# nally radial direction by sanmelody

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```									                                  Chapter 17: Temperature and Heat-4/5/10

It is crucial that you clearly distinguish between temperature and heat. Temperature is basically what a
“thermometer reads” in a defined scale (such as the Celsius or the Fahrenheit scales) and it measures the
average kinetic energy of the particles in a substance. Heat cannot be measured directly, but can be calculated
from its effect on a substance (such as a temperature change), or from energy conservation principles. Heat is
commonly thought of as a form of energy, but in calorimetry, it’s defined as the transfer of energy between
substances at different temperatures or in different states of matter. Heat is in reality the net amount of
microscopic work done to transfer energy across a boundary between substances. Just like with work, the units
of heat are Joules or any other energy unit.

The Constant-Volume Gas Thermometer and the Kelvin Scale

1. Thermometers measure their own temperature but they can be used to measure the temperature of other
objects because of the “Zeroth Law of Thermodynamics”.
a) Explain this statement. How does a thermometer affect the temperature that it’s measuring?
b) What would be the difficulty in measuring the temperature of a very small amount of fluid, such as a
thimbleful of water?

2. In principle, any gas can be used in a constant-volume gas thermometer.
a) What are the practical limitations of this principle? For example, is it possible to use oxygen for
temperatures as low as 15 K? (Look at the data on phase change temperatures in the textbook). Is there a gas
that could be used at such low temperatures?
b) A constant-volume gas thermometer registers a pressure of 0.062 atm when it is at a temperature of 450 K.
(i) What is the pressure at the triple point of water (0.16 ºC)? (ii) What is the temperature when the pressure

a) A substance is heated from -12°F to 150°F. What is its change in temperature on (i) the Celsius scale and
(ii) the Kelvin scale?
b) How many degrees Fahrenheit (ºF) are there in one degree Celsius ºC)?
c) At what temperature are the readings from a Fahrenheit thermometer and Celsius thermometer the same?

Thermal Expansion of Solids and liquids
You may need to refer to the Table of Coefficients of Expansion in the textbook

4. Markings to indicate length units are placed on a steel tape in a room at 22°C. Are measurements made with
the tape on a day when the temperature is 27°C too long, too short, or accurate? Defend your answer.

5. One surprising feature of thermal expansion is the cavities inside materials expand at the same rate as the
surrounding material. Explain how that happens. The following questions all highlight that fact.
a) Metal lids on glass jars can often be loosened by running hot water over them. How is this possible?
b) A flat metal ring has a gap in it. If the ring is heated, will the gap get smaller, larger, or stay the same?
c) In a class demonstration, when the metal ring and sphere are both at room temperature, the sphere fits
through the ring, but when heated, the sphere will not fit through the ring. After the ring is also heated, the
sphere can be passed through the ring again. Explain.
d) If the ball is slightly bigger that the hole in the ring hole at room temperature, can BOTH be heated or
cooled to another temperature so that the ball can fit through the ring? Remember that both are made of the
same material.

6. At 20.0°C, an aluminum ring has an inner diameter of 5.000 cm and a brass rod has a diameter of 5.050 cm.
(a) To what temperature must the ring be heated so that it will just slip over the rod? (b) To what temperature
must both be heated so that the ring just slips over the rod? Would this latter process work?
7. A copper rod and steel rod are heated. At O°C the copper rod has a length of Lc, the steel one has a length
Ls. When the rods are being heated or cooled, a difference of 5.0 cm is maintained between their lengths.
Determine the values of LC and Ls.

8. A pendulum clock with a brass suspension system has a period of 1.000 s at 20.0°C. If the temperature
increases to 30.0°C, (a) by how much does the period change and (b) how much time does the clock gain or
lose in one week? Hint: Recall that the period of a simple pendulum is T=2π(L/g)1/2 and take the derivative
dT/dL. Refer to the table in the textbook for expansion coefficients.

9. A rectangular plate has an area A equal to its length times its width (lw).
a) If the temperature increases by ∆T, show that the increase in area is ∆A = (2A∆T, where is the average
coefficient of linear expansion. What approximation does this expression assume?
b) Similarly, you can show that the volume expansion coefficient  = 3.

