Plant Physiology
Water Relations Problem Set
Semi-interactive Key
And Explanations
for use with PowerPoint XP
How To Use this Program
R
• Start with an unmarked copy of problem set
• Go through the program
–
–
–
–
–
–
Use mouse, not keyboard (push roller to go back)
Read the text, clicking to get more
Fill in values on your problem set as asked
Check your answers with the program
Click on links for further explanation
Click R in the upper right corner of the
explanation page to return to where you were
www.uni.edu/berg
UNI Plant Physiology
6
Need to review Water Potential?
•
•
•
•
What it is
Click here for next line
Units
Scale (values)
Factors that make up WP
– Pressure potential
– Solute potential
– Matric potential
• OK? Just click to continue to problems
www.uni.edu/berg
UNI Plant Physiology
7
Problem #1
Here you have two cells. Your job is to fill in the missing values for the
water potential factors.
Some values are given. Some values you know from the
kind of cell. Others you know from the information given.
Still further values you get from arithmetic.
mesophyll
xylem
We’ll work through each problem slowly,
with your participation, filling in values
on your copy. Just click when you want
to go onward, starting now.
-6
-10
www.uni.edu/berg
Humid morning; cells at
equilibrium with each other.
UNI Plant Physiology
22
Problem #1 - a
Strategy: First fill in what you know because of what kind of cell it is.
Then we will fill in values based on further information given.
Lastly, we will do the arithmetic to get the remaining values.
The left cell is a (live) leaf mesophyll cell. This tells you its
matric potential. Fill it in now on your copy. Click to check
your answer on the diagram.
The matric potential of all really wet things is zero.
mesophyll
xylem
-6
-10
0
0
www.uni.edu/berg
0
0
The right cell is a xylem vessel element, and it
is filled with xylem sap, mostly water. This tells
you its matric potential. Write in the matric
potential for that cell on your copy now then
click to check your answer on the diagram.
The matric potential here is zero, too,
because the solution in the cell is wet.
UNI Plant Physiology
25
Problem #1 - b
Strategy: First fill in what you know because of what kind of cell it is.
(Still doing this.)
The right cell is a xylem vessel element, and it is filled with
xylem sap, almost pure water. This tells you its solute
potential. Fill it in now on your copy. Click to check your
answer.
The solute potential of pure water is zero.
mesophyll
xylem
-6
-10
0
www.uni.edu/berg
0
0
0
This is all the information you can get
from the type of cells present. The next
step is to use the other information
included with the problem.
UNI Plant Physiology
28
Problem #1 - c
Strategy: First we filled in values based on the kinds of cells.
Now we will fill in values based on information given with the problem.
Lastly, we will do the arithmetic to get the remaining values.
The cells are at equilibrium with each other. This means
they have the same water potential. Fill in the missing
water potential now on your copy. Click to check.
mesophyll
xylem
-6
-6
-6
-10
0
0
0
www.uni.edu/berg
We would expect this type of equilibrium
on a humid morning, because the cells
would have had plenty of time for the
water to move if there was any difference
in water potential.
Humid morning; cells at
equilibrium with each other.
UNI Plant Physiology
31
Problem #1 - d
Strategy: First we filled in values based on the kinds of cells.
Then we filled in values based on information given.
Lastly, we will do the arithmetic to get the remaining values.
WP = PP + SP + MP or
Ψ = ψp + ψs + ψm
Fill in the missing values for the left cell now on your copy.
Click to check your answer.
mesophyll
xylem
-6
-6
+4
+4
-6
-6
-10
0
0
0
-6 = +4 + (-10) + 0
Now fill in the missing values for the
right cell on your copy. Click to check.
-6 = -6 + 0 + 0
Check the arithmetic again and then…
www.uni.edu/berg
UNI Plant Physiology
32
Are we finished?
Well, not quite. We should go through the final answer and make sure that
the proposed numbers are sensible, likely ones. So here we go.
Normal cells have water potentials that are between about
-1 and -15 bars. Our value looks reasonable.
Next check the pressure potential: Typically it should be positive
for live cells and negative for xylem. So our answers look good.
mesophyll
xylem
-6
-6
+4
-10
-6
0
www.uni.edu/berg
0
0
The solute potential of mesophyll cells is
always negative, and typically several bars
In magnitude. Ours still looks good.
