Science and Cooking: Problem Set 2
Due on Canvas by 11 PM on Thursday, September 18th
You are encouraged to work in groups, but all submitted work must be your own. If you
work with others, please note who they are on the front page of your problem set.
Please type or write your answers in a separate document. Your work must be
organized and legible – if your TF can’t understand what you wrote, they won’t give you
credit.
Show your work for derivations and calculations. Be sure to calculate all results fully
(don’t leave numbers in fraction form, or in terms of pi, etc) and to provide answers in
the requested units, if applicable.
Equations of the Week
Q = mcpΔT
Q = mLf
Q = mLv
Concept
cp
Lf
Lv
Description
Heat input required to raise the temperature of a
substance by a specified amount
Heat capacity
Latent Heat of Fusion
Latent Heat of Vaporization
Power
Amount of energy output per second
Calorie
Energy Density
The energy required to heat 1L of water 1°C
Energy liberated by burning a substance
Q
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
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http://en.wikipedia.org/wiki/Simmering
2
Units
J (kgm2/s2)
J/g·°C
J/g
J/g
Watts or W
(J/s)
J/g
Problem 1: Your Favorite Recipe (10 points.)
We ask you to try many recipes at home for this course, but we’d like to know what you
like to make at home as well. Attach your favorite recipe and ask three brief questions
about things you would like to understand scientifically about it.
Problem 2: Lab 1 Follow-up (45 points).
For this problem, it will be very helpful to read On Food and Cooking pages 571-577 as
well as Harold McGee’s NY Times article:
http://www.nytimes.com/2009/02/25/dining/25curi.html
In this week’s lab you dissected one of the simplest recipes: pasta! This question
elaborates on the lab by asking you to think more about the scientific properties of
pasta, and to compare your measurements in lab to theoretical predictions based on the
equation of the week. If you haven’t done the lab for this week yet, you may want to
skip to Problems 3 and 4 and come back to this one later.
a) How much heat does it take to bring 1 L of room temperature water to boiling (i.e.
to 100oC)? How much heat is needed for 2L? For 4L? (6pts)
1 L: (1 L) * (1000 mL/L) * (1 g/mL) * (4.18 J/g K) * (100 oC-25 oC) * (1 K/ oC) =
313,500 J
2 L: (2 L) * (1000 mL/L) * (1 g/mL) * (4.18 J/g K) * (100 oC-25 oC) * (1 K/ oC) =
627,000 J
4 L: (4 L) * (1000 mL/L) * (1 g/mL) * (4.18 J/g K) * (100 oC-25 oC) * (1 K/ oC) =
1,254,000 J
b) Assume the power of your stove’s burner is 1000W. Calculate how long it should
take to boil 1L, 2L, and 4L of water. (3pts)
(heat to boil water) = (power of burner) * (time to boil water)
1 L: 313,500 J = 1000 W * (x min) * (60 s/min)  x = 5.2 min
2 L: 627,000 J = 1000 W * (x min) * (60 s/min)  x = 10.5 min
4 L: 1,254,000 J = 1000 W * (x min) * (60 s/min)  x = 20.9 min
c) How do the experimental boiling times from your lab data compare to your
calculations in (b)? Why would the experimental and calculated boiling times
differ? (2pts)
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
3
http://en.wikipedia.org/wiki/Simmering
2
Experimental boiling times are longer than the calculated boiling times. Here are a few
possible reasons why:
- The power of the burner might be less than 1000 W. The older or more poorly
calibrated the burner, the less reliable the advertised power.
- Some energy from the burner is always lost to waste heat. The lower the
efficiency of heat transfer from the burner to the water in the pot, the longer it
takes to add enough energy to the water to get it to boil
- Note: Adding only small amounts of salt to the water (as we did) does not
elevate the boiling point enough to make a huge difference in how much energy
(and how long) it takes for the water to boil!