10. A liquid has a density . (a) Show that the fractional change in density (∆) for a change in temperature
∆T is (∆) ~ - ∆T. What does the negative sign signify? (b) Fresh water has a maximum density of 1.000
g/cm3 at 4.0°C. At 10.0°C and its density is 0.9997 g/cm3. What is the  for water over this temperature
interval?

11. In designing structures engineers must make allowances for the thermal expansion of metals otherwise the
structures could experience catastrophic stresses. Consider this example: Steel rails for a rapid transit train
form a continuous track that is held rigidly in place in concrete. (a) If the track was laid when the temperature
was O°C, what is the stress (F/A) in the rails on a warm day when the temperature is 25°C? (b) What fraction
of the yield strength* of 52.2 X 107 N/m2 does this stress represent? Hint: You need to relate the linear
expansion coefficient  to the Young’s modulus through the fractional change in length (∆L/L). c) Does this
help you understand why gaps are part of the design of structures? Note: For steel Y=20 x1010 N/m2 and  =
12 x10-6 1/ºC.
* Yield strength is the maximum stress (F/Aperp) that a material can withstand before it begins to fail.

12. A cylindrical pot of cross-section A contains a volume V or water. The pot and water are heated by ∆T.
a) Derive a formula for the increase in the height of the water if the expansion of the pot is neglected. and
b) Redo (a) including the expansion of the pot.
c) For a numerical version, assume the pot is made of aluminum. The pot is initially at 4.0°C, at which
temperature it has an inside diameter of 28.00 cm. Initially the pot contains 3.000 gal (1 gal ~ 3785 cm3) of
water at 4.0°C. and then heated to 90.0 ºC. Allowing for the expansion of the water but ignoring the expansion
of the pot, what is the change in depth of the water? (The density of water is 1.000 g/cm3 at 4.0°C and 0.965
g/cm3 at 90.0°C.)
d) Modify your solution to allow for the expansion of the pot. Is it necessary to consider the expansion of the
pot in this problem?

13. Explain why a column of mercury in a thermometer first descends slightly and then rises when placed in
hot water. Now try these problems, which have similarities to the one above.
a) A typical mercury thermometer is made up of a thin, cylindrical
capillary tube with a diameter of 0.0040 cm, and the spherical bulb with
a diameter of 0.25 cm. Neglecting the expansion of the glass, find the
change in height of the mercury column for a temperature change of 30°C.                     ∆h
b) If you don’t neglect the expansion of the glass, what is the change in
height of the mercury column above? Was it justified to ignore the glass?
T         T+∆T
14. The relationship L ~ Lo(1 + ∆T) is an approximation that works when the average coefficient of
expansion and/or the change in temperature is small. If  is sizable, the relationship dL/ dT =  L must be
integrated to determine the final length.
a) Assuming the “” is constant, determine a general expression for the final length L.
b) Plot a graph of L vs. Temp. What does the slope of this graph represent?
c) Given a rod of length 1.00 m and a "temperature change of 100.0 ºC, determine the % error caused by the
approximation when  = 2.00 X 10-5 (ºC)-1 (the normal value for common metals) and when  = 0.020 (ºC)-1
(an unrealistically large value just for comparison purposes).

15. A bimetallic bar is made of two thin strips of dissimilar metals bonded together. As they are heated, the
one with the larger average coefficient of expansion expands more than the other, forcing the bar into an arc,
with the outer radius having a larger circumference.                                      ∆r
(a) Derive an expression for the angle of bending ø in terms of the initial length
of the strips L, their average coefficients of linear expansion, the change in
temperature, and the separation of the centers of the strips (∆r=r2-r1)
(b) Show that the angle of bending goes to zero when ∆T goes to zero or when                   ø
the two coefficients of expansion become equal.
(c) What happens if the bar is cooled rather than heated?

Heat and Thermal Processes
You may need to refer to tables in the text for values of heat capacities and latent heats.