The matric potential in wet cells (or wet
anything else) is zero. Looks OK here.
Well, it all looks good, so we can go to
the next problem.
UNI Plant Physiology
33
Problem #2- a
Strategy: First fill in what you know because of what kind of cell it is.
Then we will fill in values based on further information given.
Lastly, we will do the arithmetic to get the remaining values.
The left cell is a (live) leaf mesophyll cell, and the right cell is a xylem
vessel element. This tells you their matric potentials. Fill both in now
on your copy. Click to check your answer on the diagram.
Wet things have a matric potential of zero.
mesophyll
xylem
-6
-10
0
0
www.uni.edu/berg
0
0
0
Now figure out the solute potential. The left
(live) cell is given, so you just have to do the
xylem cell. Write in its solute potential now on
your copy, then click to check your answer on
the diagram.
Transpiration starting
(sunrise).
Same cells
as roll
If you need
this explained
again,
above.
the mouse
back to Problem 1 – b.
UNI Plant Physiology
34
Problem #2- b
Strategy: First we filled in values based on the kinds of cells.
Now we will fill in values based on information given with the problem.
Lastly, we will do the arithmetic to get the remaining values.
If transpiration is taking place, there must be water flow from the xylem to
the mesophyll cell. This means the mesophyll cell has a lower water
potential than the xylem. Typically the differences are just a couple of
bars for nearby cells. Draw an arrow to show the direction of water flow
between the cells, then fill in a likely value
mesophyll
xylem
now on your copy.
Click to check your answer on the diagram.
-8
-6
-10
0
0
0
www.uni.edu/berg
0
If you need more explanation on water
movement, click here.
Transpiration starting (sunrise).
Same cells as above.
UNI Plant Physiology
39
Problem #2- c
Strategy: First we filled in values based on the kinds of cells.
Then we filled in values based on information given.
Lastly, we will do the arithmetic to get the remaining values.
WP = PP + SP + MP or
Ψw = ψp + ψs + ψm
Fill in the missing values for the left cell now on your copy.
Click to check your answer.
mesophyll
-8
+2
+2
-10
0
xylem
-6
-6
-6
-8 = +2 + (-10) + 0
0
Now fill in the missing values for the
right cell on your copy. Click to check.
-6 = -6 + 0 + 0
0
0
Check the arithmetic again and then…
www.uni.edu/berg
UNI Plant Physiology
42
Does it make sense?
We should go through the final answer and make sure that the proposed
values are sensible, likely ones. So here we go.
Normal cells have water potentials that are between about -1 and -15
bars. Our value looks reasonable. There must be a difference in water
potentials if transpiration is taking place. We’ve got that, too.
mesophyll
-8
+2
-10
0
www.uni.edu/berg
Next check the pressure potential: Typically it should be positive
for live cells and negative for xylem. So our answers look good.
xylem
If you got a negative pressure potential for the
mesophyll, you know that for that cell you
chose a value for the water potential that was
-6
0
too low.
-6
0
0
starting
(sunrise).
TheTranspiration
matric potential
should
be zero. OK.
Same cells as above.
We’re ready for the next problem.
UNI Plant Physiology
43
Problem #3 - a
Strategy: First fill in what you know because of what kind of cell it is.
Then we will fill in values based on further information given.
Lastly, we will do the arithmetic to get the remaining values.
The left cell is a (live) leaf mesophyll cell, and the right cell is a xylem
vessel element. This tells you their matric potentials. Fill both in now
on your copy. Click to check your answer on the diagram.
Wet things have a zero matric potential.
mesophyll
xylem
-10
-8
0
00
0
0
0
0
www.uni.edu/berg
Now figure out the solute potential. The only
one you can guess is the xylem cell (though the
mesophyll cell has probably not changed
much). Write in its solute potential now on your
copy, then click to check your answer on the
Transpiration exceeds water
diagram.
uptake by roots. Leaf
Xylem solute
potentials
typically
0 because
mesophyll
justare
wilting.
Same
xylem sapcells
is typically
almost
as above,
but pure
later. water.
UNI Plant Physiology
44
Problem #3 - b
Strategy: First we filled in values based on the kinds of cells.
Now we will fill in values based on information given with the problem.
Lastly, we will do the arithmetic to get the remaining values.
The mesophyll cell is just wilting, which tells you what its
pressure potential is. Fill it in now on your copy. Click to check
your answer on the diagram.