d) Theoretically, how much is the temperature of 1 L of boiling water supposed to
drop when you add 200 grams of room temperature pasta? How much is it
supposed to drop for 4 L? The specific heat of water is 4.18 J/g˚C, and the
specific heat of pasta is 1.8 J/g˚C. (6pts)
Q = 0 = m1c1(Tf-Ti1) + m2c2(Tf-Ti2), Thus the Q for the pasta is equal to the
negative Q for the water and vice-versa, so:
m1c1(Tf-Ti1) = -m2c2(Tf-Ti2) where subscript 1 is for pasta, subscript 2 is for water
(Tf * m1c1) – (Ti1 * m1c1) = (Ti2* m2c2) – (Tf * m2c2)
(Tf * m1c1) + (Tf * m2c2) = (Ti2* m2c2) + (Ti1 * m1c1)
Tf * (m1c1 + m2c2) = (Ti2* m2c2) + (Ti1 * m1c1)
Tf = [(Ti2* m2c2) + (Ti1 * m1c1)]/ (m1c1 + m2c2)
Tf = [(100˚C * 1000g * 4.18 J/g˚C) + (23˚C * 200g * 1.8 J/g˚C)]/[(200g * 1.8 J/g˚C)
+ (1000g * 4.18 J/g˚C)] = (418000 + 8280)/(4180 + 360) = 93.89˚C
Thus, the drop in temp for 1L of water is 100 – 93.89 = 6.11˚C
For 4 Liters, just change the mass of water to 4000g in the final calculation.
Should give you an answer of:
Tf = (1672000 + 8280)/(16720 + 360) = 98.38˚C
Thus, the drop in temp for 4L of water is 100 – 98.38 = 1.62˚C
e) How does this compare to the temperature drop you observed in the lab? (2pts)
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
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http://en.wikipedia.org/wiki/Simmering
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f) Theoretically, how much heat is required to bring the pasta + water back up to a
boil for the 1 L pot? For the 4 L pot? (6pts)
Q = m1c1(Tf-Ti1) + m2c2(Tf-Ti2) where subscript 1 is for pasta, subscript 2 is for
water (4pts)
Since both Tf and Ti are the same for the pasta and water, this becomes much
simpler to calculate:
Q = (100˚C - 93.89˚C) * (m1c1 + m2c2) = 6.11˚C * [(200g * 1.8 J/g˚C) + (1000g *
4.18 J/g˚C)] = 6.11˚C * 4540 J/˚C = 27739 Joules (1pt for 1L and 4L each)
The answer is THE SAME for the 4 liter pot! Since the amount of heat lost by the
water and gained by the pasta is a function of how much pasta is added, the
amount of heat needed to make up for the heat lost in heating the pasta up is the
same no matter how much water you use. Alternatively, you can run the same
calculation as above but change the inputs accordingly (the answer will differ by
about 200 Joules due to rounding errors).
g) Using the values you recorded from your lab for how much the temperature of 1
L and 4 L pots of water dropped upon adding pasta, how much heat did it take for
you to bring the water and pasta back to a boil? (Hint: this is the same calculation
as f). How do these numbers compare to the theoretical value in (f)? (6pts)
Same calculation as in f, just with different values.
h) Based on the water content in the cooked pasta from your lab, do you think it
would work to cook pasta in 0.5 L of water? Why or why not? (3pts)
Figure out how much water total the pasta absorbed and cite that mass here –
and then compare it to 500g, which is the mass of the water in 0.5L. If the total
amount of water absorbed is greater than 500g, then 0.5L of water will not work.
i) From Wolfram Alpha1, a serving of “dry macaroni” (140g in total) contains 2g of
fat, 102g of carbohydrates, and 21g of protein. Use the 4,4,9 rule to calculate the
number of calories you would expect to find in this serving of dry macaroni (show
your work!). (6pts)
2g fat • 9Cal/g = 18Cal
102g carbs • 4Cal/g = 408Cal
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
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http://en.wikipedia.org/wiki/Simmering
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21g protein • 4Cal/g = 84Cal
18Cal fat + 408Cal carbs + 84Cal protein = 510 Calories
j) What happens to the starch granules in pasta while you are cooking it? Be sure
to mention the roles of both heat and water (recall the McGee article). (5pts)
Pasta contains starch molecules that aggregate into large granules. When they
are in a moist environment such as a pot of water, they begin to absorb water
and swell until they burst. Heat is also required for the starch to absorb water.
The pasta must reach a certain temperature before the starch granules will begin
to swell. Once the starch granules burst, they release starches into the water. As
more starches swell and burst, more water is absorbed into pasta, and it
becomes softer and more edible.
