MasteringPhysics: Assignment Print View
A S ip ip h o n a t t h e B a r Jane goes to a juice bar with her friend Neil. She is thinking of ordering her favorite drink, 7/8 orange juice and 1/8 cranberry juice, but the drink is not on the menu, so she decides to order a glass of orange juice and a glass of cranberry juice and do the mixing herself. The drinks come in two identical tall glasses; to avoid spilling while mixing the two juices, Jane shows Neil something she learned that day in class. She drinks about 1/8 of the orange juice, then takes the straw from the glass containing cranberry juice, sucks up just enough cranberry juice to fill the straw, and while covering the top of the straw with her thumb, carefully bends the straw and places the end over the orange juice glass. After she releases her thumb, the cranberry juice flows through the straw into the orange juice glass. Jane has successfully designed a siphon. Assume that the glass containing cranberry juice has a very large diameter with respect to the diameter of the straw and that the cross-sectional cross-sectional area of the straw straw is the same same at all points. Let the the atmospheric atmospheric pressure pressure be and assume that the cranberry juice has negligible viscosity. Part A
Consider the end of the straw from which the cranberry juice is flowing into the glass containing orange juice, and let be the distance below the surface of cranberry juice at which that end of the straw is located: . What is the initial velocity velocity of the cranberry juice as it flows flows out of the straw? Let denote the magnitude of the acceleration due to gravity.
Hint A.1
H o w t o a p p r o a c h t h e p r ob ob l e m
Hint not displayed
Hint A.2
Apply Bernoulli's principle
Hint not displayed Express your answ er in terms of
and
.
ANSWER:
= Correct
The speed of fluid flowing from the outlet of a siphon tube is the same as the speed that a body would acquire in falling from rest rest through a distance . This result is valid also for fluid flowing from an opening in a container at distance
below the surface of the fluid.
Part B
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MasteringPhysics: Assignment Print View
Given the information found in Part A, find the time
it takes to Jane to transfer enough cranberry juice into the the
orange juice glass to make her favorite drink if centimeters. Assume that the flow rate of the liquid is constant, and that the glasses are cylindrical with a diameter of 7.0 centimeters and are filled to height 14.0 centimeters. Take the diameter of the straw to be 0.4 centimeters. Hint B.1
H o w t o a p p r o a c h t h e p r ob ob l e m
Hint not displayed
Hint B.2
Find the volume flow rate
Hint not displayed E x p r e s s y o ur u r a n s w e r n u m e r i c a l l y i n s ec e c o n d s t o t w o s i g n i fi fi c a n t f i g u r e s .
ANSWER:
= 3 .8 Correct
A Su Su b m e r g e d B a l l A ball of mass and volume is lowered on a string into a fluid of density to the bottom if it were not supported by the string.
. Assume that the object would sink
Part A
What is the tension figure? Hint A.1
in the string when the ball is fully submerged but not touching the bottom, bottom, as shown in the
E qu qu i l i b r i u m c o n d i t i o n
Hint not displayed
Hint A.2
Find the m agnitude of the buoyant force
Hint not displayed E x p r e s s y o u r a n s w e r i n t e r m s o f a n y o r a l l of o f t h e g i v en en q u a n t i t i e s a n d acc eleration due to gravity.
ANSWER:
=
Correct
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, the magnitude of the
MasteringPhysics: Assignment Print View
A W a t e r T a n k T h a t N e e d s Cl Cl e a n i n g A cylindrical open tank needs cleaning. The tank is filled with water to a height meter, so you decide to empty it by letting the water flow steadily from an opening at the side of the tank, located near the bottom. The crosssectional area of the tank is square meters, while that of the opening is square meters. Part A
How much time Hint A.1
does it take to empty half the tank? (Note: A useful antiderivative is
.)
How to approach the problem
Hint not displayed
Hint A.2
Find the discharge rate
Hint not displayed
Hint A.3
F i n d t h e o u t f l o w s p e e d a s a fu fu n c t i o n o f t h e f l u i d s p e e d a t t h e s u r f a c e
Hint not displayed
Hint A.4
R ew e w r i t e t h e e x p r e s s i o n f o r t h e o u t f l o w s p e e d a s a f un u n c t i o n o f t h e c r o s s -s -s e c t i o n a l areas of the tank and the opening
Hint not displayed
Hint A.5
Find the rate of c hange of the level of w ater in the t ank
Hint not displayed
Hint A.6
H o w t o s o l v e a s e p a r a b l e f i r s t - o r de d e r O DE DE
Hint not displayed
Hint A.7
The limits of integration
Hint not displayed E x p r e s s y o u r a n s w e r n u m e r i c a l l y i n s e c o n d s . T a k e t h e f r e e -f -f a l l a c c e l e r a t i o n d u e t o g r a v i t y t o b e met ers per second per second.
ANSWER:
= 51.9 Correct
B r e a k i n g a G l a s s So So d a B o t t l e
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A glass soda bottle is emptied of soda and filled to the very top with water. A cork is i s carefully fitted into the top of the bottle, leaving no air between the cork and the water. The top of the bottle has a diameter of = 2.00 and the bottom of the bottle has a diameter of
= 6.50
. The
glass breaks when it is exposed to = 70.0 of pressure. A student hits the cork sharply with her fist and the bottom of the bottle breaks. The student's fist has a mass of = 0.480 and moves downward at a speed of
= 5.00
. It
collides elastically with the cork and rebounds with the same speed. The collision lasts for = 1.20×10−4 . In this problem, the positive direction is upward.
Part A
What is the force that the fist exerts on the top of the bottle? Hint A.1
H o w t o a p p r o a c h t h e p r ob ob l e m
Hint not displayed
Hint A.2
Find the acc eleration of the fist during the collision
Hint not displayed
Hint A.3
Find the force the bott le exerts on the fist
Hint not displayed
Hint A.4
N e w t o n ' s 3 r d l aw aw
Hint not displayed E x p r e s s y o ur ur a n s w e r i n n e w t o n s .
ANSWER:
4 . 0 0 × 1 04 Correct −
Part B
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MasteringPhysics: Assignment Print View
What is the magnitude of the force exerted on the bottom of the bottle? Hint B.1
H o w t o a p p r o a c h t h i s p ro ro b l e m
Hint not displayed
Hint B.2
Find the area of the top of the bot tle
Hint not displayed
Hint B.3
F i n d t h e c h a n g e i n p r e ss ss u r e a t t h e t o p o f t h e b o t t l e
Hint not displayed
Hint B.4
F i n d t h e i n c r e a s e i n p r e s s u re re a t t h e b o t t o m o f t h e b o t t l e
Hint not displayed
Hint B.5
Find the area of the bottom of the bot tle
Hint not displayed
Hint B.6
R el e l a t i o n s h i p b e t w e e n p r e s s u r e a n d fo fo r c e
Hint not displayed E x p r e s s y o u r a ns n s w e r i n n e w t o n s t o f o u r s i g n i f ic ic a n t f i g u r e s .