-10 = 0 + (-10) + 0
mesophyll
-10
00
-10
0
00
www.uni.edu/berg
xylem
-8
-8
0
0
00
0
All that is left now is the arithmetic. Fill in the
missing values for both cells now on your copy.
Click to check your answer.
-8 = -8 + 0 + 0
Transpiration exceeds water
uptake by roots. Leaf mesophyll
Checkcell
thejust
arithmetic
wilting. again
Sameand
cellsthen…
as
above, but later.
UNI Plant Physiology
49
Are the values reasonable?
It is reasonable for a transpiring mesophyll cell to have a lower water
potential than the adjacent xylem.
A wilting cell always has 0 turgor (pressure potential), and xylem has
a negative pressure potential (sap under tension) during transpiration.
Xylem sap usually has very low levels of solutes, so it has
a solute potential of zero.
mesophyll
-10
0
-10
0
00
www.uni.edu/berg
xylem
-8
-8
0
0
00
0
Matric potential of both cells is zero, as it
should be for wet things.
So everything looks OK.
Time for the next problem.
Transpiration exceeds water
uptake by roots. Leaf mesophyll
cell just wilting. Same cells as
above, but later.
UNI Plant Physiology
50
Problem #4 - a
Here you have a view of two adjacent root cells and the soil next to one of them.
As usual, your job is to fill in the missing values for the water potential factors.
Strategy: First fill in what you know because of the cell type or soil.
Then values based on other information. Lastly, the arithmetic.
Matric potential is easy for the two cells. Fill the values for the cells in
now on your copy. For soil, we’ll have to do some more work (later.)
root xylem root cell
soil
-5
0
-8
0
0
0
0
www.uni.edu/berg
Matric potential is zero for wet things.
Now on to filling in what we know
about the soil, just because it is
normal soil.
Cells at equilibrium with soil.
Not same as cells above.
UNI Plant Physiology
51
Problem #4 - b
Soil has different water relations properties than cells do. Because it is open to
the air, the water is just like that sitting on the lab bench. This tells you its
pressure potential, so write the pressure potential for the soil on your copy now.
Because we use open water as our reference (= zero pressure),
water in soil would have the same pressure potential (zero).
The water solution between the particles of most soil is typically quite
dilute, with very low concentrations of minerals. This tells you what the
solute potential is for the soil water. Fill the
root xylem root cell
soil
values for the cells in now on your copy.
-5
0
-8
0
0
www.uni.edu/berg
0
0
0
0
Very dilute solutions have solute
potentials of essentially zero.
Now on to using the other information
Cells at equilibrium with soil.
included in the problem.
Not same as cells above.
UNI Plant Physiology
54
Problem #4 - c
Strategy: First fill in what you know from the material. Now figure out the
values based on other information. Lastly, the arithmetic.
If the cells both at equilibrium with the soil, they must also be at
equilibrium with each other. Fill in the water potential value for both
cells on your copy now.
The water potential for all 3 parts of the system is the same: -5.
root xylem root cell
-5
-5
-5
-5
soil
-5
Now all we have left is the arithmetic
and the check of whether the values
are reasonable.
0
0
-8
0
0
www.uni.edu/berg
0
Cells at equilibrium with soil.
Not same as cells above.
UNI Plant Physiology
55
Problem #4 - d
Strategy: First fill in what you know from the material. Now figure out the
values based on other information. Lastly, the arithmetic.
Fill in the missing values for each part. Check your answer after each cell.
root xylem
root cell
-5 = -5 + 0 + 0
soil
-5 = +3 + (-8) + 0
root xylem root cell
soil
-5
-5
-5
-5
+3
+3
0
0
-8
0
0
0
-5
-5
www.uni.edu/berg
-5 = 0 + 0 + (-5)
-5
The check: All the cell values are
reasonable, and similar to others we
have seen. For soil, the action is
normally in the matric potential,
which is what we see here. The
water potential of the whole system
is at equilibrium.
UNI Plant Physiology
56
Problem #5 - a
Here you have two adjacent root cells in shallow salty water (tide flat).
As usual, your job is to fill in the missing values for the water potential factors.
Strategy: First what you know because of the type of cell or surroundings.
Then values based on other information. Lastly, the arithmetic.
