Problems for Chapter 1
1. True or false: Isotopes are forms of the same element with the same number of
neutrons and electrons, but different numbers of protons.
2. True or false: Working with stable isotopes of the HCNOS elements needs special
permits and licensing due to radioactivity hazards.
3. True or false: The heavy isotopes of an element are always rarer than their light
isotope counterparts.
4. True or false: Most modern instrumentation is not equipped to detect isotopes,
with the exception of mass spectrometers and some laser systems.
5. True or false: If the human in Fig. 1.3 were composed exclusively of light
isotopes, that human would be 10% lighter than the human depicted.
6. True or false: The element fluorine has only one stable isotope form, while the
element chlorine has two stable isotopes.
7. True or false: Sulfur has more stable isotopes than does hydrogen or nitrogen.
8. True or false: Early scientists who discovered stable isotopes in the late 1880s
were awarded Nobel prizes.
9. True or false: Stable isotopes are similar forms of the same element, differing in
the number of neurons and spin orientation.
10. True or false: Ecologists can freely add stable isotopes to field experiments with
little cost or effort, but interpretation of the isotope addition experiments is not
always simple.
Chapter 2 Problems
1. Write out the equation that defines . Explain why this equation is sometimes
called a ratio of ratios.
2. Using the definition of , explain why standards have values of 0o/oo.
3. Does 0o/oo mean 0% heavy isotope, that no heavy isotope is present at all? Why
or why not? How much heavy isotope is actually present in the standards for the
HCNOS elements whose  values are 0o/oo? Table 2.1 may be of help.
4. Most natural isotope  values are 1 to 50o/oo different than isotope compositions
of standards. Use the  definition to calculate the % heavy isotope (atom percent)
for carbon samples that are -1o/oo and -50o/oo vs. the carbon standard listed
Table 2.1. (Conversion 4 in the printed Appendix gives help with this
calculation). Do these values of % heavy isotope seem similar or very different to
you? Why? Do you think it is easier to see the isotope differences using the 
notation, or when isotope values are expressed as % heavy isotope? (Hint: reading
section 2.2 may help you with this problem).
5. Explain the difference between o/o and o/oo. Why are  values given in o/oo units
and not the more familiar o/o units?
6. If a sample is enriched in heavy isotopes relative to a standard, will the  value be
positive or negative, greater than 0o/oo or less than 0o/oo? If a sample is enriched
in light isotopes relative to a standard, will the  value be positive or negative,
greater than 0o/oo or less than 0o/oo? Explain your answer.
7. If the amount of heavy isotope in a carbon sample is 0.1 percent (%) more than
that of the carbon standard listed in Table 2.1, what is the  value of the sample?
What are the  values for samples that have 1% and 10% more 13C than the
standard? What are  values for samples that have 0.1%, 1% and 10% less 13C
than the standard?
8. Commercial laboratories can separate isotopes and prepare isotope-enriched
samples. If a sample has twice the heavy isotope content of the carbon isotope
standard given in Table 2.1, what is the 13C/12C isotope ratio of this enriched
sample and what is the value for this sample?
Chapter 3 Problems
1. True or false: The carbon stable isotope value of CO2 in the atmosphere is
currently about -8o/oo.
2. True or false: Some plants have 15N values of 0o/oo, signifying that they are
composed of N2 gas, the nitrogen isotope standard that also has a 0o/oo value.
3. True or false: The sulfur isotope value of sulfate in seawater is -21o/oo.
4. True or false: When you dry plant samples, all the hydrogen isotopes disappear
from the sample as water evaporates.
5. True or false: Oxygen isotopes are rare in humans.
6. True or false: Stable isotope records of past ecologies are present in many
historical materials, including forest soils.
7. True or false: In species invasions in lakes, newly arrived fish rapidly acquire the
same isotope values as the older, resident fish.
8. True or false: When animals move and migrate, they can retain their old isotope
values in tissues such as hair and feathers.
9. True or false: The isotope values of CO2 in the atmosphere are often inversely
related to CO2 concentrations.
10. True or false: Human diet planning needs to include a consideration of stable
isotopes for a healthy balanced diet.