ANSWER:
4 . 2 3 × 1 05 Correct
Part C
Which of the following explains why the bottle breaks? H i n t C .1 .1
How t o solve this problem
Hint not displayed C h oo oo s e t h e c o r r e c t e x p l a n a t i o n .
ANSWER:
T h e p r es e s s u re r e t h a t t h e w a t e r e x e r t s o n t h e w a l l s o f t h e g l a ss ss b o t t l e i s g r e a t e r t h a n t h e p r e s s u r e r e q ui ui r e d t o b r e a k t h e g l a s s . T h e f or o r c e t h a t t h e w a t e r e x e r t s o n t h e b o t t o m o f t h e b o t t l e i s g r ea ea t e r t h a n the failure pressure of the glass. T h e f or o r c e t h a t t h e w a t e r e x e r t s o n t h e b o t t o m o f t h e b o t t l e i s g r ea ea t e r t h a n the force required to break t he glass. W he he n t h e h a n d s t r i k e s t h e b o t t l e , i t s e n d s a s h o c k w a v e t h r o u g h t h e b o t t l e , causing the w ater molec ules to get very close together. When the mass of e x t r a w a t e r m o l e c u l e s s t r ik i k e t h e b o t t o m o f t h e b o t t l e , i t b r e ak ak s . Correct
Fo r c e o n a G o l d f i s h V e c t o r D r a w i n g
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A fishbowl contains a single goldfish and is filled with water to the level indicated. Part A
At each of the designated points, rotate the given vector to indicate the direction di rection of the force exerted by the water on either the inside of the fishbowl (for points A and B ) or the outside of the goldfish (for points C , D , and E ). ). Hint A.1
Di r e c t i o n o f t h e f o r c e d u e t o t h e w a t e r
Hint not displayed The orientation of your vectors w ill be graded.
ANSWER:
View All attempts used; correct answer displayed
P r e s s u r e i n a Ce Ce n t r i f u g e A test tube tube filled with liquid of uniform density , as shown in the figure, is spun in a centrifuge with angular frequency . The test tube lies perpendicular to to the axis of rotation of the centrifuge. The pressure pressure in the fluid fluid at the distance from the axis of rotation is . You may ignore the variation in pressure with depth; asume that it is much smaller than the variation in pressure with radius.
Part A
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MasteringPhysics: Assignment Print View
What is the pressure Hint A.1
of the fluid in the test tube at an arbitrary distance
from the axis of rotation?
How to approach the problem
Hint not displayed
Hint A.2
More on how to approach the problem
Hint not displayed
Hint A.3
C en en t r i p e t a l f o r c e
Hint not displayed
Hint A.4
Find the change in pressure
Hint not displayed
Hint A.5
R ew ew r i t e t h e c h a n g e i n p r e s s u r e
Hint not displayed
Hint A.6
So l v i n g d i f f e r e n t i a l e q u a t i o n s
Hint not displayed
Hint A.7
A helpful integral
Hint not displayed Express your answ er in term s of
,
,
,
, and
.
ANSWER:
= Correct
Pressure on a Dam As the reservoir behind a dam is filled with water, the pressure that the water exerts on the dam increases. Eventually, the force on the dam becomes substantial, and it could cause the dam to collapse. There are two significant issues to be considered: First, the base of the dam should be able to withstand the pressure , where is the density of the water behind the dam, is its depth, and is the magnitude of the acceleration due to gravity. This means that the material of which the dam is made needs to be strong enough so that it doesn't crack (compressive strength). The second issue has to do with the strength of the foundation of the dam. The water pressure exerts a clockwise torque on the dam, as shown in the figure. The foundation of the dam should be strong enough so that the dam does not topple. The material has to be strong enough that the dam does not snap (shear strength). To study this phenomenon, consider the simple model of a dam depicted in the diagram. A reservoir of water
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MasteringPhysics: Assignment Print View
(density ) behind the dam is filled to a height . Assume that the width of the dam (the dimension pointing into the screen) is .
Part A
Consider a horizontal layer of the dam wall of thickness the magnitude Hint A.1
located a distance
above the reservoir floor. What is
of the force on this layer due to the water in the reservoir?
F i n d t h e p r e s s u re re a s a f u n c t i o n o f d e p t h
Hint not displayed Express your answ er in terms of , , the magnitude of the acc eleration due to gravity a n y q u a n t i t i e s f r o m t h e p r o b l em em i n t r o d u c t i o n .
, and
ANSWER:
= All attempts used; correct answer displayed
Part B
The force of the water produces a torque on the dam. In a simple model, if the torque due to the water were enough to cause the dam to break free from its foundation, the dam would pivot about its base (point P ). ). What is the magnitude of the torque about the point P P due due to the water in the reservoir? Hint B.1
How to approach the problem
Hint not displayed
Hint B.2
D ef ef i n i t i o n o f t o r q u e
Hint not displayed
Hint B.3
Find the torque on a horizontal layer of the dam
Hint not displayed
Hint B.4
So m e h e l p f u l i n t e g r a l s
Hint not displayed E x p r e s s y o ur u r a n s w e r i n t e r m s o f q u a n t i t i e s g i v e n in in t h e p r o b l e m i n t r o d u c t i o n a n d t h e m a g n i t u d e of the acc eleration due to gravity . http://session.masteringphysics.co http://session.masteringphysics.com/myct/assignm m/myct/assignmentPrintView?as entPrintView?assignmentID=148 signmentID=1488671 8671 (8 of 38) [12/13/2010 7:11:42 PM]
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ANSWER:
= Answer Requested
Thus the average moment arm is
. Therefore, since the water exerts a greater force at the
bottom of the dam than at the top, the net effect is as if the the force were acting at a height of
from the bottom.
A Wa Wa t e r T a n k o n M a r s You are assigned the design of a cylindrical, pressurized water tank for a future colony on Mars, where the acceleration due to gravity is 3.71 meters per second per second. The pressure at the surface of the water will be 105 , and the depth of the water will be 13.8 . The pressure of the air in the building outside the tank will be 95.0 . Part A
Find the net downward force on the tank's tank's flat bottom, of area area 2.10
, exerted exerted by the water water and air inside the tank tank
and the air outside the tank. Hint A.1
The net force
Hint not displayed
Hint A.2
What is
(a Pascal)?
Hint not displayed
Hint A.3
D en en s i t y o f w a t e r
Hint not displayed
Hint A.4
Find the force exert ed on the tank's bottom by the air outside the tank
Hint not displayed
Hint A.5
F i n d t h e f o rc rc e e x e r t e d o n t h e t a n k ' s b o t t o m b y t h e w a t e r
Hint not displayed E x p r e s s y o u r a ns n s w e r n u m e r i c a l l y i n N e w t o n s , t o t h r e e s i g n i fi fi c a n t f i g u r e s .
ANSWER:
1 . 2 9 × 1 05 All attempts used; correct answer displayed
H e a t F l o w i n g t h r o u g h a Se S e c t i o n e d Ro Ro d
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A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheri atmo spheric c pressure) pressure) at one end and and with an an ice-water ice-water mixtur mixture e at the other other . The The rod consist consists s of a 1.00 1.00 section secti on of copper (with one end in the boiling water) joined end-toend to a length of steel (with one end in the ice water). Both sections of the rod have cross-sectional areas of 4.00 . The temperature of the copper-steel junction is 65.0 after a steady state has been reached. Assume that the thermal conductivities of copper and steel are given by and
.