Matric potential is easy for all three. Fill the values in now on your copy.
shallow salty
root xylem root cell
water
-5-15
0-2
000
0
www.uni.edu/berg
-8
-20
0
0
0
0
0
0
Matric potential is zero for wet things.
Now on to filling in more that we
know about the cells and the water.
Transpiration is going on.
UNI Plant Physiology
57
Problem #5 - b
Strategy: Still working on what you know because of the type of cell or
surroundings. Then values from other information. Lastly, arithmetic.
There’s nothing more we can do with the cells, but the water is open
to the atmosphere, so it has the same pressure potential as water in
soil or on the lab bench. Fill in this value for the pressure potential of
the water now on your copy.
shallow salty
root xylem root cell
water
-5-15
0
0
0-2
000
0
www.uni.edu/berg
-8
-20
0
0
0
0
0
The pressure potential of open
water is zero, just as it is in soil.
Now on to filling in more by using the
extra information given with the
problem.
Transpiration is going on.
UNI Plant Physiology
60
Problem #5 – c
Strategy: First working on what you know because of the type of cell or
surroundings. Then values from other information. Lastly, arithmetic.
Transpiration will move water from the surroundings to the root cell, and
then into the root xylem. This tells you that the water potential gets lower
along that path (that’s why the water moves). Draw in arrows showing
the water movement, and put in likely values for the cell water potentials
now on your copy.
root xylem root cell
-19
-19
-17
-17
shallow
salty water
-5-15
Water moves from higher to
lower water potential.
Now all we have left is the arithmetic.
0
0-2
000
0
www.uni.edu/berg
-8
-20
0
0
0
0
0
Transpiration is going on.
UNI Plant Physiology
61
Problem #5 – d
Strategy: First working on what you know because of the type of cell or
surroundings. Then values from other information. Lastly, arithmetic.
On your copy, now fill in the missing values for the water potential
factors, one item at a time, moving from left to right. Then click to check.
root xylem
root cell
-19 = -17 + (-2) + 0
root xylem root cell
-19
-17
-17
0-17
-2
0
00
0
0
+3
+3
-8
-20
0
0
0
0
www.uni.edu/berg
shallow salty water
-17 = +3 + -20 + 0
-15 = 0 + (-15) + 0
salty water
-5
-15
0
All we have left to do is
checking that the numbers
make sense.
-15
-15
0
0
0
UNI Plant Physiology
62
Does it make sense?
Xylem typically has negative pressure potential. The solute potential here is
not zero, but that can be explained by the salty water the plant is in. The
xylem sap has taken in some of that salt. Everything looks OK.
Live cells usually have positive pressures. OK here.
In shallow water the pressure should be zero. In salty water the
solute potential should be negative. Looks good.
root xylem root cell
-19
0-17
-2
0
00
0
0
-17
+3
-8
-20
0
0
0
0
Matric potential should be
shallow salty water zero in all the wet parts of the
system.
-5
-15
0
-15
0
0
0
There should be a gradient of
water potential if there is
transpiration. OK here.
Time for theisnext
problem.
Transpiration
going
on.
www.uni.edu/berg
UNI Plant Physiology
63
Problem #6 - a
Here you have two adjacent root cells in saline (salty) soil. As usual, your job
is to fill in the missing values for the water potential factors.
Strategy: What you know because of the type of cell or surroundings.
Matric potential is easy for the cells. We’ll deal with the soil later. Fill
the values of the matric potentials for the cells in now on your copy.
root xylem root cell
saline soil
-5-8
0
0
0
-8
-4
000
0
0
0
0
0
www.uni.edu/berg
Matric potential is zero for wet things.
(The soil isn’t wet.)
Now fill in a likely value for the soil
pressure potential.
Water in soil, open to the atmosphere,
always has a pressure potential of zero.
Transpiration is going on.
Time for the next step.
UNI Plant Physiology
64
Problem #6 - b
Still working on what you know because of the type of cell or
surroundings.
We know something about the pressure potential of live cells,
even if we don’t know the exact value. Fill in a reasonable
number for the root cell pressure potential.
root xylem root cell
saline soil
-8
+2
+2
Other values are possible, but live
cell pressure potential is always
zero (for wilting cells) or positive
(the rest).
0
-4
0
www.uni.edu/berg
0
UNI Plant Physiology
We have more work to do,
still based on the kinds of cells.
65
Problem #6 - c
Still working on what you know because of cell type or surroundings.