Chapter 4 Problems
1. Using the chocolate isotopes I Chi model in workbook 4.1 (Chapter 4 folder on
this CD, I Chi Spreadsheets), what is the shape of the curves that result when
isotope “pickiness” or fractionation is set at 1 so that there is no preference for
heavy or light isotopes? Which part of the curves would you label “turnoverdominated” and which parts “close to equilibrium”? If you change the mixing
resupply or replacement rate to values of 75, 150, 500, 700, and 900, how do the
curves change, and why? (A good way to view the dynamics is to enter the
numbers in this list one-by-one, typing them in, then use the “undo” arrows to
2.
3.
4.
5.
replay the list forwards and backwards until you see what is going on). Is there
always a “turnover-dominated” portion of the curves, and one that is “close to
equilibrium”? What controls whether the system is at equilibrium or in the more
dynamic turnover stage?
Continuing problem #1 with Chocolate Isotopes, set the replacement rate to an
average value of 500 out of 1000 chocolates eaten, then manipulate the
“pickiness” factor or  fractionation. What curves result with  fractionation
values of 1,2, 4, 8,16 and 18? (Technical note: in theory, maximum possible
fractionations for the HCNOS isotopes are for hydrogen isotopes and approach
18; see Science 110:14-16,1949). What happens to the ratio eaten, light/darks, as
you change the fractionation factor? Why? (Hint: because the total number of
chocolates is constant in this system, steady state eventually occurs where inputs
must equal outputs in terms of both light and dark chocolates. Most models in this
book are not steady state models, but this chocolate isotope model is a steady state
model once amounts reach equilibrium).
Continuing problems 1 and 2 with Chocolate Isotopes, you may have the sense
now that fractionation and mixing sometimes oppose one another. See if this is
really true. First set the  fractionation to 1 (no selectivity) and resupply to 500.
Note that these values give a final result at the end of the year of about equal
amounts of light and dark chocolates. Now set the resupply to 750, and type in
fractionation values until you find a value that gives this same final result
approaching of an even mix of light and dark chocolates. What is this value for
fractionation, and how do you explain this effect that fractionation and mixing
(via resupply in this case) seem to cancel each other out? Give your answer in
both qualitative and quantitative terms. (Hint: as in problem #2, think about
equality between inputs and outputs in developing your quanitative answer).
Using the oxygen cycling I Chi model (see workbook 4.2 in Chapter 4 folder on
this CD, I Chi Spreadsheets), first locate the three master variables at the upper
left of the worksheet for: daytime photosynthesis, continuous respiration, and
continuous isotope fractionation during respiration. Change these variables one at
a time and note the effects to answer the following questions. What are the effects
of doubling each of these master variables on the a) oxygen concentrations with
time, b)  values with time and c) the trajectories of oxygen dynamics in the
(concentration, ) graph? What are the effects of halving the values? Finally,
using the original values for the three variables, what is the effect of changing the
fourth master variable, the isotope value of oxygen from photosynthesis, from 0 to
-10o/oo and then from 0 to +10o/oo? Considering the overall results from this
exercise, would you agree or disagree with the idea that isotopes can help
constrain ideas about how oxygen is produced and consumed in aquatic systems?
Explain your answer.
This problem addresses whether oxygen isotope measurements are really useful
for studying oxygen dynamics. Originally, oxygen isotope models were
developed for the ocean to study production/respiration ratios or P/R (Global
Biogeochemical Cycles 1:49-59, 1987). When P/R = 1, production and respiration
are balanced and there is no net change in oxygen concentrations. But when
photosynthesis due to algal growth is greater than respiration, oxygen
accumulates and P/R>1. Here we compare two growth scenarios where P/R>1,
considering only the first 12 hours when oxygen is increasing. Using the oxygen
cycling I Chi model in workbook 4.2 (see Chapter 4 folder on this CD, I Chi
Spreadsheets), change the fractionation factor to 15o/oo, the value used in the first
modeling study cited above in this problem. Using your cursor to click on the
graphs in the workbook, what are the values at 12 hours (0.5 days) for oxygen
concentrations and 18O values? What is P/R at 12 hours? Next change the P rate
from 4 to 6 and R rate from 2 to 4 in the workbook and repeat your observations
about oxygen concentrations, 18O, and P/R at 12 hours (0.5 days). Make a table
of your results for both scenarios showing initial and 12 hour values for oxygen
concentrations, 18O, and calculated P/R. Now that you have compiled the results,
look back at the experiment and answer these questions: Can you use the oxygen
concentrations alone to estimate the P/R ratio, the rates of photosynthesis and the
rates respiration? If not, does adding the 18O information help with these three
estimates?