Key points
Granules absorb water and swell until they burst
Heat is required for the pasta to absorb water
Starch is released into the water
Problem 3: Olive Tomato Cream Sauce (35pts)
This sauce was served in the Harvard dining halls last week. For all questions, assume
that room temperature = 23˚C. This table may be useful in finding the specific heats of
different foods2:
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
3
http://en.wikipedia.org/wiki/Simmering
2
a) The power output of your stove is 2kW. Butter melts at 35˚C, and the heat
capacity of solid butter is 1.42 J/g˚C. If the butter starts at room temperature, how
long will it take to melt the butter on the stove? Don’t forget the latent heat of
fusion! (6 pts)
Q = mcpΔT
Q = P•t
t = mcpΔT/P
2kW = 2000W
0.9931lbs butter = 450.5g
t = (451.4g)(1.42J/g•°C)(35°C-23°C)/(2000W) = 3.8seconds
b) Starting out with a cold pan, you find that it takes longer to melt the butter. What
are some reasons for the differences you observed between the theoretical and
actual melting times? (5pts)
Some possible reasons:
- Heat is being used to heat the pan as well as the butter
- Heat is being lost to the air/environment
- The butter is being heated from only one side
c) Based on the heat capacity in part b, how much energy is required to melt the
amount of butter in this recipe (assume 1g = 0.0022lbs)? (5pts)
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
3
http://en.wikipedia.org/wiki/Simmering
2
Q = mcpΔT (2pts)
Q = (451.4g)(1.42J/g•°C)(35°C-23°C) = 7691.9J (3pts)
d) How long will it take? (5pts)
Same as a, free points.
e) If you did not have a stove to make this cream sauce, you would have to burn
wood to heat it. The energy density of wood is 14MJ/kg (capital M stands for
“Mega” and is 106 of whatever unit it’s placed in front of). How much wood do you
need to bring the cream to a simmer? The optimal simmering temperature is
under debate, but for this case let’s assume it is 82˚C3. Assume the cream is at
room temperature before you add it to the pot and that 1 cup cream = 240g. The
heat capacity of cream is 3.1J/g˚C. (14pts)
First: find the amount of energy needed to bring the cream to a simmer:
Q = mcpΔT
22.6972quarts = 90.7888cups = 21789.3g
Q = (21789.3g)(3.1J/g•°C)(82°C-23°C) = 3985263 or ~ 3990000 Joules
Second: find the amount of wood needed to generate this heat:
14MJ/kg = 14000000 J/kg
3990000 Joules * 1kg/14000000 Joules = 0.285kg
Problem 4: Cooking Tuna in Exhaust (10 points+10points Extra Credit)
A student in a previous year of this class, Taylor Reiter, told Michael Brenner a story
about a Japanese fisherman who had caught a huge tuna fish, and dragged it back to
port behind the boat. Unfortunately, by the time the fisherman got back, he found that
the fish had cooked in the heat from the exhaust.
a) Is this possible? What information would you need to know to figure this out?
Write the variables needed below and label them clearly (e.g. T i = the initial
temperature of the fish, etc.). (5pts)
Assuming that the power output of the boat is constant, and assuming all the
heat from the exhaust goes directly in to the tuna
- final temperature of tuna required for cooking
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www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
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http://en.wikipedia.org/wiki/Simmering
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- power of the exhaust or amount of heat released by the exhaust per unit time
- speed of the boat and distance to home or time the boat was being driven
- mass of the tuna
- heat capacity of tuna
- temperature of ocean
b) Reorganize the equation(s) of the week to determine the answer to this question:
Assuming power output of the exhaust is constant, how long does the boat have
to run to cook an average-sized tuna? Use the variables you defined in part (a) in
place of actual numbers. (5pts)
t = mcpΔT/P
t = time boat is running or distance/speed
t = (m of tuna)(cp of tuna or other cooked fish)(Tcooked tuna Tocean)/(Pboat exhaust)
mass of tuna
Cp of tuna
Change in temp
P of boat exhaust
c) Extra Credit (optional): Find reasonable numbers for the values in part b (and cite
your sources), and answer the question. (10pts)
1
www.wolframalpha.com
http://www.engineeringtoolbox.com/specific-heat-capacity-food-d_295.html
3
http://en.wikipedia.org/wiki/Simmering
2