Chapter 5 Problems
1. You are working in the laboratory analyzing samples for nitrogen isotope
compositions (15N), and find that smaller samples seem to give more trouble
than larger samples. Investigation shows that some small amounts of nitrogen
enter the samples unavoidably as part of the oxygen used to combust samples.
This small amount of extra nitrogen contributes contamination that you can
measure as 1 g of N. You investigate the effect of this blank by analyzing a size
series of standards with 1,2,4,8,16,32,64 and 128 g of N. The standards have a
known isotopic composition of 0o/oo, and you estimate the blank isotope
composition is 20o/oo. What are the predicted values of these samples that are a
mixture of blank and pure sample? Make a table and graph of your results, then
explain them.
2. Continuing with problem 1, plot your  isotope results vs. 1/(total amount of N in
sample) as (x,y) data in the form (1/mass, ). Note that this is a form of the
“Keeling plot” discussed in section 3.5. Fit a line to the data; what is the intercept
of the line? Explain this result. Consult section 5.7 and give the equations that
apply for mixing when one source is fixed, then rearrange these equations to show
how they describe a straight line with an interesting intercept. Which of the
following names gives the correct description of the curve for the original (mass,
) data of the previous problem: linear, polynomial, power, or exponential?
3. You are sitting in a closed, non-ventilated room with other students. You get
sleepy, then start to wonder about the rising amount of CO2 in room air. This
begins to interest you, so you make some calculations. If there are 3 people each
adding 0.5 millimoles of CO2 to room air every hour, what are the concentrations
and isotope compositions of room CO2 you expect to see after 1, 3 and 10 hours?
Assume that the room initially contains 1 millimole of CO2, that initial air CO2
has a -8o/oo 13C value, and that human breath CO2 has an isotope composition of
-20o/oo. a) Give the results of your calculations in a table, b) make a graph
showing your time course results, and c) explain the results in a few sentences.
4. Here is a problem you should do first in an experiment at your kitchen table. Fill
two clear bottles to 100mls each with tap water. Add one drop of yellow food
coloring to one bottle, and one drop of blue food coloring to the other bottle.
What are the colors of solutions you make by mixing 1 part blue water and 1 part
yellow water, a 1:1 mixture? And more generally, what are the colors for the
blue:yellow series that are mixed in the following ratios: 1:0, 1:0.1, 1:0.5, 1:1,
0.5:1, 0.1:1 and 0:1? Keep samples of your mixtures for reference. Now make a
third solution, this time with a more concentrated yellow solution, 100mls of tap
water with 10 drops of yellow food coloring. What will be the colors of the
mixtures in the same dilution series where you mix the same blue solution with
the more concentrated yellow solution in the ratios 1:0, 1:0.1, 1:0.5, 1:1, 0.5:1,
0.1:1 and 0:1? Make a table and graph that compares the color results for both
experiments and explain why they are different. With the mixture series, calculate
“fraction blue solution by volume” and put this in your table and use it as your xaxis in your graph. Finally, add table salt (NaCl) to the blue bottle so that it easy
5.
6.
7.
8.
9.
to taste the saltiness, mimicking salty-tasting seawater that has an average salt
concentration of 35g/l. Using the concentrated yellow solution without salt as
freshwater, what is the saltiness of samples in the same mixture series? (Taste the
water to evaluate “saltiness”). Add your results to your graph. Once you have
done these experiments at home, repeat them in a spreadsheet mixing model
where isotope values represent the different blue and yellow sources, e.g. 5o/oo =
blue and 15o/oo = yellow. Give the isotope values of the mixtures in a table and
as a graph. For the last salt experiment, use 35g/l as the salt concentration in the
blue solution and 0g/las the salt concentration in the yellow solution. Give your
results in tabular and graphic form, then summarize the results in words. Overall,
this experiment mimics well what happens in estuaries where salt water mixes
with freshwater, and isotope colors change in dramatic fashion for HCNOS
elements. Consult the following paper, and explain the general effects of
concentrations on isotope mixing in estuarine settings: Estuaries 25:264-271
(2002).
You are mixing two samples in equal amounts, 100g and 100g. The first sample
has a carbon isotope value of -10o/oo and the second sample a carbon isotope
value of -20o/oo. What is your predicted isotope value for the mixture? Then you
find out that the carbon concentration of the first sample is 10g/100g, but the
second sample has a carbon concentration of 1g/100g. What is the actual isotope
value of the mixture? Does the isotope value reflect proportional mixing by
weight or by amount of carbon? Why is this a weighted average problem? Read
the article Oecologia 130: 114-125 (2002) and answer this question: When is a
50/50 mixture not really 50/50?
You are working on a mixing problem for food webs with 2-dimensional isotope
data, using C and N for samples and three potential sources in a biplot with
the (C, N) data. In your graph, you draw in a mixing triangle by connecting
isotope values of three sources, but see that some of the samples have isotope
values outside the mixing triangle. Is this a difficulty for the three-source mixing
model? If so, give one or more reasons that samples can have isotope values
outside the mixing envelope defined by the measured sources.
Develop an I Chi mixing model in spreadsheet form for an animal that grows
without any losses. Start with an initial animal mass of 100 units and a  value of
10o/oo. At each time increment, add one unit of mass growth with an isotope
value of 0o/oo. Set up a spreadsheet that tracks the mass and isotope changes over
1000 time increments, until the animal grows to 1100 units. Calculate the isotope
values at each time increment, and graph the results for mass and  vs. time.
