MathBench- Australia Measurement: Straight Lines 1 Measurement: Straight Lines/ Standard Curves URL: http://mathbench.org.au/measurement/straight-lines-standard-curves/ 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.
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Measurement: Straight Lines/ Standard Curvesmathbench.org.au/.../2015/11/Measurement-Straight-Line1.pdf · 2016-09-22 · As you are now aware, line graphs are not always straight
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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).
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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 recognised 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?
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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)
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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 realise 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.
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
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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.
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.
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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 (e.g. 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.
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’)