6. Following the step-by-step approach I Chi modeling approach outlined in section
4.4 and in the Workbook 4.4 (Chapter 4 folder on this CD, I Chi Spreadsheets),
develop a spreadsheet model for CO2 dynamics in a closed greenhouse full of
large plants maintained 24 hours a day in well-lit conditions. The plants fix CO2
and remove it from the atmosphere, but also respire CO2 and return CO2 to the
atmosphere. Incorporate the following initial assumptions into your model: a) the
initial 13C value of CO2 is -8o/oo, the current 13C value of CO2 in the
atmosphere, b) the initial concentration of CO2 is 375 ppm, c) during
photosynthetic CO2 uptake, plants fractionate by 20o/oo and use a constant
amount of 5 ppm CO2 per hour, and d) during continuous respiration, plants do
not fractionate carbon isotopes but add back CO2 with a constant value of -28o/oo
at a rate of 4 ppm per hour. What are the predicted concentrations and isotope
values after 10 days, at 240 hours? Show your spreadsheet model and graphs of
your time-course prediction for CO2 concentrations and isotope values. Also,
compare your model to the oxygen model of problem 4 and sections 4.2-4.5.
What are the similarities and differences between the oxygen and carbon dioxide
models?
7. Use the gain-loss model for cows presented in section 4.7 and modify it to explore
carbon isotope dynamics for cows feeding in a pasture. Assume that the
fractionation involved in carbon loss, primarily as respired CO2, is 1o/oo, but that
there is no fractionation during assimilation of carbon from the diet. (This same
assumption of no fractionation during assimilation was discussed in section 4.7
for N assimilation, and also likely applies to carbon isotopes). Use the spreadsheet
cow model in Workbook 4.7a (Chapter 4 folder on this CD, I Chi Spreadsheets) to
work through various growth and starvation scenarios for carbon, parallel to those
given already for nitrogen. If animals in nature generally have carbon isotope
compositions about +0.5o/oo vs. their diets, what kind of gain-loss dynamics do
the models predict for these free-living animals? Why?
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
to taste the saltiness, mimicking salty-tasting seawater that has an average salt
5.
6.
7.
8.
9.
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?
Chapter 6 Problems
1. Consider five separate HCNOS samples from an isotope separation lab that each
have been enriched to 10% heavy isotope. What are the  values for these 5
samples? (Hint: you will need to use the definition of  and the values of the
standards given in Table 2.1 of Chapter 2 to answer this question).
2. From the same commercial lab just mentioned above in problem 1, you find you
can also buy 15N-enriched samples that have been enriched from natural
abundance 15N levels to 5%, 10%, 20%, 60%, 90% and 99% 15N. What is the
natural abundance level for 15N in the nitrogen standard (see Table 2.1), and what
are the  values of these enriched samples?
3. A commercial lab that does isotope separations will supply carbon samples with
no heavy isotope at all. What would be the  value of such heavy-isotope depleted
samples?
4. The same laboratory will supply carbon samples with 99% pure “13C-enriched”
samples that contain 99% 13C and only 1% 12C. What is the  value of such
samples?
5. You want to start an isotope addition experiment adding 13CO2 to the air in a
greenhouse, seeking to label plants and root exudates, then isolating bacteria in
the root zone to trace label transfer into soil food webs. If the natural isotopic
composition of CO2 in air is -8o/oo, the CO2 concentration in air is 375 ppm and
you raise the concentration by a small amount to 394ppm, what will be the 
values of CO2 if you use 99% 13C-CO2 or 0% 13C-CO2? Given your results, would
you choose to use 13C-enriched or 13C-depleted CO2 for the tracer addition? Why?