Explain the results. If you plot model results for each time point as (1/mass, ),
what shape or curve do the points describe, and what is the extrapolated  value at
the y-intercept? Explain your results.
Modify the model of Problem 7 to allow for loss of 0.5 units of mass at each time
step, with no fractionation or change in  value during loss. Graph the results and
compare with the Problem 7 results. What is the main effect you can see of
allowing losses in the models? Why does this effect occur?
Modify the model of problem 7 so that the initial mass is 10 instead of 100. How
do results change and why? If you leave the initial mass at 100, but increase the
loss to 0.8 from 0.5 at each time increment, what do you observe and why?
Lastly, if you plot model results for each time point as (1/mass, ), what curve
results, and what is the extrapolated  value at the y-intercept? Explain your
results, especially in comparison to the simpler results from Problem 7.
Chapter 5, Problem #10
Analyzing a Food Web with C and N Stable Isotopes
Practice helps, so here we practice interpreting C and N isotopes, using a
terrestrial food web example. Think of this as exercising your isotope powers, working
out or “isosizing for fun and fitness” in six steps below to plot up some food web data
and then interpret the plots. You will need to make some graphs and tables in this
intellectual work-out, then turn to the answers in the latter part of this Section to see how
you did. Ready?
We begin by gathering a few clues about a possible food web – a few leaves on
the ground, mushrooms, a worm and a centipede from the soil, spiders from the bushes
and bird feathers from sparrows and eagles. We dry the samples overnight at 60oC, grind
them to a fine powder with mortar and pestle, then send them off for analysis. The N and
C isotope results come back 3 months later, and now the work begins to interpret the
data.
Step 1. Make a table and an isotope biplot (x,y) graph with the stable isotope data
samples that have these respective 15N and 13C values: tree leaves 2o/oo and -27o/oo,
mushroom 3o/oo and -24o/oo, worm 4.2o/oo and -26o/oo, centipede 8o/oo and -21o/oo,
spider #1 5o/oo and -24o/oo, spider #2 10o/oo and -21o/oo, sparrow feather 8.5o/oo and 23o/oo, eagle feather 12o/oo and-18o/oo. Questions: Look at your (x,y) plots – are the
axes correctly labeled? Which is the x-axis, 15N and 13C, and why did you choose one
or the other? Can you make an initial assessment of who is eating whom?
Step 2. Assign trophic levels to your plot, using a 3o/oo 15N increase to indicate a 1
trophic level (TL) increase. Use the plant isotope value as TL 1, and see how many
consumer animal TLs beyond 1 are indicated. (Note: there are usually less than 8 TLs in
natural food webs). The general formula for calculating trophic levels starting with plants
at TL1 is:
TL = 1 + (15NCONSUMER - 15NPLANT)/3
Note that in this formula the “3” is the o/oo increment in 15N that occurs on average per
each trophic level, and the “1” in the equation ensures that trophic levels start at TL = 1 at
the plant level. Questions: How many trophic levels are there? Is there any sample whose
trophic level does not make sense?
Step 3. Consider the 13C data more closely. More fieldwork and sampling gives you two
more data points; add these to your food web picture. Here are the respective 15N and
13C data: corn 2 o/oo and -13o/oo and grasshoppers 4.2o/oo and -13o/oo. Update your
table and make another isotope biplot graph with the combined data. Questions: Does
including these new points change your impressions of which plant sources support the
food web? Do you think the tree leaves are the only source of nutrition in this food web?
Which food sources are most important?
Step 4: Make corrections to the animal 13C data, factoring out fractionation effects
related to TL, and leaving source inputs as the sole reason for the animal 13C variations.
This correction gives you the 13C value of the inferred plant diet for each consumer,
needed for direct comparison to the possible plant foods. The increase in 13C per trophic
level averages near 0.5o/oo, so use the following formula to calculate 13C values
corrected for trophic level effects:
Corrected 13C = Measured 13C - 0.5 *(TL -1)
where TL is estimated from the 15N data in step 2 above. Add the corrected 13C values
to your data table. Questions: Do these TL corrections change the 13C data very much?
Which food sources are most important for the overall food web?
Step 5. Calculate source contributions in terms of % tree leaves and % corn using the
corrected 13C values. This is done with the following mixing equations (explained in
section 5.3) that give contributions of sources 1 and 2 as:
Fraction of source 1 = (SAMPLE - SOURCE2)/(SOURCE2 - SOURCE1)
Fraction of source 2 = 1 – Fraction of source 1
The SOURCE1 and SOURCE2 values are the 13C values for the plants: tree leaves and corn,
respectively. Multiply the fractional contributions by 100 to obtain % contributions, and
add these % contributions to your data table. Question: Does corn or tree leaf material
provide most of the nutritional support for the food web?
Step 6: Combining results for the TL of each sample (based on 15N) and its source
contribution (based on 13C), make a (x,y) graph that shows your interpretations of TL
vs. source, and compare this to your original graph from step 1. Questions: What is your
overall interpretation of the food web in terms of both TL and sources? Which is better to
use as the y-axis, TL or source?