Part A
How much heat per second Hint A.1
flows from the boiling water to the ice-water mixture?
How to approach the problem
Hint not displayed
Hint A.2
E qu qu a t i o n f o r h e a t c o n d u c t i o n
Hint not displayed Express your answer in w atts.
ANSWER:
= 5.39 Correct
Because the system is assumed to have reached a steady state, the heat flowing out of the boiling water must be equal to the heat flowing through the copper section of the rod.
Part B
What is the length Hint B.1
of the steel section?
How to approach the problem
Hint not displayed E x p r e s s y o u r a ns ns w e r i n m e t e r s .
ANSWER:
= 0.242 Correct
Extending the analysis from the first part, we see that, since a steady state has been reached, the heat flowing through the copper section must be equal to the heat flowing through the steel section, which is also equal to the heat flowing into the ice water.
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H o t Ro d s Two circular circular rods, both of length and having the same diameter, are placed end to end between rigid supports with no initial stress in the rods. The coefficient of linear expansion and Young's modulus for rod A are and respectively; those for rod B are and
respectively. Both rods are "normal" materials with
The temperature of the rods is now raised by
.
.
Part A
After the rods have been heated, which of the following statements is true? C h oo oo s e t h e b e s t a n s w e r .
ANSWER:
The length of each rod is still . The length of each rod changes but the c ombined length of the rods is still . Correct
The length of the combined rod remains the same, but because the rods have different expansion coefficients, the lengths of the individual rods change. In other words, even though and .
Part B
After the rods have been heated, which of the following statements is true? C h oo oo s e t h e b e s t a n s w e r .
ANSWER:
The stress in eac h rod remains zero. A compressive stress arises that is the sam e for both rods. A compressive stress arises that is different for the tw o rods. A tensile stress arises that is the same for both rods. A tensile stress arises that is different for the tw o rods. A c o m p r e s s i v e s t r e s s a r i s e s i n o ne ne r o d a n d a t e n s i l e s t r e s s a r i s e s i n t h e other rod. Correct
Stress is a force per unit area. By Newton's 3rd law, the force on rod A due to rod B is the same as that on rod B due to rod A. Since the rods have the same diameter, their cross-sectional area is the same. Therefore, the stress on each rod must be the same.
Part C
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What is the stress H i n t C .1 .1
in the rods after heating?
How to approach the problem
Hint not displayed
H i n t C .2 .2
C ha ha n g e i n l e n g t h f o r e a c h r o d
Hint not displayed Express the stre ss in term s of
,
,
,
, and
.
ANSWER:
= Correct
Another way of thinking about this is that the combination of rods has a net thermal expansion coefficient and a net Young's modulus given by .
H o w M u c h C o o l a n t D o Y o u N e ed ed ? In a bizarre experiment devised by a physics teacher and an industrial arts teacher at a local high school, a cast-iron engine block of mass with coolan coolantt of mass is heated to 100 ( ) in a water bath. The engine block is then quickly submer quickly submerged ged in an insula insulated ted conta container iner holdin holding g a mass of 1000 of water water,, at a tempe temperatur rature e of 15 to find the mass of the coolant in the engine. The final tempe temperatur rature e of the water water,, engine engine,, and coolan coolantt is found to be 18 after heat from the engine is transferred to the water. The engine coolant used is pure ethylene glycol
, which has a specific heat
,
of
. (Usually you would use a mix of ethylene glycol and water as coolant.) The mass of the engine block used is 275 is 4187
and the specific heat of cast iron
is
. The specific heat of water
.
Assume that this is a closed, isolated system. Part A
For this experiment, the students were asked to find the mass of the coolant. If pure ethylene glycol is used as the coolant, what is the mass of coolant used in the experiment? Hint A.1
How to approach the problem
Hint not displayed
Hint A.2
Find the heat added to the w ater
Hint not displayed
Hint A.3
Find the heat removed from t he engine alone
Hint not displayed
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Hint A.4
The equation for the heat removed from the c oolant
Hint not displayed
Hint A.5
Find the heat removed from t he engine and coolant
Hint not displayed F in in d t h e m a s s o f e t h y l e n e g l y c o l
ANSWER:
i n k i l o g r a m s t o t h r e e s i g n i f ic ic a n t d i g i t s .
= 12.3 Correct
Adding Ice to Water An insulated beaker with negligible mass mass contains liquid water water with a mass mass of 0.275
and a temperature of 64.0
. Part A
How much much ice at a temperatur temperature e of -22.7 -22.7 system will be 40.0 Hint A.1
must be droppe dropped d into the the water water so that that the final final temperatu temperature re of the the
?
How to approach the problem
Hint not displayed
Hint A.2
Ca l c u l a t e t h e h e a t l o s t b y t h e w a t e r
Hint not displayed
Hint A.3
How t o calculate the heat gained by the ice
Hint not displayed
Hint A.4
Heat gained by the ice
Hint not displayed Take the specific heat of liquid w ater to be 4190 , and the heat of fusion for w ater to be 334
ANSWER:
= 5.03×10 Correct
, the specific heat of ice to be 2100 .
2
−
A n O v e r f l o w i n g Fl a s k
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A glass flask whose volume is 1000
at a tempe temperatur rature e of 0.600
is compl completely etely filled with mercu mercury ry at the same
temperatur temp erature. e. When the flask and mercu mercury ry are warme warmed d toget together her to a tempe temperatur rature e of 52.0
, a volume of 8.35
of
mercury overflows the flask. Part A
If the coefficient of volume expansion of mercury is
= 1.80×10−4
, compute
, the coefficient of volume
expansion of the glass. Hint A.1
H o w t o a p p r o a c h t h e p r ob ob l e m
Hint not displayed
Hint A.2
C al al c u l a t e t h e c h a n g e i n v o l u m e o f t h e m e r c u r y
Hint not displayed
Hint A.3
C al al c u l a t e t h e c h a n g e i n v o l u m e o f t h e f l a s k
Hint not displayed Express your answer in inverse kelvins.
ANSWER:
= 1.75×10 Correct
5
−
It is important to remember that when an object with a cavity inside of it is heated, the cavity will expand at the same rate that the object is expanding owing to the thermal expansion of the object. This is because the linear dimensions of the object will expand outward at all points equally in the object, whether on the outside surface or the inside surface of the cavity. Therefore, the volumes of the object and the cavity will expand by the same factor.
Co m p r e s s i o n o f a J a g u a r X K 8 C y l i n d e r A Jaguar XK8 convertible has an eight-cylinder e ight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contain ins s 499
of air at atmospheric pressure (
of the stroke, the air has been compressed to a volume of 46.2 to
) and a temperature of 27.0
and the gauge pressure has increased
.
Part A
Compute the final temperature Hint A.1
.