16. In as experiment you put heat into a 500 g solid sample of a material at the rate of 10 kJ/min, while
recording temperature as a function of time.
You plot your data on a graph.                                                 T(ºC)
40

a) What is the melting/freezing temperature and the latent heat of               30
fusion for this solid?
20
c) What are the specific heats of the solid and liquid phases of the material?
d) Explain why the unit of specific heats can be equally expressed as            10

J/kg ºC or J/kg K.                                                              0
1   2         3   t(min)

17. A student inhales 22°C air and exhales 37°C air. The average volume of air in one breath is 200 cm3.
a) Neglecting evaporation of water into the air, estimate the amount of heat absorbed in one hour by the
air breathed by the student. The density of air is approximately 1.25 kg/m3, and the specific heat of air is
1000 J/kg· °C.
b) Consider a classroom of dimension 10 m x10 m x4 m with 30 students. During the course of a two-
hour lecture, how much would the room’s temperature rise due to the breathing of the students?
c) If you consider that the human body is mostly water and a typical body mass is 60 kg, how much would
the body temperature drop in one hour due to the breathing? So how does our body keep its temperature?

18. A 300-g aluminum vessel contains 200 g of water at 10°C.
a) 100 g of water at 100°C is poured into the container, what is the final equilibrium temperature of the
system?
b) 100 g of steam at 100°C is poured into the container, what is the final equilibrium temperature of the
system?
c) Why can you get a more severe burn from steam at 100°C than from water at 100°C?

19. A 1.0-kg block of copper and a 1.0-kg block of lead at 20°C are dropped into a large vessel of liquid
nitrogen at 77 K and some of the nitrogen boils away. (The specific heat of copper is 0.092 cal/g °C and
of lead it’s 0.031 cal/g °C. The latent heat of vaporization of nitrogen is 48 cal/g.)
a) Which one will boil away the greatest amount of nitrogen? Explain.
b) How many kilograms of nitrogen boil away by the time the blocks reach 77 K?
c) Which of these metals would be safer to hold if they were at 100ºC?

20A. In an insulated vessel, 250 g of ice at 0°C is added to 600 g of water at 18°C.
a) What is the final temperature of the system?
b) How much ice remains when the system reaches equilibrium?
c) Draw a T vs. ±Q graph for the ice and the water for the entire process
d) What if the ice started out at -10ºC?
e) Would it be possible to freeze the added water?

20B. Redo the problem above but adding only 100 g of ice at 0ºC to the 600g of water at 18°C.

21. A cooking vessel on a slow burner contains 10.0 kg of water and an          T(ºC)
unknown mass of ice in equilibrium at 0ºC at time t = 0. The temperature        3.0

of the mixture is measured at various times, and the result is plotted in the   2.0
graph. During the first 50 min, the mixture remains at 0ºC. From 50 min
1.0
to 60 min, the temperature increases to 2.0°C. Neglecting the heat capacity
of the vessel, determine the initial mass of the ice.                           0.0

20     40      60     t(min)
22. A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique is to
measure the temperature difference between the input and output points of a flowing stream of the liquid
while adding heat at a known rate. In one particular experiment, a liquid of density 0.78g/cm3 flows
through the calorimeter at the rate of 4.0 cm3/s. At steady state, a temperature difference of 4.8°C is
established between the input and output points when heat is supplied at the rate of 30 J/s. What is the
specific heat of the liquid?

23. In a famous experiment John P. Joule demonstrated that a change in potential energy could generate a
change in temperature in water, thus establishing the fact that “heat was a form of energy”. Joule was on
the right track although we now think of heat as a “transfer of energy”. In the following problems a
change in mechanical energy leads to a change in temperature or phase change. The key is to use “energy
conservation” and make the proper unit changes. Remember 1 J = 1 Nm =4.18 cal.

a) A 3.0-g copper penny at 25°C drops 50 m to the ground. If 60% of its initial potential energy goes
into increasing the internal energy, determine its final temperature. Does the result depend on the
mass of the penny? Explain.
b) A 3.0-g lead bullet at 25ºC is traveling at 240 m/s when it embeds in a block of ice at 0°C. If all
the heat generated goes into melting ice, what quantity of ice is melted? (The latent heat of fusion
for ice is 80 kcal/kg, and the specific heat of lead is 0.030 kcal/kg ºC.)

