relationships are shown between one or more
variables in experiments involving numbers. They allow us to immediately determine the effect of changing one
of these numerical variables on another numerical variable you can measure. Though this sounds complicated
don’t panic, you have come across this before.
For example, on the “The Biggest Loser” TV show, it would be interesting to show the rate of weight loss of
contestants. This could be easily displayed by plotting the weight loss of the contestants at the various “weigh
ins”. As all the data collected is numerical (i.e., time, since starting the show and weight) a line graph could be
used to show the differences in the rate of weight loss between contestants.
As the show’s producers decide when “the weigh ins” are going to happen, time is known as the “independent
variable”. That is, the variable that the experimenter controls. The weight of the contestant is dependent on
what day is chosen so is known as the “dependent variable”. Convention says that we plot the independent
variable (in this case, time) on the bottom or “x- axis” and the person’s weight loss on the “y - axis” (dependent
variable).
Ok now, we have all been on diets. I personally would want my graph of weight loss to be as steep as possible
and never flattening out until my goal weight, but of course this does not happen. Line graphs can therefore show
when a person is losing weight quickly (fast rate) and when they are plateauing (zero rate). Line graphs are used
because it is very easy to interpret rates of change between variables.
MathBench- Australia
Straight Lines
December 2015
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As you are now aware, line graphs are not always straight lines. If you do collect data points that allow you to
draw a straight line, you can say there is a correlation between the two variables or a linear relationship. In
science, there are lots of linear relationships, for example the length of the femur (a leg bone) and the height of
humans, the excretion of Vitamin B2 in urine vs dose etc. In biology, often the first time you deal with linear
graphs is using spectroscopy where at low concentrations of chemicals there is a linear relationship between the
absorbance of light through a solution and its concentration.
Straight Lines-you have met them before!
A linear relationships means that as one thing increases or decreases, the other variable increases or decreases
by the same proportion. If you are paid an hourly rate in your job you are acutely aware of this relationship
because:
If you work:
• twice as many hours in a week, your pay is twice as much.
• only 50 % of the hours, you get 50% of your usual pay.
• no hours, you don't get paid.
The relationship between the hours worked (independent variable) and your pay (dependent variable) is linear
and if graphed would give you a straight line. The steeper the graph, the higher your hourly rate. Have a look at
the graph to the right:
In all cases, the person’s pay is directly proportional to the work hours – double the hours, double the pay. But
some people are accumulating money a lot faster than other people. That is reflected in the steeper slope/gradient
of some lines (like Bill Gates).
MathBench- Australia
Straight Lines
December 2015
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In fact, you can look at a graph such as the one below and work out how much each person makes per hour. (In
other words, “the change in money per time” or “rate of pay). From the graphs above can you work out the
hourly rate of a Pizza guy and a PC tech:
Pizza Guy: $ 20 per hour
PC tech: $ 45 per hour
You should have recognized that the "change in money per time" as the same as the change in y value divided by
the change in x, which is the classic formula for slope or gradient of a line.
If you can engrave this on your brain, you'll have a head start interpreting lots of graphs whether they are straight
lines or not. The slope at any part of a graph tells you how fast the process is happening right then.
This graph represents the sales of pizza slices over a period of 4 hours. At what point in time is pizza income
increasing at the fastest rate?
Click the buttons to see how slope and rate are related:
MathBench- Australia
Straight Lines
December 2015
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A straight line has only one slope or gradient
A straight-line graph is a special type of curve. It has the same slope everywhere so when determining the slope
it does not matter where in the line you calculate it.
For example, to find out how much a Lab Tech makes per hour, there are several ways to do it.

You could measure how much they made in any single hour – such as they made $60 in two hours
therefore $30 dollars an hour.

If you want to get fancy, you could pick a different hour, but it seems more work for the same answer.