How to approach the problem
Hint not displayed
Hint A.2
Mass of air in the cylinder
Hint not displayed
Hint A.3
. At the end
R el el a t i o n b e t w e e n t h e i n i t i a l a n d f i n a l s t a t e s
Hint not displayed
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Hint A.4
Gauge pressure
Hint not displayed ANSWER:
= 503 Correct
The increase in gas temperature caused by this compression stroke is one of the reasons why a car engine gets so hot when it is running.
V o l u m e o f C op op p e r Part A
What is the volume
of a sample of 2.40
density of copper is Hint A.1
of copper? The atomic mass of copper (Cu) is 63.5
.
How to approach the problem
Hint not displayed
Hint A.2
W h at at i s t h e m a s s o f t h e c o p p e r ?
Hint not displayed
Hint A.3
Using the density in calculations
Hint not displayed Express your answer in cubic cent imeters.
ANSWER:
= 17.09 Correct
P r e s s u r e Co Co o k e r A pressure cooker is a pot whose lid can be tightly sealed to prevent gas from entering or escaping. Part A
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, and the
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If an otherwise empty pressure cooker is filled with air of room temperature and then placed on a hot stove, what would be the magnitude of the net force on the lid when the air inside the cooker had been heated to ? Assume that the temperature of the air outside the pressure cooker is of the pressure cooker lid is
. Take atmospheric pressure to be
(room temperature) and that the area
.
Treat the air, both inside and outside the pressure cooker, as an ideal gas obeying Hint A.1
.
C al al c u l a t e t h e p r e s s u r e i n s i d e
Hint not displayed
Hint A.2
R el el a t i n g p r e s s u r e a n d f o r c e
Hint not displayed
Hint A.3
D et et e r m i n e t h e r o l e o f t h e o u t s i d e p r e s s u r e
Hint not displayed Express the force in t erms of given variables.
ANSWER:
= Correct
Part B
The pressure relief valve on the lid is now opened, allowing hot air to escape until the pressure inside the cooker becomes equal to the outside pressure . The pot is then sealed again and removed from from the stove. Assume Assume that when the cooker is removed from the stove, the air inside it is still at . What is the magnitude of the net force Hint B.1
on the lid when the air inside the cooker has cooled back down to
How to approach the problem
Hint not displayed
Hint B.2
W h a t s t a y s c o n s t a n t w h e n t h e c o o k e r i s o p e n ed ed ?
Hint not displayed
Hint B.3
C al al c u l a t e t h e p r e s s u r e i n s i d e
Hint not displayed E x p r e s s t h e m a g n i t u d e o f t h e n e t f o r c e i n t e r m s o f g i v e n v a r ia ia b l e s .
ANSWER:
= Correct
G a s Pr Pr e s s u r e G i v e n T e m p e r a t u r e v s . V o l u m e G r a p h R a n k i n g T a s k
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?
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The figure shows representations of six thermodynamic states of the same ideal i deal gas sample.
Part A
Rank the states on the basis of the pressure of the gas sample at each state. Hint A.1
The ideal gas law
Hint not displayed
Hint A.2
Determining pressure
Hint not displayed R a nk n k p r e s s u r e f r o m h i g h e s t t o l o w e s t . T o r a n k i t e m s a s e q u i v a l e n t , o v e r la la p t h e m .
ANSWER:
View Correct
A v e r a g e Sp Sp a c i n g o f Ga s M o l e c u l e s
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Consider an ideal gas at 27.0 degrees Celsius and 1.00 atmosphere pressure. Imagine the molecules to be uniformly spaced, with each molecule at the center of a small cube. Part A
What is the length Hint A.1
of an edge of each small small cube if adjacent cubes touch touch but don't overlap? overlap?
How to approach the problem
Hint not displayed
Hint A.2
C al al c u l a t e t h e v o l u m e p e r m o l e
Hint not displayed
Hint A.3
C al al c u l a t e t h e v o l u m e p e r m o l e c u l e
Hint not displayed
Hint A.4
The edge length of a cube
Hint not displayed E x p r e s s y o u r a ns ns w e r n u m e r i c a l l y i n m e t e r s .
ANSWER:
= 3.45×10 Correct
9
−
For carbon dioxide gas, the physical volume of a molecule (based on the van der Waals equation constant ) is approximately is only about
. This implies that the linear dimension of the molecule
.
Comparing this number to the result of your calculation above, you can see that the size of the molecule is is less than one eighth of the length of the cube edge that surrounds the molecule, which is also the average distance separating one molecule from the next. A small molecular size, as compared to the distance between molecules, is a necessary assumption in the kinetic-molecular model of an ideal gas.
A n Ex Ex p a n d i n g M o n a t o m i c G a s We start start with with 5.00 moles moles of an ideal monat monatomic omic gas gas with an an initial initial temperatur temperature e of 133 the process, process, absorbs an amount of heat equal to to 1280
and does an amount amount of work equal to to 2180 .
Part A
What is the final temperature Hint A.1
of the gas?
F i r s t l a w o f t h e r m o d y n am am i c s
Hint not displayed
Hint A.2
Find the change in internal energy
Hint not displayed
Hint A.3
. The gas gas expands expands and, in
C al al c u l a t e t h e c h a n g e i n t e m p e r a t u r e
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Hint not displayed Use
= 8 .3 .3 1 4 5
for the ideal gas constant .
ANSWER:
119 = All attempts used; correct answer displayed
A Flexible Balloon A flexible balloon contains 0.380 volume volum e of 6850
of an unknown polyatomic gas. Initially the balloon containing the gas has a
and a tempe temperatur rature e of 21.0
. The gas first expand expands s isobar isobarically ically until the volum volume e double doubles. s. Then
it expands adiabatically until the temperature returns to its initial value. Assume that the gas may be treated as an ideal gas with and . Part A
What is the total heat Hint A.1
supplied to the gas in the process?
W h ic ic h p a r t s o f t h e p r o c e s s h a v e h e a t f l o w ?
Hint not displayed
Hint A.2
R el el a t i o n o f h e a t c a p a c i t y a n d h e a t
Hint not displayed
Hint A.3
Find the c hange in temperature
Hint not displayed ANSWER:
3720 = All attempts used; correct answer displayed
Part B
What is the total change in the internal energy Hint B.1
of the gas?
D ep ep e n d e n c e o f i n t e r n a l e n e r g y o n t e m p e r a t u r e
Hint not displayed ANSWER:
=0 Correct
Part C
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What is the total work H i n t C .1 .1
done by the gas?
How to approach the problem
Hint not displayed ANSWER:
= 3720 Correct
Part D
What is the final volume H i n t D .1 .1
?
R el el a t i o n b e t w e e n v o l u m e a n d t e m p e r a t u r e
Hint not displayed ANSWER:
= 0.110 Correct
Wo r k f r o m a n A d i a b a t i c E x p a n s i o n In this problem you are to consider an adiabatic adiabatic expansion expansion of an ideal diatomic gas, which means that the gas expands with no addition or subtraction of heat. This applet shows the adiabatic compression and expansion of an ideal monatomic gas with . It will help you to see the qualitative behavior of adiabatic adiabatic expansions, though your actual calculations will use a slightly different . Assume that the gas is initially at pressure , volume , and temperature . In addition, assume that the temperature of the gas is such that you can neglect vibrational degrees of freedom. Thus, the ratio of heat capacities is . Note that, unless explicitly stated, the the variable should not appear in your answers--if answers--if needed use the fact that for an ideal diatomic gas. Part A
Find an analytic expression for Hint A.1
, the pressure as a function of volume, during the adiabatic expansion.