24. An iron plate is held against an iron wheel so that there is a sliding frictional force of 50.0 N acting
between the two pieces of metal. The relative speed at which the two surfaces slide over each other is 40.0
m/s. (a) Calculate the rate at which mechanical energy is converted to thermal energy. (b) The plate and
the wheel have a mass of 5.00 kg each, and each receives 50% of the thermal energy. If the system is run
as described for 10.0 s and each object is then allowed to reach a uniform internal temperature, what is the
resultant temperature increase?

Heat Transfer processes

25. It is important to distinguish between specific heat and heat conductivity. The following questions are
designed to clarify the difference.
a) A tile floor in a bathroom may feel uncomfortably cold to your bare feet, but a carpeted floor in an
adjoining room at the same temperature will feel warm. Why?
b) So called “fire walkers” walk over burning coals seemingly unharmed. How are they able to do
this?
c) Concrete has a higher specific heat than soil. Use this fact to explain (partially) why cities have a
higher average night-time temperature than the surrounding countryside. If a city is hotter than the
surrounding countryside, would you expect breezes to blow from city to country or from country
to city? Explain.
d) If you hold water in a thin paper cup over a flame, you can bring the water to a boil without
burning the cup. How is this possible?

26. Two conceptual questions for those who drink coffee:
a) Suppose you pour hot coffee for your guests, and one of them chooses to drink the coffee after it has
been in the cup for several minutes. In order to have the warmest coffee, should the person add the cream
just after the coffee is poured or just before drinking? Explain.
b) Two identical cups both at room temperature are filled with the same amount of hot coffee. One cup
contains a metal spoon, while the other does not. If you wait for several minutes, which one of the two
will have the warmer coffee? Which heat transfer process explains your answer?

27. A bar of gold (Au) is in thermal contact with a bar of silver (Ag) of the
same length and area. One end of the compound bar is maintained at 80 ºC           80 ºC  Au      Ag      30ºC

while the opposite end is at 30.0 ºC.                                                             Insulation
a) When the heat flow reaches steady-state, find the temperature at the junction.
b) Determine the temperature gradient (dT/dx) in each metal.
c) Draw a graph of T vs. x from end to end.
d) What would change if the two materials were interchanged?
e) If the two bars were replaced with a single bar of the same length and cross-section what would be the
thermal conductivity constant of the material that would have the same heat current as the two bars?

27B. For more practice, repeat problem above, but with the gold segment being L/3 and the silver being
2L/3. Does the slope of the temperature gradient depend on the length of the material?

27C. Two rods of the same length L, but different materials and cross-sectional areas Thot              1           Tcold
are placed side-by-side as shown. a) Determine the rate of heat flow in terms of the               2
thermal conductivities, areas, and temperature difference. Generalize to several rods.                      Insulation
b) Determine the equivalent k’ of the material of a single rod that could replace the two given here.

28. A box with a total surface area of 1.20 m2 and a wall thickness of 4.00 cm is made of an insulating
material. A 10.0-W electric heater inside the box maintains the inside temperature at15.0°C above the·
outside temperature.
a) Find the thermal conductivity k of the insulating material.
b) If you were using two or more layers of insulation, would the order of the layers matter? For example,
would the heat flow rate be different if the higher k material were closer to the colder temperature or
viceversa?

29. More and more windows are made with “double-pane” glasses to save energy. Calculate the “R” value
of (a) a window made of a single pane of glass 1/8 in. thick and (b) a thermal pane window made of two
single panes each 1/8 in. thick and separated by a l/4-in. air space. (c) By what factor is the heat loss
reduced by using the thermal window instead of the single pane window?

The following problems require calculus.