6. Fractionation occurs in tracer experiments, but is often of little consequence when
tracer additions are made at high levels and sample  values are 1000o/oo or
more. But in many experiments, lower enrichments occur, and fractionation can
be more important. This problem and the following problem explore the effects of
fractionation within tracer addition experiments. If fractionation during carbon
fixation is 1.02, a 2% faster reaction for 12C-CO2 than 13C-CO2, what will be the
isotope composition of plants at the natural abundance isotope level, with
atmospheric CO2 at -8o/oo? (Calculate the fractionation exactly using the isotope
ratio RCO2 from the  definition, i.e., RCO2 = RSAMPLE = RSTANDARD*(+
1000)/1000. After fractionation the new RSAMPLE for the plant is RPLANT =
RCO2/1.02. Using the  definition, this calculated RPLANT value, and the RSTANDARD
value from Table 2.1, what is the 13C value of the plant? Also, what is the simple
isotope difference,  between the -8o/oo CO and the plant 13C value?). Now
repeat these calculations for three enrichment experiments where the 13C value
of CO2 has been raised to 100o/oo in experiment 1, 1,000o/oo in experiment 2,
and 10,000o/oo in experiment 3. What are isotope differences between plants and
CO2 using the ratio-based precise calculations vs. the more approximate simple
( difference calculated for these three experiments? How do these values
compare with the differences calculated for the first natural abundance
experiment? Does fractionation seemingly grow, shrink or stay the same in the
enrichment experiments? Why? Looking at your combined results, do you agree
with the first sentence of this problem? Why?
7. Repeat problem 6, but instead of using 13C-enriched CO2, use 13C-depleted CO2,
so that the three tracer experiments have CO2 whose 13C value has been lowered
to -100, -500, and -950o/oo. Assume again that fractionation during carbon
fixation is 1.02, a 2% faster reaction for 12C-CO2 than 13C-CO2, and calculate the
exact isotope composition of plants growing with the -100, -500, and -950o/oo
CO2. Does fractionation seemingly grow, shrink or stay the same in these “12C
enrichment” experiments that are also “13C-depletion” experiments? Why?
Overall, considering results of your calculations for problems 2, 6 and 7, is there a
 value for enrichment experiments above which you don’t have to worry about
fractionation?
8. You are working with a stream ecosystem, and decide to perform a tracer addition
experiment over 6 weeks during a summer field season. The stream has a very
low natural ammonium concentration of 0.1 mmol m-3 and flows downhill rapidly
at a rate of 5 m3/sec. To bring the ammonium isotope value to 1000o/oo, how
many grams of 15N isotope do you have to add during the 6 weeks (42 days),
assuming that the background ammonium has the same  value as the atmospheric
air standard (see Table 2.1 in Chapter 2) and that you use 10% 15N-labeled
ammonium that also contains 90% 14N? If the 10% 15N-labeled ammonium is
commercially available as ammonium sulfate at $220 for 50g, how much will you
have to spend for the summer experiment? If the calculated amount is too
expensive, what could you do to lessen costs but still have an effective labeladdition experiment with added 15N-labeled ammonium? (Hint: you may want to
consult Technical Supplement 6A on this CD for strategies about calculating
isotope amounts added to field experiments).
9. What are the HAP (atom percent heavy isotope values) values for samples with 
values of -30o/oo for each of the HCNOS elements? What are the atom percent
values for samples with  values of 1000o/oo for each of the HCNOS elements?
(Table 2.1 and Conversion 4 in the printed Appendix to the book can be helpful
for this problem). For the enriched +1000o/oo values, what are the “atom %
excess” values after subtracting the natural background atom percent values of the
standards in Table 2.1?
10. You make a 1:1 mixture of two samples that have equal carbon concentrations,
and  values of -20 and -40o/oo. Calculate the atom percent values for 13C content
for the -20 and -40o/oo values, give the atom percent value for the mixture, then
use that value to calculate the true 13C value of the mixture. How different is this
value that simply averaging the -20 and -40/oo  values? Repeat this experiment
with  values of -20 and +100o/oo, -20 and +1000o/oo, and -20 and +10,000o/oo.
Given your results, when do you think using  values is acceptable for calculating
mixing dynamics, and when do you think you need to use atom % values.
Chapter 7 Problems
Please note that several of the spreadsheet problems 7.5-7.12 below are difficult, and will
take some time and effort to solve.
1. Investigate fractionation by typing in fractionation values of zero in the I Chi
worksheets found on this CD, especially in folders for Chapters 4 and 5. This
turns off fractionation so that only mixing is active. Then use the “undo” button to
reverse your action, turning fractionation back on. What are the basic effects you
observe about fractionation?