F i n d t h e c o n s e r v e d q u an an t i t y i n a n a d i a b a t i c p r o c e s s
Hint not displayed Express the pressure in terms of
and any or all of the given initial values
,
, and
.
ANSWER:
= Answer Requested The fact that
is a constant derives from the definition of an adiabatic process as one in which no heat
flow into or out of the system occurs. Setting
in the first law of thermodynamics (
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) gives
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the starting point for this derivation:
. See your textbook for more details.
Part B
At the end of the adiabatic expansion, the gas fills a new volume the gas on the container during the expansion. Hint B.1
, where
. Find
, the work done by
How to approach the problem
Hint not displayed
Hint B.2
Help w ith the math
Hint not displayed Express the w ork in terms of
,
, and
. Your answ er should not depend on tem perature.
ANSWER:
= Answer Requested
Part C
Find
, the change of internal energy of the gas during the adiabatic expansion from volume
H i n t C .1 .1
to volume
Find the initial internal energy
Hint not displayed
H i n t C .2 .2
Find the final internal energy
Hint not displayed Express the ch ange of intern al energy in term s of
,
, a n d /o /o r
.
ANSWER:
= Answer Requested If you used the the hints hints to solve this part, they lead you on a long, but instructive, path to calculate simpler method you might have used is to apply the first law of thermodynamics,
. A much
, together with the fact that for an adiabatic process by definition. Basically, the molecules transfer some of their energy and thus also momentum to the walls of the container, causing the expansion. So the temperature decreases, while the volume increases. Of course, some external force will eventually stop the expansion.
H e a t i n g a R oo oo m
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.
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Imagine you've been walking outside on a cold winter's day. When you arrive home at your studio apartment, you realize that you left a window open and your room is only slightly warmer than the outside. You turn turn on on your your 1.01.0space spa ce heate heaterr right right away away and and wait wait impat impatien iently tly for for the the room room to warm warm up. up. In this problem, make the following assumptions: The entire output of the space heater goes into warming the air in the room. ●
The air in the room is an ideal gas with five degrees of freedom per particle (three translational degrees of freedom and two rotational degrees of freedom—about right for nitrogen and oxygen). The air in the room is at a constant pressure of 1.00 . ●
●
●
At room temperature and atmospheric pressure, 1
of air fills a volume of 23
. This is slightly larger than the
volume volum e of air air at standar standard d temperat temperature ure and press pressure, ure, because because room tempera temperature ture is hotter hotter than than 0
.
Part A
How long long will will itit be befor before e the heat heater er warms warms the the air air in the the room room by by 10. Hint A.1
?
How to approach the problem
Hint not displayed
Hint A.2
F i n d t h e e n e r g y n e e de de d t o r a i s e t h e t e m p e r a t u r e
Hint not displayed E x p r e s s y o ur u r a n s w e r i n m i n u t e s t o t w o s i g n i f ic ic a n t f i g u r e s .
ANSWER:
16 time = Correct
In practice practice,, it would would probably probably take take more than than an hour hour to heat heat the room room by 10. 10. because becau se the walls walls and any items in the room are in thermal contact with the air and would have to be warmed up also.
P S S 1 9 .1 .1 T h e F i r s t L a w o f T h e r m o d y n a m i c s L e a r n i n g G o a l : To practice Problem-Solving Strategy 19.1 The First Law of Thermodynamics.
A 1.60
sample of carbon dioxide undergoes a two-step process. First, at a constant volume, the gas behaves
idea id eall lly y as it is co cool oled ed fr from om 43 43.0 .0
to -7 -78. 8.5 5
wher wh ere e it re reac ache hes s a pr pres essu sure re of 1
. Se Seco cond nd,, at -7 -78. 8.5 5
and an da
constant pressure of 1 the gas undergoes a phase change to dry ice. What is the change in internal energy of carbon dioxide for this entire process? The constant volume molar heat capacity of carbon dioxide gas is 28.46 , the heat of sublimation of carbon dioxide is 6030
, and the density of dry ice
is
.
P r o b l e m S o l v i n g S t r a t e g y : T h e f i rs r s t l a w o f t h e r m o d y na na m i c s IDENTIFY the relevant concepts: The first law of thermodynamics is the statement of the law of conservation of energy in its most general form. You can apply it to any situation in which you are concerned with changes in the internal energy of a system, with heat flow into or out of a system, and/or with work done by or on a system. SET UP the problem using the following steps: 1. Carefully define what the thermodynamic system is. 2. For problems with more than than one step, identify the initial and final states for each step. 3. Identify the known known quantities and the target variables. 4. The first law, can be applied just once to each step in a thermodynamic process, so you will often need additional equations. These often include http://session.masteringphysics.co http://session.masteringphysics.com/myct/assignm m/myct/assignmentPrintView?as entPrintView?assignmentID=148 signmentID=1488671 8671 (22 of 38) [12/13/2010 7:11:43 PM]
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for the work done in a volume change and the equation of state of the material that makes up the thermodynamic system (for an ideal gas, ). EXECUTE the solution as follows: 1. Consistent units are essential. If
is in
and
is in
, then
is in joules. If a heat capacity is given in
terms of calories, usually the simplest procedure is to convert it to joules. 2. The internal energy change in any thermodynamic process or series of processes is independent of the path, whether the substance is an ideal gas or not. Since is the same for every possible path between the same two states, you can then relate the various energy quantities for other paths. 3. When a process consists of several distinct steps, it often helps to make a table showing , , and for each step. Put these quantities for each step on a different line, and arrange them so the ’s, ’s, and ’s form columns. Then, you can apply the first law to each line; in addition, you can add each column and apply the first law to the sums. 4. Using steps 1 –3, solve for the target variables. EVALUATE your answer: Check your results for reasonableness. In particular, make sure that each of your answers has the correct algebraic sign. Recall that a positive means that heat flows into into the the system, and a negative means that heat flows out of the of the system. A positive negative
means that work is done by by the the system on its environment, whereas a
means that work is done on on the the system by its environment.
I D EN EN T I FY FY t h e r e l e v a n t c o n c e p t s
In this problem, carbon dioxide undergoes a two-step process in which work is done and heat flows between the gas and its environment. The resulting change in internal energy of the gas is governed by the first law of thermodynamics. S ET ET U P t h e p r o b l e m u s i n g t h e f o l l o w i n g s t e p s
Part A
Identify the initial and final states of carbon dioxide for each step in the process. D ra ra g t h e a p p r o p r i a t e i t e m s t o t h e i r r e s p e c t i v e b i n s .
ANSWER:
View Correct The figure illustrates this two-step process.
