chapter 12 the laws of thermodynamics 12 the laws of thermodynamics. principles of thermodynamics...
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Chapter 12Chapter 12
The Laws of ThermodynamicsThe Laws of Thermodynamics
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Principles of ThermodynamicsPrinciples of Thermodynamics
Energy is conservedo FIRST LAW OF THERMODYNAMICSo Examples: Engines (Internal -> Mechanical)
Friction (Mechanical -> Internal)All processes must increase entropyo SECOND LAW OF THERMODYNAMICSo Entropy is measure of disordero Engines can not be 100% efficient
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Converting Internal Energy to Converting Internal Energy to MechanicalMechanical
VPQU ∆−=∆Work done by expansion
AVxPAFxFW /,, ∆=∆=∆=
VPW ∆=
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ExampleExampleA cylinder of radius 5 cm is kept at pressure with a pistonof mass 75 kg.
a) What is the pressure inside the cylinder?
b) If the gas expands such that the cylinder rises 12.0 cm, what work was done by the gas?
c) What amount of the work went into changing the gravitational PE of the piston?
d) Where did the rest of the work go?
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SolutionSolution
Given: M =75, A = π⋅0.052, ∆x=0.12, Patm = 1.013x105 Pa
a) Find Pgas
atmgas PAMgP += = 1.950x105 Pa
b) Find Wgas
VPW ∆= = 183.8 JxAPW ∆= gas
c) Find Wgravity
= 88.3 JmghW =
d) Where did the other work go?
Compressing the outside air
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Quiz ReviewQuiz ReviewA massive copper piston traps an ideal gas as shown to the right. The piston is allowed to freely slide up and down and equilibrate with the outside air. Pick the most correct statement?
A. The pressure is higher inside the cylinder than outside.B. The temperature inside the cylinder is higher than outside the cylinder.C. If the gas is heated by applying a flame to cylinder, and the piston comes to a new equilibrium, the inside pressure will have increased.D. All of the above.E. A and C.
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V
V
V
V
Some VocabularySome Vocabulary P
Isobarico P = constant
Isovolumetrico V = constant
Isothermalo T = constant
Adiabatico Q = 0
P
P
P
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Outside Air: Room T, Atm. P
A massive piston traps an amount of Helium gas as shown. The piston freely slides up and down. The system initially equilibrates atroom temperature (a)Weight is slowly added to the piston, isothermally compressing the gas to half its original volume (b)
A. Pb ( > < = ) PaB. Tb ( > < = ) TaC. Wab ( > < = ) 0D. Ub ( > < = ) UaE. Qab ( > < = ) 0 VPW
VPQU∆=
∆−=∆
ExampleExample
Vocabulary: Wab is work done by gas between a and bQab is heat added to gas between a and b
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Outside Air: Room T, Atm. P
A massive piston traps an amount of Helium gas as shown. The piston freely slides up and down. The system initially equilibrates atroom temperature (a)Weight is slowly added to the piston,adiabatically compressing the gas to half its original volume (b)
A. Pb ( > < = ) PaB. Wab ( > < = ) 0C. Qab ( > < = ) 0D. Ub ( > < = ) UaE. Tb ( > < = ) Ta
Vocabulary: Wab is work done by gas between a and bQab is heat added to gas between a and b
VPWVPQU
∆=∆−=∆
ExampleExample
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Outside Air: Room T, Atm. P
A massive piston traps an amount of Helium gas as shown. The piston freely slides up and down. The system initially equilibrates atroom temperature (a)The gas is cooled, isobaricallycompressing the gas to half its original volume (b)
A. Pb ( > < = ) PaB. Wab ( > < = ) 0C. Tb ( > < = ) TaD. Ub ( > < = ) UaE. Qab ( > < = ) 0
Vocabulary: Wab is work done by gas between a and bQab is heat added to gas between a and b
VPWVPQU
∆=∆−=∆
nRTPV =
ExampleExample
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Work from closed cyclesWork from closed cycles
Consider cycle A -> B -> A
WA->B
-WB->A
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Work from closed cyclesWork from closed cycles
Consider cycle A -> B -> A
WA->B->A= Area
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Work from closed cyclesWork from closed cycles
Reverse the cycle, make it counter clockwise
-WB->A
WA->B
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ExampleExample a) What amount of work is performed by the gas in the cycle IAFI?
b) How much heat was inserted into the gas in the cycle IAFI?
c) What amount of work is performed by the gas in the cycle IBFI?
area enclosed=WWIAFI = 3Patm= 3.04x105 J
WQU −=∆
∆U = 0 Q = 3.04x105 J
V (m3) W = -3.04x105 J
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One More ExampleOne More ExampleConsider a monotonic ideal gas which expands according to the PV diagram.
a) What work was done by the gas from A to B?b) What heat was added to the gas between A and B?c) What work was done by the gas from B to C?d) What heat was added to the gas beween B and C?e) What work was done by the gas from C to A?f) What heat was added to the gas from C to A?