2. In the Chocolate Isotope example of Section 4.1 of Chapter 4, what rate equations
describe the faster consumption of light vs. dark chocolates? (Hint: remember that
the chocolate isotope example is special in that a steady state develops over time,
so that inputs = outputs).
3. In the oxygen model of Section 4.2 of Chapter 4, mixing and fractionation often
oppose on another in their effects on isotope values. Common fractionations used
for respiration during oxygen consumption range between 10 and 25o/oo. Type in
a 10o/oo fractionation into the list of master variables in the oxygen model of
Workbook 4.2 (found in the Chapter 4 folder on this CD, I Chi Spreadsheets). As
you do this, watch the graph showing oxygen concentration vs. isotopes, and note
how changing this fractionation factor from 24.2o/oo to 10o/oo changes the
direction of the purple line that shows the day-night oxygen dynamics over 96
hours. Also note changes in the simpler graphs of oxygen concentration and
isotopes vs. time. With the fractionation at 10o/oo, can you find another value for
oxygen added or mixed in from photosynthesis that will offset the effects of
fractionation, so that each day, the system returns approximately to the same
starting point of 100% saturation at 250 mmol O2 m-3 and 24.2o/oo? What is the
isotope value of new oxygen mixed in that resets the system? Repeat this exercise
for fractionations of 15o/oo, then 20o/oo, each time adjusting the isotope value of
the oxygen mixed until the system is reset to 100% saturation and 24.2o/oo. Why
are there seemingly paired values for fractionation and mixing that particularly
return the system to very nearly the same starting point each day?
4. This is a weighted average, mixing problem that also helps show how to calculate
a fractionation factor from isotope differences observed in field studies. The
problem is to revise the fractionation estimate used in the cow model of Section
4.7 to include all the various form of N loss. Detailed data unfortunately do not
exist for cows, but we can use the following average data from 4 llamas that were
fed alfalfa and studied by Sponheimer et al. (see references in Section 4.7).
Dietary N input was 49.6g N/day and the  value of the diet was 0.4o/oo. This
input must be balanced with outputs that included fecal N (9.5g/d and  =
3.3o/oo), urinary N (31.8g and  = 0.1o/oo), and remaining N (8.3 g/d) that
includes animal protein N represented by hair ( = 4.7o/oo). The first task is to
create a mass balance isotope equation. Using the given values, can you write a
balanced equation for isotopes so that inputs = outputs? You will probably find
that the answer is no. To make the balance, assume that of the 8.3g N/d that was
not excreted, only 6.3g N/d was in fact gained by the animal, but a small and
hard-to-measure amount of 2g N/d was lost as N2 gas via denitrification, an
anaerobic respiration occurring in the guts of the animals. Now recalculate the
mass balance, and determine the 15N value of nitrogen lost during denitrification.
Next, using the value of 6.3g N/d assimilated into animal protein, what is the
overall N retention efficiency and what is the fraction lost? Using the top line of
the diagram Fig. 7.5, the value just calculated for the fraction lost, and the 4.7o/oo
15N value for hair that represents the “residue” not lost but retained by the
llamas, what is the average fractionation value  for the total losses? You can
substitute this newly calculated  value for the 9o/oo value used in the cow model
of section 4.7 for more realistic simulations of cow isotopes during pasture
grazing.
5. Imagine a whale that each year migrates from the arctic in the summer to the
equator in the winter and then back again to the arctic. Some whales actually
make such annual migrations. Further, whale food had 13C values of -25o/oo in
the arctic and -15o/oo at the equator. Draw a diagram and develop a day-by-day
spreadsheet model of how isotopes in food will change over time during the
course of an annual migration, then replicate this cycle for 5 years. Use this 5-year
food isotope projection and I Chi modeling to calculate isotope values in whale
blood (a low-mass, fast turnover tissue) and whale blubber (a high mass, slow
turnover tissue) during the annual migration cycle. (Hint: section 5.9 shows a
tissue turnover example). Allow for tissue turnover, mass loss and fractionation
during tissue turnover, even as the whale is growing with a net gain mass.