30. When a warm object cools to room temperature its temperature decreases exponentially over time.
This is called “Newton’s law of cooling” : T= (To – Troom)e-Ct + Troom , where To is the initial temperature
of the body and “C” here is a constant that depends on the physical properties of the object (we will be
testing this law in the lab). Show that this law can be derived from the formula for the heat current.

31. The following problems have similar solutions to the example worked out in class for a thermal
conductor with a varying cross-section. In these problems the heat current flows radially outward and the
solution starts with the differential form of the heat current formula: (dQ/dt)= kA(-dT/dr), where (dT/dr)
is the temperature gradient. The varying area “A” is identified and expressed in terms of “r”. Once the
proper integral is set-up the derivation is straightforward.                               r1          r2
a) The example worked out in class featured a heat conductor of length L and                     L        H
conductivity constant k that is shaped like a truncated cone. The radii of the         T1         T2
ends are r1 and r2. Show that the heat current is H= kπ(r1r2)∆T/L.
b) A vessel in the shape of a spherical shell has an inner radius a and outer radius b.
The wall has a thermal conductivity k. If the inside is maintained at a temperature Ta          a
and the outside is at a temperature Tb , show that the heat current between                 Ta        b

Tb
dQ 4 kab
the spherical surfaces is:             
 Ta  Tb 
dt  b  a 

c) The inside of a hollow cylinder is maintained at a temperature T1 while the outside
T2        r2
is at a lower temperature, T2. The wall of the cylinder has a thermal conductivity k.

Neglecting end effects, show that the rate of heat flow from the inner wall (radius r1)           T1   r1
dQ          T  T 
to the outer wall (radius r2) in the radial direction is:        2Lk 1 2 
dt         ln r2 /r1 

Special challenge problems

32. For a numerical version of the problem above   consider a Thermos bottle in the shape of a cylinder
that has an inner radius of 4.0 cm, outer radius of 4.5 cm and length of 30.0 cm. The insulating walls have
a thermal conductivity equal to 2.0 X 10-5 cal/s-cmºC. One liter of hot coffee at 90ºC is poured into the
bottle. If the outside wall remains at 20°C, how long does it take for the coffee to cool to 50°C? (Assume
that coffee has the same properties as water.)

33. A pond of water at O°C is covered with a layer of ice 4.0 cm thick. Heat flows from the warmer water
below to the colder air above through the ice layer. If the air temperature stays constant at – l0°C, how
long will it be before the ice thickness is 8.0 cm? (Hint: To solve this problem, write the conductivity
formula as dQ/dt= kA(∆T/x) and note that the incremental heat dQ extracted from the water through the
thickness x of ice is the amount required to freeze a thickness dx of ice.         Air at -10ºC
That is dQ = LAdx, where is the density of the ice, A is the surface
area of the ice layer, and L is the latent heat of freezing).You will need
Ice                    x
the following: kice=2 W/mCº, L=335 kJ/kg, ice=917 kg/m3.
freezing layer of ice             dx

Water at 0ºC

34. Two problems about radiation. Stefan’s law can used to determine the heat flow in or out of bodies
a) Consider the human body. Typically the skin surface area is ~1.2 m2 and the skin temperature is ~30ºC.
Considering the body and the surrounding air at ~20ºC, determine the net heat loss rate of our warm
bodies. Assume the emissivity of our bodies “e”=1. If you eat 2000 kcal/day, what % of that amount is
heat loss?
b) The intensity of sunlight (the “solar constant”) arriving at the earth’s surface has been measured to be
~956 W/m2 (this takes into account the 30% or so reflected sunlight due to clouds, etc…; above the
atmosphere the solar constant is larger, ~1366 W/m2). Only half the earth’s surface can absorb sunlight at
one time and, and since the earth is round, you have to consider the perpendicular area component to the
sunlight. This amounts to a collection area of πR2 (were R is the radius of the earth, 6.4 x 106 m).
However, the earth radiates energy outward in all directions over its entire surface so the radiation area is
4πR2. If the earth is in thermal equilibrium with sunlight (so the earth is not heating up or cooling down),
what is the earth’s average temperature over its surface? Assume that the emissivity of our earth is 0.6.

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