Xylem solute potential probably isn’t zero, because the plant is in
saline soil. Now fill in a reasonable value for the xylem solute potential.
Xylem sap in plants in saline soil is likely to be around -1 bar (it
might be a bit lower), because of the soil salt level.
root xylem root cell
saline soil
-8
+2
-1
-1
0
www.uni.edu/berg
At last it is time to use the other
information to get the remaining
values.
0
-4
0
Transpiration is going on.
UNI Plant Physiology
66
Problem #6 - d
Strategy: Now the values from other information.
Transpiration sets up a gradient of water potential. The water will be
moving from the soil to the root cell to the xylem. Fill in arrows showing
water movement, and reasonable values of water potential for the cells.
root xylem root cell
-12
-12
-10
-10
-8
+2
0
-1
0
saline soil
Other values are possible, but
water always flows from higher
to lower water potential.
Time for the arithmetic.
-4
0
Transpiration is going on.
www.uni.edu/berg
UNI Plant Physiology
67
Problem #6 – e
Now the arithmetic.
Fill in the missing values for each part. Check your answer after each cell.
root xylem
root cell
-12 = -11 + (-1) + 0
-10 = +2 + (-12) + 0
root xylem root cell
-12
-10
-8
+2
0
-1
-12
-12
-4
-4
-4
www.uni.edu/berg
0
-8 = 0 + (-4) + (-4)
saline soil
-11
-11
0
saline soil
Time for the final check.
Try your own before
proceeding.
UNI Plant Physiology
68
Does it make sense?
Xylem typically has negative pressure potential. The solute potential here
is not zero, but that can be explained by the saline soil the plant is in. The
xylem sap has taken in some of that salt. Everything looks OK.
Live cells usually have positive pressures. OK here.
In any soil the pressure should be zero. In saline soil the solute
potential should be negative. In soil that isn’t really wet, the matric
potential should be negative.
Looks good.
root xylem root cell saline soil
-12
-11
-1
0
-10
-8
+2
0
-12
0
-4
-4
There is a gradient of water
potential that drives transpiration.
Other values are possible, but
water always flows from higher to
lower water potential.
Transpiration is going on.
www.uni.edu/berg
UNI Plant Physiology
69
Problem #7 - a
Here you have two adjacent root cells in a highly saline environment.
Strategy: What you know because of the type of cell or surroundings.
Matric potential is easy for the cells. Fill the values for the cells in now.
Matric potential is zero for wet things.
root xylem
-8
root cell
-8
-8
+5
Now use the information given to put in the
water potential for the root cell.
Cells are often in water potential
equilibrium (same) before dawn.
-3
0
0
www.uni.edu/berg
0
0
Plant growing in very saline soil.
Cells
equilibrium
Timeare
foratthe
arithmetic.with
each other (before dawn).
UNI Plant Physiology
70
Problem #7 - b
Now the arithmetic. Fill in the missing values for each part. Check
your answer after each cell.
root xylem
root cell
-8 = -5 + -3 + 0
-8 = +5 + (-13) + 0
root xylem
root cell
-8
-8
-5
-5
-3
+5
-13
-13
0
0
www.uni.edu/berg
Now check the values to make sure they
make sense, given the type of cell and the
conditions.
Plant growing in very saline soil.
Cells are at equilibrium with each
other (before dawn).
UNI Plant Physiology
71
Does it make sense?
Xylem typically has negative pressure potential. The solute potential here
is quite low, but that can be explained by the extremely salty environment
the plant is in. The xylem sap has taken in quite a bit of that salt.
Everything looks OK.
The live cell has a positive pressures, and a quite low solute potential,
which we might expect in the highly saline environment. OK here.
root xylem
root cell
-8
-8
-5
-3
+5
-13
0
0
www.uni.edu/berg
The cell water potentials aren’t very low, so
the soil must be quite moist, even thought it
is quite salty. This often happens in areas
with ample irrigation with saline water.
Plant growing in very saline
soil. Cells are at equilibrium
with each other (before dawn).
UNI Plant Physiology
72
Problem #8 - a
Here you have two adjacent root cells in a highly saline environment.
What you know because of the type of cell or surroundings. Fill in the
values for matric potential of the cells now.
root xylem
root cell
-3
-3
0
0
-13
-13
0
0
www.uni.edu/berg
Now use the information given to figure out
more. These are the same cells as above, so
solute potentials will be about the same. Put
these in now.