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Carbon dioxide starts at a known temperature but unknown pressure the gas drops to -78.5
and volume
, the press pressure ure of the gas also decrea decreases ses until it reach reaches es
the gas changes phase to become a solid, so the volume decreases to stay the same.
. As the temperature of . At this point point,,
while the the pressure pressure and temperature temperature
Part B
What equations will you need in the process of solving for the change in internal energy is the heat of sublimation,
is the heat of fusion, and
of the carbon dioxide?
is the constant volume heat capacity.
C h ec ec k a l l t h a t a p p l y .
ANSWER:
Correct
E X EC EC U T E t h e s o l u t i o n a s f o l l o w s
Part C
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What is the change in internal energy H i n t C .1 .1
of carbon dioxide for this entire process?
H o w t o a p p r o ac ac h t h i s p r o b l e m
This process consists of two steps. During step 1, the carbon dioxide gas cools at a constant volume from an initial pressure and temperature to a final pressure and temperature. During step 2, the carbon dioxide changes phase and decreases its volume while at a constant temperature and pressure. The change of internal energy of a system is given by . Because this is a two-step process, the overall change in internal energy becomes . To simplify the expression, look for terms that are equal to zero. For each step in the process, determine the heat flow into and work done by the carbon dioxide. Keep in mind that , ,
H i n t C .2 .2
, and
.
Find the heat transferred during the first step of t he process
Hint not displayed
H i n t C .3 .3
Find the w ork done during the first step of the process
Hint not displayed
H i n t C .4 .4
Find the heat transferred during the second st ep of the process
Hint not displayed
H i n t C .5 .5
Find the w ork done during the second step of the proc ess
Hint not displayed Express your answer in joules.
ANSWER:
4 . 3 3 × 1 04 = Answer Requested −
EVALUATE your answ er
Part D
What are the signs associated with each of the following quantities? D ra ra g t h e a p p r o p r i a t e i t e m s t o t h e i r r e s p e c t i v e b i n s .
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ANSWER:
View All attempts used; correct answer displayed The change in internal energy of a system only depends upon the heat flow into or out of the system and the work done by the system, . Getting the signs correct in this equation is essential. Both heat flow into and work done by the system are positive quantities.
PSS PS S 20.1 Heat Engi nes L e a r n i n g G o a l : To practice Problem-Solving Strategy 20.1 Heat Engines.
Steam at a tempe temperatur rature e
= 345
and
= 1.00
enters enter s a heat engine at an unknow unknown n flow rate. Afte Afterr passi passing ng
through throug h the heat engine, it is releas released ed at a tempe temperatur rature e of the engine is 490
and
= 1.00
, and the exiting steam has a heat transfer rate of
the engine and the molar flow rate for steam is 37.47
= 100
. The measu measured red power outpu outputt
= 3000
. Find the efficiency
of
of steam through the engine. The constant pressure molar heat capacity
.
Problem Solving Strategy: Heat engines IDENTIFY the relevant concepts: A heat engine is any device that converts heat partially to work. SET UP the problem using the following steps: 1. Carefully define what the thermodynamic system is. 2. For multi-step processes with more than one step, identify identify the initial and final states for for each step. 3. Identify the known known quantities and the target variables. 4. The first law, , can be applied just once to each step in a thermodynamic process, so you will often need additional equations. The equation
is useful in situations for which the thermal efficiency of the engine is i s relevant. It's helpful to sketch an energy-flow diagram. EXECUTE the solution as follows: 1. Be very careful with the sign conventions for and the various 's. is positive when the system expands and does work; is negative when the system is compressed. Each is positive if it represents heat entering the system and is negative if it represents heat leaving the system. 2. P ow ower is work per unit time ( ), and heat current is heat transfer per unit time ( ). 3. Keeping steps 1 and 2 in mind, solve solve for the target variables. EVALUATE your answer: Use the first law of thermodynamics to check your results, paying particular attention to algebraic signs. I D EN EN T I FY FY t h e r e l e v a n t c o n c e p t s
This heat engine partially converts heat from the incoming steam into work, so the problem solving strategy for heat engines is applicable. S ET ET U P t h e p r o b l e m u s i n g t h e f o l l o w i n g s t e p s http://session.masteringphysics.co http://session.masteringphysics.com/myct/assignm m/myct/assignmentPrintView?as entPrintView?assignmentID=148 signmentID=1488671 8671 (26 of 38) [12/13/2010 7:11:43 PM]
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Part A
Which of the following quantities are known? C h ec ec k a l l t h a t a p p l y .
ANSWER:
The temperat ure of steam as it leaves the engine,
The power out put of the engine, T h e m o l a r f l ow ow r a t e o f s t e a m ,
T h e c o n s t a n t p r e s s u r e m o l a r he he a t c a p a c i t y o f s t e a m , The efficiency of the engine,
The heat transfer rate for steam leaving the engine, The heat transfer rate for steam ent ering the engine, The temperature of the steam as it ent ers the engine,
Correct
The efficiency and the molar flow rate of steam through the engine are both target variables in this problem. Even though the heat transfer rate of the steam entering the heat engine is not given in the problem statement, it is not a target variable. The energy-flow diagram for this system is shown. Heat flows into the engine at a rate of as steam at a temperature
. Work leaves the engine at a rate
as steam at temperature
. The remaining heat leaves the engine at a rate of
.
E X EC EC U T E t h e s o l u t i o n a s f o l l o w s
Part B
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What is the efficiency of the heat engine? Hint B.1
H o w t o a p p r o a c h t h e p r ob ob l e m
You are asked to find the efficiency of the heat engine. To find the efficiency of the heat engine, first find the heat transfer rate of the incoming steam from the relationships between work and power and between heat and heat transfer rate. Once you know the incoming heat transfer rate, you can then calculate the efficiency of the heat engine using the power delivered by the engine.
Hint B.2
F i n d t h e h e a t t r a n s f e r r at at e o f t h e i n c o m i n g s t e a m
What is the heat transfer rate of the incoming steam Hint B.2.1
?
Find an expression for the heat transfer rate of the incom ing steam
Hint not displayed Express your answer num erically in joules per second.
ANSWER:
= Answer not displayed
Hint B.3
F i n d a n ex ex p r e s s i o n f o r t h e e f f i c i e n c y o f t h e h e a t e n g i n e
Find an expression for the efficiency of the heat engine . Start with the expression , and replace
and
with expressions involving the power
generated by the engine, the heat transfer rate
of the steam entering the engine , and time . Refer to EXECUTE step 2 if you need help recalling the relationships between these variables. Hint B.3.1
H o w t o p u t t h e e f f i c i e n c y e q ua u a t i o n i n t o t e r m s o f p o w e r a n d h e at a t t r a n sf sf e r r a t e s
Hint not displayed Express
in terms of
ANSWER:
and
.
= Answer not displayed
E x p r e s s t h e e f f i c i e n c y n u m e r i c a l l y t o t h r e e s i g n if if i c a n t f i g u r e s .