For example, between hours 2 and 3, the Lab Tech earned $90 - $60 = $30 extra dollars: Remember
when people want to find a slope, they often draw a figure that looks like that indicated in the graph
below. This is will result in the same value for the rate of pay.
Hint: As long as the line goes through (0,0), it’s much easier to find the slope by just finding the value of y
at a particular x value and dividing y by x. Remember what you are determining is the slope or gradient of the
line.
What is the pay rate for the forest conservation worker? How about the veterinarian?(All wage data taken from
Monster.com)
MathBench- Australia
Straight Lines
December 2015
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You may remember from high school that the equation of a straight line is:
The slope (known as ‘m’) is that rate or gradient we were talking about. The intercept (b) is just the value for y
where the line crosses the y axis.
Linear relationships and Spectros
A common linear relationship used in biology is correlation of the amount of light that some chemical solutions
absorb with their concentration.
You see examples of the linear relationship of light absorbance and concentration all the time. If a parent makes
up a red cordial drink at a low concentration, children are very quick to complain. That’s because children
immediately use their eyes as a spectrophotometer and realize that the amount of light absorbed by the drink is
different to what they were expecting. In fact, if the parents have halved the cordial concentration, then the
absorbance of the solution would be half, with the drink looking less dark.
A spectrophotometer (affectionately known as “a spectro”) does this same job as the children’s eyes but puts a
number on the absorbance (ahh, a numerical variable we can measure!!). The higher the absorbance value, the
less light that can get through the solution.
MathBench- Australia
Straight Lines
December 2015
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Key Knowledge: A spectrophotometer is a device that measures how much light a solution absorbs,
which often has a linear relationship to the concentration of some chemicals in the solution.
For a standard curve (i.e., a line graph that can used to find out the concentration of an unknown solution), the
experimenter decides on the concentrations to be measured (independent
variable) and the output from the spectrophotometer is the value for the dependent variable (called Optical
Density or OD). Thus you end up with a graph like this:
Yeah, a linear relationship!! If at higher concentrations the curve does flatten off, don’t panic as this is often is
due to the limitations of the instrumentation used at high absorbance values. You should not use data in this
region of the graph for calculations.
So there you go, if your parents are trying to rip you off when making up your favorite drink, you now have the
science to do an experiment and come back with some objective data to prove your case!
In other words, the OD increases by 0.2 units for every additional 1 mM of cordial. What is the slope of the graph
above?
Before you can even start to answer this, you need to think about units. So, y is measured in OD units, and x is
concentration, which ranges from 0 to 5 mM. A useful unit for x would be "mM". Thus the units of the slope are
"OD / mM"
Don’t stress if you are lost at this point–below is the nuts and bolts of using standard curves which does not
involve much maths. You don’t have to understand all the maths behind standard curves to be able to use
them!! As biologists, we must be able to be able to construct or set up experiments to generate data for a
standard curve and use them appropriately.
MathBench- Australia
Straight Lines
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Standard curve: Choosing a wavelength of light
Before we use the spectro there is one other thing we need to know: the wavelength of light the compound (in
this case the cordial) absorbs. In most experiments, this is known and is provided in the methods. The
wavelength can be different for different chemicals. Do you remember the spectrum of light?? Water doesn't
absorb any light, so all wavelengths (colors) get through, and mixed together, they look like white (= colorless)
light. Our cordial solutions look red because the solution absorbs more of the other colors of light. The more
concentrated the drink, the more non-red light that gets absorbed–and correspondingly, what does get through
looks redder.
The online version of this module contains an
interactive applet that allows you to practice with
virtual spectrophotometer. To find this applet, go to:
http://www.mathbench.umd.edu/mod105_1b_beerslaw
_TOC/page06.htm
To test the concentration of our cordial, you would need to set your spectro to measure something other than a red
wavelength. So in general, before you use a spectro, you have to figure out what wavelength works the best. If
you remember from first year chemistry you could run an absorbance scan of the solution across all wavelengths
to select the one that absorbs the best, or use a trial and error approach. Remember, unlike our cordial solution,
some chemicals absorb UV light, so by the naked eye solutions look clear, however they can be measured by
selecting a wavelength in the UV range. After you've figured out the best wavelength to use, you can proceed to
the next step, actually gathering data to construct a standard curve.