P (kPa)
V (m3)
A
BC
75
50
25
0.2 0.4 0.6
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V (m3)
A
BC
•First find UA and UB
PVUnRTPVnRTU
23
23
:Gas Monotonic
=
=
=
UA = 22,500 J, UB = 22,500 J, ∆U = 0
•Finally, solve for Q
WQU −=∆ Q = 20,000 J
SolutionSolution P (kPa)75
a) Find WAB
b) Find QAB
50WAB = Area = 20,000 J
25
0.2 0.4 0.6
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SolutionSolution
V (m3)
A
BC
P (kPa)75
c) Find WBC
d) Find QBC
WAB = -Area = -10,000 J 50
25
0.2 0.4 0.6•First find UB and UCPVU
23
=UB = 22,500 J, UC = 7,500 J, ∆U = -15,000
•Finally, solve for QWQU −=∆Q = -25,000 J
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SolutionSolution P (kPa)e) Find WCA
f) Find QCA
V (m3)
A
BC
75
WAB = Area = 0 J 50
25
0.2 0.4 0.6•First find UC and UAPVU
23
=UC = 7,500 J, UA = 22,500 J, ∆U = 15,000
•Finally, solve for QWQU −=∆Q = 15,000 J
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Continued ExampleContinued ExampleTake solutions from last problem and find:
a) Net work done by gas in the cycleb) Amount of heat added to gas
WAB + WBC + WCA = 10,000 JQAB + QBC + QCA = 10,000 J
This does NOT mean that the engine is 100% efficient!
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Review QuizReview Quiz Outside Air: Room T, Atm. P
A massive piston traps an amount of Helium gas as shown. The piston freely slides up and down. The system initially equilibrates atroom temperature, T = Ta = 20 ºC (a)The gas is then slowly heated until T = Tb = 100 ºC (b)
A. Pa = PbB. Wab > 0C. Qab > WabD. All of the above.E. A and C.
Vocabulary: Wab is work done by gas between a and bQab is heat added to gas between a and b
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EntropyEntropyMeasure of Disorder of the system(randomness, ignorance)S = kBlog(N) N = # of possible arrangements for fixed E and Q
py
px
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Why do we use log(N)?Why do we use log(N)?
BA NNN ⋅=
# of ways to arrange entire system
Consider two systems A and B :
BA
BASSS
NNN+=
+= )log()log()log(
Entropy of both systems
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EntropyEntropyTotal Entropy always rises! (2nd Law of Thermodynamics)Adding heat raises entropy
TQS /=∆
Defines temperature in Kelvin!
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Why does Q flow from hot to cold?Why does Q flow from hot to cold?
Consider two systems, one with TA and one with TB
Allow Q > 0 to flow from TA to TB
Entropy changed by:
∆S = Q/TB - Q/TA
If TA > TB, then ∆S > 0
System will achieve more randomness by exchanging heat until TB = TA
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Efficiencies of EnginesEfficiencies of EnginesConsider a cycle described by:W, work done by engineQhot, heat that flows into engine from source at ThotQcold, heat exhausted from engine at lower temperature, Tcold
Efficiency is defined:
Qhot
engine
Qcold
W
hot
coldhot
QQQ −
=hot
cold
−=1hotQWe =:engine
Since , 0/ >=∆ TQS
⇒>⇒>hot
cold
hot
cold
hot
hot
cold
cold
TT
TQ
TQ
hot
cold
engines
TTe −<1
:
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1st and 2nd Laws of Thermodynamics1st and 2nd Laws of Thermodynamics
1. Energy is conservedYOU CAN’T WIN
2. Entropy always increasesYOU CAN’T BREAK EVEN
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CarnotCarnot EnginesEnginesIdealized engineMost efficient possible
hot
cold
hot TT
QWe −== 1
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CarnotCarnot CycleCycle
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ExampleExampleAn ideal engine (Carnot) is rated at 50% efficiency when it is able to exhaust heat at a temperature of 20 ºC. If the exhaust temperature is lowered to -30 ºC, what is the new efficiency.
SolutionSolutionGiven, e=0.5 when Tcold=293 ºK, Find e when Tcold= 243 ºK
First, find Thot
eTT coldhot −
=1
1
hot
cold
TTe −=1 = 586 ºK
Now, find e given Thot=586 ºK and Tcold=243 ºK
hot
cold
TTe −=1 e = 0.586
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RefrigeratorsRefrigerators
Given: Refrigerated region is at TcoldHeat exhausted to region with Thot
Find: EfficiencyQhot
engine
Qcold
W
WQe cold=
:orrefrigerat
1/1
−=
−=
coldhotcoldhot
cold
QQQQQ
0/ >=∆ TQSSince ,
1/1
:orrefrigerat
−<
coldhot TTe⇒>⇒>
cold
hot
cold
hot
cold
cold
hot
hot
TT
TQ
TQ
Note: Highest efficiency for small T differences
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Heat PumpsHeat PumpsGiven: Inside is at Thot
Outside is at TcoldFind: Efficiency
Qhot
engine
Qcold
W
WQe hot=
:pumpheat
hotcoldcoldhot
hot
QQQQQ
/11
−=
−=
0/ >=∆ TQSSince ,
hotcold TTe
/11
:pumpheat
−<⇒>⇒>
cold
hot
cold
hot
cold
cold
hot
hot
TT
TQ
TQ
Like Refrigerator: Highest efficiency for small ∆T
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ExampleExampleA modern gas furnace can work at practically 100% efficiency, i.e., 100% of the heat from burning the gas is converted into heat for the home. Assume that a heat pump works at 50% of the efficiency of an ideal heat pump.
If electricity costs 3 times as much per kw-hr as gas, for what range of outside temperatures is it advantageous to use a heat pump?Assume Tinside = 295 ºK.
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SolutionSolutionFind T for which e = 3 for heat pump.Above this T: use a heat pumpBelow this T: use gas
hotcold TTe
/11
:pumpheat
−<
295/115.03T
e−
⋅==
50% of ideal heat pump
C 27- K 8.24565295 °=°==T
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Real engines emit more than heat!Real engines emit more than heat!
Steam Engine Movie
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