ANSWER:
0.140 = Answer Requested
An efficiency efficiency of 0.140 indicates indicates that 14.0 remaining 86.0
of the the heat from the incoming incoming steam is converted converted to work. The
is expelled from the heat engine as heat in the outlet steam.
Part C
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What is the molar flow rate of steam into the engine? H i n t C .1 .1
H o w t o a p p r o a c h t h e p r ob ob l e m
The molar flow rate to the power (
of steam can be calculated by relating the temperature change and molar heat capacity ) delivered by the steam engine.
From the first law of thermodynamics, you know that . That is, the energy of the heat engine is conserved and the net heat flowing into the engine equals the net work done by the engine. You can use the fact that the steam is a gas held at a constant pressure to find an expression for involving .
H i n t C .2 .2
The net heat flow ing into the heat engine
The heat required for the temperature temperature change of moles of steam can be expressed in terms of the difference in temperatures of the entering and exciting steam as . The heat lost by the steam is equal to the heat gained by the steam engine, so the net heat flowing into the steam engine can be expressed as .
H i n t C .3 .3
Find an expression for the molar flow rate of steam t hrough the heat engine
Express the molar flow rate
of steam through the heat engine as a function of the generated power
constant pressure molar heat capacity
, and the change in temperature
, and since the energy of the heat engine is conserved, Express the molar flow rate in terms of
,
, and
. Keep in mind that
, the
is related to
. .
ANSWER:
= Answer Requested
Express the molar flow rate in moles per second t o three significant figures.
ANSWER:
= 5.34×10 Correct
2
−
Knowing the flow rate of steam through the heat engine is important for a few reasons. In the design stage, an estimate of the flow rate is used the size the heat engine. During the operation of the heat engine, enough steam must flow through the heat engine to produce the required amount of power.
EVALUATE your answ er
Part D
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Which of the following changes, if made individually, would cause an increase in the amount of power produced by the heat engine? C h ec ec k a l l t h a t a p p l y .
ANSWER:
higher molar flow rat e,
l o w e r m o l a r f l ow ow r a t e ,
smaller value of larger value of smaller value of
larger value of
All attempts used; correct answer displayed A heat engine can produce more power either by improving its efficiency or by increasing its throughput. If either the entering heat transfer rate increases or the exiting heat transfer rate decreases, the power generated by the engine will increase. Dividing both sides of by time makes the relationship between power and heat transfer rates clearer: . By either increasing the entering heat transfer rate or decreasing the exiting heat transfer rate, the efficiency of the heat engine is increased. If the molar flow rate increases, the amount of heat that the engine converts into work per unit time increases, but the engine engi ne is not necessarily more efficient.
A T h r e e - St St e p G a s C y c l e A monatomic ideal gas has pressure and temperature . It is contained in a cylinder of volume movable piston, so that it can do work on the outside world. Consider the following three-step transformation of the gas: 1. The gas is heated at constant volume until the pressure reaches (where ). 2. The gas is then expanded at constant temperature until the pressure returns to . 3. The gas is then cooled at constant pressure until the volume has returned to . It may be helpful to sketch this process on the pV pV plane. plane. Part A
How much heat Hint A.1
is added to the gas during step 1 of the process?
F i r s t l a w o f t h e r m o d y n am am i c s
Hint not displayed
Hint A.2
Think about
Hint not displayed
Hint A.3
H o w t o f i nd nd
Hint not displayed
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with a
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Hint A.4
H o w t o f i nd nd
Hint not displayed Express the heat added in terms of
,
, and
.
ANSWER:
= Answer Requested
Part B
How much work Hint B.1
is done by the gas during step 2?
H o w t o a p p r o ac ac h t h i s p r o b l e m
Hint not displayed
Hint B.2
Find
Hint not displayed
Hint B.3
Find the initial and final volumes
Hint not displayed Express the w ork done in terms of
,
, and
.
ANSWER:
= Answer Requested
Part C
How much work is done by the gas during step 3? If you've drawn a graph of the process, you won't need to calculate an integral to answer this question. H i n t C .1 .1
The easy way t o solve this problem
Hint not displayed
H i n t C .2 .2
Find the formula for w ork done
Hint not displayed Express the w ork done in terms of
,
, and
.
ANSWER:
= Answer Requested
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En t r o p y C h a n g e i n a F r e e E x p a n s i o n : A M i c r o s c o p i c V i e w A thin partition divides a thermally insulated vessel into a lower compartment of volume compartment of volume
. The lower compartment contains
and an upper
moles of an ideal gas; the upper part is evacuated.
Part A
When the partition is removed, the the gas expands and fills both compartments. How many moles of gas were initially contained in the lower l ower compartment if the entropy change of the gas in this free-expansion process is 17.28 ? Hint A.1
How to approach the problem
Hint not displayed
Hint A.2
F i n d t h e n u m b e r o f p o s s i b l e m i c r o s c o p i c s t a t e s o f a g a s af af t e r a f r e e e x p a n s i o n
Hint not displayed
Hint A.3
Find the m icroscopic expression for the change in entropy of a gas
Hint not displayed
Hint A.4
Relating
to t he gas constant
Hint not displayed E x p r e s s y o u r a ns n s w e r t o t h r e e s i g n if if i c a n t f i g u r e s .
ANSWER:
= 1.89 Correct
F r o m H o t t o Co C o o l : A Ch Ch a n g e i n E n t r o p y In a we wellll-i -ins nsul ulat ated ed ca calo lori rime mete ter, r, 1. 1.0 0
of wa wate terr at 20
is mi mixe xed d wi with th 1. 1.0 0
of ic ice e at 0
.
Part A
What is the net net change change in entropy
of the system from the time of mixing until the moment the ice completely
melts? The heat of fusion of ice is
.
Note that since the amount of ice is relatively small, the temperature of the water remains nearly constant throughout the process. Note also that the ice starts out at the melting point, and you are asked about the change in entropy by the time it just melts. In other words, you can assume that the temperature of the "ice water" remains constant as well. Hint A.1
How to approach the problem
Hint not displayed
Hint A.2
D es es c r i p t i o n o f e n t r o p y
Hint not displayed
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Hint A.3
H e a t n e e de de d t o m e l t t h e i c e
Hint not displayed Express your answer num erically in joules per kelvin. Use tw o significant figures in your answer.
ANSWER:
= 8.35×10 Correct
2
−
As you would expect, in this spontaneous spontaneous process process the net change in entropy is positive: The entropy increases. This is evident not just from the calculation but also from the fact that a crystal becomes liquid and hence the degree of disorder increases.
Re f r i g e r a t o r L i g h t The inside of an ideal refrigerator is at a temperature a temperature temperature
, while the heating coils on the back of the refrigerator are at
. Owing Owing to a malfunctioning malfunctioning switch, the light bulb within the refrigerator refrigerator remains remains on when the the
door is closed. The power of the the light bulb is ; assume that all of the energy generated by the light bulb goes into heating the inside of the refrigerator. For all parts of this problem, you must assume that the refrigerator operates as an ideal Carnot engine in reverse between the respective temperatures. Part A
If the temperatures inside and outside of the refrigerator do not change, how much extra power refrigerator consume as a result of the malfunction of the switch? Hint A.1
does the
Find the coeffic ient of performance
Hint not displayed
Hint A.2
H o w m u c h h e a t i s a b s o rb r b e d f r o m t h e c o l d r e s e r v o i r fo fo r t h e p r o p e r l y w o r k i n g refrigerator?