Standard curve–making and using
You need to produce a graph showing the linear relationship between the OD of the solutions (eg cordial) at
various concentrations. That means you will get a line graph similar to that shown below (it may flatten out at
higher ODs). As the experimenter, you decide what concentrations will be tested and the spectro will measure
the ODs of those samples. You should also measure the OD of the unknown (in the case the diluted cordial drink
made up by the parents) at the same time. Then plot the OD values for the concentrations you have chosen on a
graph (do not use the data on your unknown sample at this point). You need to draw a “line of best fit” through
the data points.
MathBench- Australia
Straight Lines
December 2015
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Now, when I say "draw a line of best fit", please DON'T think this means simply connecting the dots as if you
were doing a dot-to-dot puzzle. Instead, you want a single straight line that goes approximately though the center
of your group of dots (see graph above), so some dots are above and some are below the line, but all are as close
as possible. (In statistical jargon, this is known as "doing a linear regression’)
So back to constructing a standard curve for cordial. As we don’t know the concentration of the cordial in
molarity, and the manufacturers say you should be using a 20% (v/v) solution of the concentrate, the units of the
“x axis” in this case is in %.
You would take a sample of cordial, dilute it to 10% (v/v), read the absorbance using the spectro, and mark
your data point on the graph. Then you need to do the same with a 20% (v/v) sample, a 30% (v/v) sample, and a
40% (v/v) sample. Your graph should look something like that shown below.
So, for every 20% (v/v) change in concentration, OD increases by about 1.0 unit. The slope of the line is 1.0/20,
or 0.05, which is also the value of "e", the extinction coefficient (more on that later).
So if the kids want to accuse their parents of diluting the cordial too far, they can take their drinks, measure and
OD and use the standard curve to determine the concentration of the kid’s drinks.
MathBench- Australia
Straight Lines
December 2015
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Using a standard curve–minimum maths method
So if the kids want to accuse their parents of diluting the cordial too far, they can take their drinks, measure and
OD and use the standard curve to determine the concentration of the kids' drinks.
For example, if the OD of the drink were 0.8, you would move along the “y axis” to an OD of 0.8 and then draw
a line across to the standard curve, then a perpendicular line and read the value of the “x axis”. You will get the
answer approx 15% (v/v).
So if you have a spectro in your garage and do the above experiment you now can make a very solid case that
your parents are over diluting your drink.
Beer's Law: explains the relationship between cordial concentration and
light
Beer’s Law uses a maths approach to explain the experimentally obtained standard curve. Luckily, the
relationship is linear, and all we need to figure out is the slope from the standard curve.
So let's write the equation of a line,
keeping in mind that ...




Instead of x, we say C (for concentration).
Instead of y, we say OD (for optical density, which is the technical way of saying “how much light the
cordial absorbed”).
Instead of m, we say e (which stands for “extinction coefficient”, i.e., the amount of light that is
extinguished by each extra bit of cordial).
We know that if there is no cordial in the solution then there is no absorbance so the intercept (b) equals
zero
Just to make it a little trickier, the amount of light absorbed also depends is the distance the light has to pass
through the cordial solution. So if I had a glass of 10 cm diameter, then the absorbance would be twice that of a
glass with 5 cm diameter (if I was looking through the side of the glass). Another linear relationship!
MathBench- Australia
Straight Lines
December 2015
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So, to be really correct, we have to write the equation like this:
where "l" stands for the “length of the path” that light follows.
Luckily, those people who designed the spectros were pretty smart: they keep the distance that light has to travel
constant at 1 cm, which means in practice we can just use the easier equation, OD = ec (that is why the cuvettes
have a 1 cm width!) The equation above is known as Beer's Law. Yes, there was a person named Beer (Herr
Professor Beer, actually).