6. Use the oxygen model presented in Section 4.2 and in Workbook 4.4, worksheet
10 (see Chapter 4 folder on this CD, I Chi Spreadsheets), and link a CO2 model to
the oxygen dynamics. That is, when photosynthesis produces one unit of oxygen,
it also consumes one unit of dissolved CO2. And conversely, when respiration
consumes one unit of oxygen, one unit of CO2 is generated. The linkage should
thus exist at a molar ratio of 1:1 for O2: CO2 in respiration and in net
photosynthesis. Assume that the CO2 pool is represented as DIC (dissolved
inorganic carbon) in the sea with an initial concentration of 2000 mmol m-3, and
the carbon isotope composition of the initial DIC is 1.0o/oo DI13C.
7. You are working in a lake ecosystem, trying to understand dynamics of plant
production by studying oxygen isotopes (18O). You begin thinking about a
closed system with just photosynthesis during the day, and respiration day and
night. Assume that the isotope value of newly-produced photosynthetic oxygen is
0o/oo, the initial isotope value of oxygen in the water is 24.2o/oo, and that the
initial concentration of oxygen is 250 mmol m-3. Develop an I Chi model for 24
hours of oxygen cycling (a day-night model), with midnight as your initial time
point. Make graphs that show a) oxygen concentrations over the day-night cycle,
and b) isotope compositions of that oxygen. Use a fractionation factor of  =
18o/oo for fractionation during respiration. (Hint: see Section 4.4 for help with a
basic I Chi model for oxygen dynamics). The rest of this problem involves
expanding this basic I Chi model to include three more processes: equilibrium
with the atmosphere, exchange with the atmosphere, and oxygen injection during
storms via bubbles.
a. Equilibrium with the atmosphere. When plants produce oxygen in the day
by photosynthesis, this often leads to oxygen supersaturation and loss of
oxygen to the atmosphere. But respiration also consumes oxygen, so that
oxygen can become undersaturated especially at night, and oxygen will
begin invading from the atmosphere. Overall, the atmosphere buffers the
lake system, bringing it back towards an equilibrium or saturated oxygen
concentration value, assumed to be 250 mmol m-3 in this example. Add the
processes of invasion and evasion to your model, assuming that invasion
introduces oxygen with an isotope composition of 24.2o/oo and that
evasion occurs without isotope fractionation ( = 0o/oo). Then explore
daily results for concentrations and isotopes in a eutrophic system with
abundant photosynthesis and respiration vs. in an oligotrophic system that
is fairly unproductive for both photosynthesis and respiration. What are
the effects of the higher rates of invasion and evasion for the eutrophic
system?
b. Exchange. Think about the effects of winds that stir the surface of the lake
and lead to exchange of oxygen between the water and atmosphere, with
exchange occurring continually when oxygen concentrations are at
saturation. Concentrations don’t change, but isotopes do change during
this exchange process. Assume that exchange introduces oxygen with 18O
= 24.2o/oo into the water, and that exchange only occurs at saturation,
with evasion and invasion dominating when concentrations are above or
below saturation. What happens when photosynthesis = respiration in the
presence and absence of exchange?
c. Oxygen injection. Storms re-equilibrate oxygen concentrations and
isotopes in the dissolved oxygen pool, actually leading to a mild 103%
supersaturation characteristic of many low-productivity aquatic systems.
Add the possibility of bubble injection into the model, then explore two
scenarios for stormy (large amounts of bubble injection) and calm (low
amounts of bubble injection) conditions.
8. You are working on climate models and want to understand the history of carbon
isotopes (13C) in CO2. Analyses of air trapped in ice cores show an historical
overview of this dynamic. Especially, you learn that in the year 1800, the isotope
composition of CO2 in global air was -6.5o/oo and CO2 at that time had a
concentration of 260ppm. Today you measure the outside air and find a
concentration of 375ppm, and you read that combustion of fossil fuel is
responsible for the increase in concentration. If fossil fuel (mostly oil) has a 27o/oo 13C value, and there is no fractionation during various combustion
scenarios (in cars, factories, etc.) that produce CO2 from fossil fuels, what would
you expect is the 13C value of CO2 in today’s air? Next, you read that CO2 in air
exchanges with a much larger amount of CO2 and bicarbonate in the ocean,
buffering large changes in atmospheric CO2 concentration. Exchange involves
losing CO2 from the air to the ocean, but also regaining CO2 to the atmosphere
from the ocean, in a balanced way that does not change amounts, but could affect
isotopes. The large, buffered ocean will emit CO2 that is currently mostly
equilibrated with pre-industrial conditions, and this CO2 emitted from the ocean
will have a -6.5o/oo value. You begin to think about what this exchange might
mean for isotopic values, use I Chi modeling to show the time course of CO2
concentrations and isotopic values of CO2 over time. Show these graphs for 2
scenarios, a) assuming no exchange with the ocean, and b) assuming that the
atmosphere exchanges with the oceanic carbon pool each year, so that the
predicted outcome of the modeling for the year 2004 is 375ppm atmospheric CO2
with a value of -8o/oo.