Same plantTime
and soil
as cells
in previous
for the
arithmetic.
problem. Transpiration is going on.
UNI Plant Physiology
73
Problem #8 - b
Now use the information given to figure out the effects of transpiration.
You have to figure out which way the water is moving as transpiration is
going on, and what are likely values for the water potential. These are the
same cells as the previous problem, but the transpiration may have
dragged their water potentials down a little lower.
Draw an arrow showing the direction of transpiration, then put
in plausible values for the water potential.
root xylem
root cell
-12
-12
-10
-10
-3
-13
0
0
www.uni.edu/berg
Water will move from soil (not shown) into the
live root cell, then into the xylem for transport
upward. There are other possible values for
water potentials, but water will always move
from higher to lower values. There are other
limits on the possible values, too. If you exceed
them, your other values won’t make sense.
Same plant and soil as cells in previous
problem.
Transpiration
is going on.
Nothing
left but arithmetic.
UNI Plant Physiology
74
Problem #8 - c
The arithmetic should be easy by now. Fill it in and check your values.
root xylem
-12 = -9 + (-3) + 0
root cell
root xylem
-12
-9
-9
Does it make sense? Well, you have a water
potential gradient, and zero matric potential,
which is what you expect.
-10 = +3 + (-13) + 0
The other thing to check is that the pressure
potential is negative in the xylem and positive
in the live cell. This is what places limits on the
root cell
water potential. You can’t have the live cell with
a water potential of -20 here, because that
-10
would give you a negative pressure potential in
+3
+3
a live cell (impossible).
-3
-13
0
0
Same plant
On toand
the soil
last as
problem.
cells in previous
problem. Transpiration is going on.
www.uni.edu/berg
UNI Plant Physiology
75
Problem #9 - a
We’re up in the top of the tree now. With the same xylem sap as in the
previous problem. The soil has now dried substantially.
What you know because of the type of cell or surroundings. Fill in the
values for matric potential of the cells now.
leaf cell
-3
0
0
www.uni.edu/berg
xylem
-3
-3
0
0
Now use the information given to figure out
more. The xylem (right here) is the same as in
the problem above (on left), so its solute
potential will be the same. Put the value in now.
Same plant and soil as in previous problem.
The soil is at -15 bars, the whole plant is at
equilibrium with the soil (night), and the leaf
is wilting.
Almost done.
UNI Plant Physiology
76
Problem #9 - b
Continue using the information given to figure out more. The equilibrium
between the cells and the soil is next. Put values for cell water potentials in now.
Use the “wilting” information to determine the leaf cell pressure potential.
Fill in the value now.
Wilting tissue always has a pressure potential of zero.
Now do the arithmetic, one cell at a time.
leaf cell
xylem
root xylem
-15
-15
-15
-15
0
0
-15
-15
-12
-12
0
-3
0
root cell
-15
= +0
+ (-15)
0 as-15
= +(-12) problem.
+ (-3) + 0
Same
plant
and+soil
in previous
The soil is at -15 bars, the whole plant is at
equilibrium with the soil (night), and the leaf
is wilting.
One last check that the values are sensible.
www.uni.edu/berg
UNI Plant Physiology
77
Does it make sense?
The wilting live cell has zero pressure potential, and the xylem has a negative
pressure potential. Everything looks OK there.
Cells in equilibrium have the same water potential. The matric potential is
zero. Looks OK here.
The low solute potential in the xylem is believable because the plant
is growing in very saline soil.
leaf cell
xylem
-15
-15
0
-12
-15
-3
0
0
www.uni.edu/berg
Everything looks fine. You’re done!
You can go back to review any thing and
any time you wish.
Same plant and soil as in previous problem.
The soil is at -15 bars, the whole plant is at
equilibrium with the soil (night), and the leaf
is wilting.
UNI Plant Physiology
78
Thank You
Thank you for testing this program. If you find any errors,
please email them to me at bergv@uni.edu. If you have any
comments, please do the same. Please share the folder
(program, problem set and viewer) with your colleagues and
students.
Because this program has my name on it, please DO NOT
modify it. I will try to correct errors and make improvements as
fast as possible, and will post the new version on my download
web page, with the date of the latest correction.
www.uni.edu/berg
UNI Plant Physiology
79