In most cases we don’t know the rate at which light gets absorbed for compounds in solution at different
concentrations (“e” value or “extinction co-efficient”). We usually experimentally determine the “e” value via a
standard curve. This is where the graph bit comes into play!! The slope/gradient or rate of a standard curve is
the “e” value.
Are you lost? Let’s recap!!


The “e” value in the equation above is commonly known as the extinction coefficient that we can use to
determine the concentration of a chemical in solution. ,
The value of e is the slope/gradient/rate of a line graph of an OD vs concentration graph ie a standard
curve.
This is why we make standard curves. The only way to find the correct rate/gradient/slope is by measuring it.
Oh, now I know why we do so many standard curves in biochemistry!!
So now we have a second way to determine the cordial concentration using the standard curve.
The “plug and chug” method.
It is as easy as determining the slope of the line from the standard curve and using it in the formula: OD=ec
So using this method the kid’s cordial had an OD of 0.8.
The slope of the line is 5 so 0.8 = 5  c so c=0.8/5 = 16% (v/v) rather than the 25% (v/v) recommended
by the manufacturer. The almost the identical result as when we read it off the graph.
MathBench- Australia
Straight Lines
December 2015
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It is exactly the same in the lab-try it yourself
John and Suzy want to measure the concentrations of a protein (Bovine Serum Albumin, BSA) in solution. After
finding the proper wavelength and the ODs of 1 mg/mL, 2 mg/mL, 4 mg/mL, and 8 mg/mL BSA solutions were
obtained. They then measured the ODs of two unknown BSA solutions. What were the concentrations of the
protein solutions?
Table1: Data collected for ODs of solutions at various BSA concentrations
BSA concentration
(mg/mL)
1
2
4
8
Unknown 1
Unknown 2
OD (280 nm)
0.3
0.6
1.2
2.4
0.7
1.5
First, what is “e”?





Remember that “e” is the rate at which stuff in the water absorbs light.
Slope = rate
Find the slope of the graph.
How much does the absorbance change as you go from 0 to 8 mg/mL protein?
Since Beer's Law says the relationship is DIRECTLY PROPORTIONAL, then the absorbance
will change 1/10 th as much from 0 to 1mg/mL as from 0 to 10 mg/mL.
Answer: e = 0.3 OD /mg/mL.
So far, we know that e = 0.3 OD/mg/mL. Fitting that into our equation, we get
OD = 0.3 c
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Straight Lines
December 2015
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Now that you have “e”, what are the concentrations of the three protein samples (their
ODs were 0.7 and1.5)
 you can write an equation for concentration based on absorbance.
 To get the equation for concentration, do this:
OD = e c
c = OD / e
c = OD / 0.3
 Use Google as a calculator!
Answer: 0.7/0.3 = 2.3 mg/mL, 1.5/0.3=5 mg/mL,
Summary

You have learnt how to construct and use a standard curve to determine the concentration of an unknown
solution.
 When you use a spectrophotometer, you need to know the wavelength of light your chemical of interest
absorbs at.
 You need to determine the “e” value for your compound. This process of calibration is also called
“creating a standard curve”. That is “standard” as in something you can measure against, and “curve” as
in a function drawn on a graph.
Once you have the standard curve, you can use it in one of two ways:
 Simply read the concentration from the curve itself, by measuring the OD of the unknown and using this
value to find the x value.
 Use Beer’s Law (OD = ecl) once you have determined the slope of your line on a standard curve
Learning Outcomes
After having completed this module you should now be able to:
 Describe the linear relationships between two parameters
 Construct and use a standard curve for determining the concentration of a unknown solution using
spectroscopy
 Use Beer’s Law and the gradient of a standard curve to determine the concentration of an unknown
solution using spectroscopy.