9. Develop an I Chi spreadsheet model for a continuous culture, flow-through
system used for growing microbes. Fluid drips in continuously to an open vat that
is well-stirred, then flows out so that inputs = outputs. Microbes grow and use up
part of the nutrients supplied, so that amounts stabilize as a balance of input and
uptake.
10. Develop a nitrogen isotope I Chi spreadsheet model for a food web in the upper
ocean. Simplify the food web to include only three components, nitrate,
phytoplankton, and zooplankton. Nitrogen can be lost from the upper ocean pools
of phytoplankton and zooplankton pools but then re-enter the nitrate pool as
regenerated nitrate. Nitrogen can also be lost to the deeper ocean via sinking of
phytoplankton and zooplankton. New sources of nitrate can include upwelling of
deep water and nitrogen fixation. Nitrate can also be removed by denitrification.
Hint: be sure to diagram this model before beginning. Also, for the “N lost” or (=
“product lost”) from zooplankton and phytoplankton compartments, you will need
to use the equation for the product lost or formed (see Sections 4.4, 7.1) when
recirculating or regenerating this N from phytoplankton and zooplankton back
into the nitrate pool. Assume no fractionation during the regeneration process.
11. Take a minute to reread the first part of Section 7.5 before you start work on this
problem. Consider the Pacific landscape of Section 7.5, where 15N values of
nitrate and particulate organic nitrogen (PON) increase as nitrate-rich water
upwells at 2oS and moves north. Develop a spatial I Chi model for predicting this
change in 15N from 2oS to 24oN, with changes in nitrate 15N modeled at 1o
intervals of latitude as water moves north from its point of upwelling. Use a
fractionation factor of  = 5o/oo for nitrate consumption, and use closed-system
equations for nitrate isotopic compositions and PON (see Fig. 7.4 for equations;
nitrate = residual reactant and PON = instantaneous product), with a) nitrate
modeling focused on upwelled nitrate and assuming no new nitrate inputs and b)
PON modeling assuming that PON formed is grazed and remains at the latitude in
which it is formed, and does not move north with water and nitrate. Also assume
that as water moves north, there is an increasing amount of new “gyre” N added
via a combination of local recycling and N fixation, with a net isotope value of
3o/oo. Can you use your model to predict 15N values of sedimentary PON,
assuming a 6o/oo increase in 15N values from surface PON to sediment PON?
12. Take a minute to reread the latter part of Section 7.5 before you start work on this
problem. Consider the muddy floor of the ocean, as presented in Section 7.5.
Think about sulfate dynamics as sulfate diffuses into the mud from a large
reservoir of sulfate in bottom waters. Bacteria in the sediments consume sulfate as
it diffuses downwards, with isotope fractionation during sulfate reduction to
sulfide. Develop an I Chi model for this process as a closed system reaction with
increasing depth, involving sequential sulfate removal and fractionation as sulfate
diffuses downwards. Then develop a second related model that is more complex
and allows for three types of mixing at larger and smaller scales: bioturbation
mixing, diffusion and exchange. At the largest scale, bioturbation by burrowing
animals will mix sulfate from one interval with sulfate from the interval above it.
Set up this mixing so that 0-100% of the sulfate in an interval can mix with the
sulfate in the interval above it. The second type of mixing is at a smaller scale, via
diffusion. Diffusion also resupplies sulfate, but with isotope fractionation that
develops in response to gradients developed by sulfate reduction (see Jorgensen
1979 and Chanton et al. 1987 references in Section 7.5). Set up this diffusional
mixing so that 0-100% of the sulfate lost to sulfate reduction in an interval can be
resupplied from the interval above it, with a characteristic isotopic fractionation.