Hint not displayed
Hint A.3
H o w m u c h w o r k i s d o n e b y t h e m a l f u n c t i o n i n g re r e f r ig i g e r at at o r ?
Hint not displayed
Hint A.4
C on on v e r t h e a t i n t o p o w e r
Hint not displayed
Hint A.5
H o w m u c h p o w e r w a s c o n s u m e d b y t h e p r o p er e r l y w o r k i n g r e fr f r i g er er a t o r ?
Hint not displayed
Hint A.6
P ut ut t i n g i t a l l t o g e t h e r
Hint not displayed Express the ext ra pow er in terms of
,
, and
.
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ANSWER:
= Answer Requested You'll note that for typical household refrigerators, this result suggests that that for a given value of power needed to run the the refrigerator with the light on is much smaller than shows this for a specific case.
, the extra
. The next part of this problem
Part B
Suppose the refrigerator has a 25-W light bulb, the temperature inside the refrigerator is
, and the temperature
of the heat dissipation coils on the back of the refrigerator is . Find the extra power consumed by the refrigerator. Keep in mind that you will need to use absolute units of temperature (i.e., kelvins). E x p r e s s y o ur ur a n s w e r n u m e r i c a l l y i n w a t t s t o t h r e e s i g n i f i c a n t f i g u r e s .
ANSWER:
3.20 = Answer Requested
R ef ef r i g e r a t o r P r o t o t y p e s R a n k i n g T a s k Six new refrigerator refrigerator prototypes prototypes are tested in the laboratory. For each refrigerator, the electrical electrical power to operate and the maximum heat energy that can be removed per second
from its interior are given.
Part A
Rank these refrigerators on the basis of their performance coefficient. Hint A.1
H o w t o a p p r o a c h t h e p r ob ob l e m
Hint not displayed
Hint A.2
D ef ef i n i t i o n o f t h e p e r f o r m a n c e c o e f f i c i e n t
Hint not displayed R a nk n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a l e nt nt , o v e r l a p t h e m .
ANSWER:
View Correct
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needed for it
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Part B
The six refrigerators are placed in six identical sealed rooms. Rank the refrigerators on the basis of the rate at which they raise the temperature of the room. Hint B.1
T e m p e r a t u r e a n d C o ns ns e r v a t i o n o f e n e r g y
Hint not displayed R a nk n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a l e nt nt , o v e r l a p t h e m .
ANSWER:
View All attempts used; correct answer displayed
S i x C a r n ot o t E n g i n e s w i t h V a r y i n g Re R e s e r v o i r s Ra Ra n k i n g T a s k Six Carnot engines operating between different hot and cold reservoirs are described below. The heat energy transferred to the gas during the isothermal expansion phase of each cycle is indicated. Part A
Rank these engines on the basis of the change in entropy of the gas during the isothermal expansion phase of the cycle. Hint A.1
C h an an g e i n e n t r o p y
Hint not displayed R a nk n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a l e nt nt , o v e r l a p t h e m .
ANSWER:
View All attempts used; correct answer displayed
Part B
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Rank these engines on the basis of the change in entropy of the gas during one complete cycle. Hint B.1
C ha ha n g e i n e n t r o p y f o r a c o m p l e t e c y c l e
Hint not displayed
Hint B.2
D oe oe s t h e s e c o n d l a w o f t h e r m o d y n a m i c s a p p l y ?
Hint not displayed R a nk n k f r o m l a r g e s t t o s m a l l e s t . T o r a n k i t e m s a s e q u i v a l e nt nt , o v e r l a p t h e m .
ANSWER:
View All attempts used; correct answer displayed
Entropy Change of an Expan ding Gas Two moles of an ideal gas undergo a reversible isothermal expansion from 1.93×10−2 temperature of of 26 26.8
to 4.89×10−2
at a
.
Part A
What is the change in entropy Hint A.1
of the gas?
How to approach the problem
Hint not displayed
Hint A.2
C al al c u l a t e t h e w o r k d o n e b y t h e g a s
Hint not displayed
Hint A.3
C al al c u l a t i n g t h e c h a n g e i n e n t r o p y
Hint not displayed Express your answer num erically in joules per kelvin.
ANSWER:
= 15.5 Correct
As the gas expands, the greater volume allows al lows the molecules of the gas to explore a greater range of positions, so the disorder of the gas (the molecules of which can have an increased randomness of position in an increased volume) will therefore be increased as well.
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M e l t i n g I c e w i t h a C a rn o t E n g i n e A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation of the engine, the heat rejected by the engine melts a mass of ice equal to 2.35×10−2 . Throughout this problem use
for the heat of fusion for water.
Part A
During this time, how much work Hint A.1
is performed by the engine?
How to approach the problem
Hint not displayed
Hint A.2
Temperature conversion
Hint not displayed
Hint A.3
C al c u l a t e t h e h e a t r e j e c t e d
Hint not displayed
Hint A.4
C al c u l a t e t h e h e a t a b s o r b e d
Hint not displayed
Hint A.5
U s i ng t h e f i r s t l a w o f t h e r m o d y n am i c s
Hint not displayed ANSWER:
= 2870 Correct
As you can see from this problem, it is very important to keep in mind the signs of the heats exchanged in an engine. When the Carnot engine absorbs heat from a reservoir, the heat will be a positive quantity since the heat is being added to the engine, before it does any work. Similarly, when the Carnot engine rejects heat to a reservoir, the heat will be a negative quantity since the heat is lost from the engine. The work done by the engine, by the first law of thermodynamics, is therefore the sum of all heat changes in the engine.
T h e Ca r n o t I c e m a k e r An ice-making machine inside a refrigerator operates in a Carnot cycle. It takes heat from liquid water at 0.0 rejects heat to a room at a temperature of 24.2 . Suppose that liquid water with a mass of 79.6 converted to ice at the same temperature. Take the heat of fusion for water to be = 3.34×105 . Part A
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at 0.0
and is
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How much heat Hint A.1
is rejected to the room?
How to approach the problem
Hint not displayed
Hint A.2
C al al c u l a t e t h e h e a t a b s o r b e d
Hint not displayed
Hint A.3
E qu qu a t i o n f o r h e a t t r a n s f e r i n a C a r n o t e n g i n e
Hint not displayed Express your answer in joules to four significant figures.
ANSWER:
= 2 . 8 9 4 × 1 07 Correct
Part B
How much energy Hint B.1
must be supplied to the device?
How to approach the problem
Hint not displayed Express your answer in joules.
ANSWER:
= 2.36×10 6 Correct
Just as a Carnot engine can be used to absorb heat from a hot room and reject it to a cold room after work is done by the engine, the process can also be reversed: The heat can be absorbed from a cold room and rejected to a hot room. However, to do this, energy (work) must instead be supplied supplied to to the device to keep it operating.
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