Lastly, exchange at the molecular level could possibly mix sulfate without
changing concentrations, only isotopes. Set up this mixing so that 0-100% of the
sulfate in an interval can exchange with the sulfate in the interval above it.
For all models, plot the sulfate concentrations and isotopes vs. depth, and
make an (x,y) plot of (-ln(concentration), 34S). This last (x,y) plot gives the
permil fractionation factor as the slope of the line (i.e., slope = ). Once the
models are constructed, the task is to compare results of the two models,
remembering that the first model is a simple closed system model, and the second
model is similar but more complex in that it also allows up to three types of extra
mixing. Can you adjust the sulfate concentration profiles so that they are equal for
the two models, but there is extra sulfate entering the system in the mixing
model? What happens to fractionation in this case measured as  values? If you
now further allow exchange, what happens to fractionation? (Note: when you
make these comparisons, the fractionation in any diffusional mixing should be the
last thing you adjust, because it is determined from the isotope gradient that
develops from the main sulfate reduction process. The fractionation here is not
independent of the main isotope gradient, and you have to adjust it to bring the
system to balance. It is the author’s experience that this adjusted fractionation
should match the overall fractionation calculated from the plot of (ln(concentration), 34S) for downward diffusing sulfate. Can you adjust the
fractionation in the diffusion step until it matches the fractionation calculated
from that plot? When you have done this, this second, more complex model is
balanced and complete).
13. Here is a list of isotope cycling papers - can you recast any one of these studies in
terms of I-Chi and get the same answers? (Note: no answers are provided in this
book for these challenging papers):
Altabet, M.A. 1988. Variations in nitrogen isotopic composition between sinking
and suspended particles: implications for nitrogen cycling and particle
transformation in the open ocean. Deep-Sea Research 35:535-554.
Cline, J.D. and I.R. Kaplan. 1975. Isotopic fractionation of dissolved nitrate
during denitrification in the eastern tropical North Pacific Ocean. Marine
Chemistry 3:271-299.
Goldhaber, M.B. and I.R. Kaplan. 1980. Mechanisms of sulfur incorporation and
isotope fractionation during early diagenesis in sediments of the Gulf of
California. Marine Chemistry 9:95-143.
Kroopnick, P. and H. Craig. 1976. Oxygen isotope fractionation in dissolved
oxygen in the deep sea. Earth and Planetary Science Letters 32:375-388.
Macko, S.A., M.L. F. Estep, M.H. Engel and P.E. Hare. 1986. Kinetic
fractionation of stable nitrogen isotopes during amino acid transamination.
Geochimica et Cosmochimica Acta 50:2143-2146.
Uhle, M.E., S.A. Macko, H.J. Spero, D.W. Lea, W.F. Ruddiman and M.H. Engel.
1999. The fate of nitrogen in the Orbulina universa foraminifera-symbiont
system determined by nitrogen isotope analyses of shell-bound organic matter.
Limnology and Oceanography 44:1968-1977.
Voss, M., J.W. Dippner and J.P. Montoya. 2001. Nitrogen isotope patterns in the
oxygen-deficient waters of the Eastern Tropical North Pacific Ocean. DeepSea Research I 48:1905-1921.
Chapter 8 Problems
1. True or False: At the present time, lasers can measure isotope compositions of
trace gases such as CO2 and so can provide an isotope scanner for some types of
field ecology.
2. True or False: Currently most stable isotope measurements are made with mass
spectrometers because these machines are 10-100x more accurate and precise than
laser-based systems.
3. True or False: Most mass spectrometer systems require little sample preparation
and can vaporize samples and measure isotopes of all the HCNOS elements in a
sample within a two-minute scan.
4. True or False: Stable isotopes can be used in multivariate analyses of niche space.
5. True or False: In the case of mangrove biology described in section 8.2, stable
isotopes were the best markers for separating different species.
6. True or False: The causes of isotope variation in plants are extremely well-known,
so that once you have measured isotope data for plants, it is very easy to infer
their in situ physiology and sources of nutrients.
7. True or False: Stable isotope values often reflect a combination of source and
processing (fractionation) information, and it takes some thought, skill and often
experimentation to understand whether source or fractionation information
dominates the measured isotope values.
8. True or False: Imagination is all it takes to be successful in science, or at least
successful in stable isotope ecology.