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Isolated systems can exchange neither energy nor matter with the environment. Closed systems exchange energy but not matter with the environment. Heat Work reservoir Open systems can exchange both matter and energy with the environment. Heat Work reservoir Thermodynamic systems
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Page 1: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Isolated systems can exchange neither energy nor matter with the environment.

Closed systems exchange energy but not matter with the environment.

Heat

Work

reservoir

Open systems can exchange both matter and energy with

the environment.

Heat

Work

reservoir

Thermodynamic systems

Page 2: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Quasi-static processes : near equilibrium

Initial state, final state, intermediate state: p, V & T well defined

Sufficiently slow processes = any intermediate state can considered as at thermal equilibrium. Thermal equilibrium means that It makes sense to define a temperature.

Examples of quasi-static processes:- isothermal: T = constant- isochoric: V = constant- isobaric: P = constant- adiabatic: Q = 0

Quasi-static processes

Page 3: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Work in thermodynamics

Expansion: work on piston positive, work on gas negativeCompression: work on piston negative, work on gas positive

Page 4: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Work during a volume change

∫=⇒

==

=

2

1

.

.

V

V

pdVW

pdV

Adxp

dxFdW

Page 5: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Work in pV diagrams

Work done equals area under curve in pV diagram

Careful with the signs…

Page 6: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

1st Law of Thermodynamics

WQU −=∆Conservation of energy

Heat is positivewhen it entersthe system

Work is positivewhen it is done bythe system

Heat is negativewhen it leavesthe system

Work is negativewhen it is done onthe system

Page 7: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

1st Law of Thermodynamics

pdVdQdU −=Conservation of energy

Heat is positivewhen it entersthe system

Work is positivewhen it is done bythe system

Heat is negativewhen it leavesthe system

Work is negativewhen it is done onthe system

Page 8: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

)( 122 VVpW −=

a. isochoricb. isobaric

a. isobaricb. isochoric

)( 121 VVpW −= ∫= f

i

V

VpdVW

isothermal

• The work done by a system depends on the initial and final states and on the path � it is not a state function.

• Amount of heat transferred also depends on the initial, final, and intermediate states � it is not a state function either.

(a) (b) (c)

State Functions

Page 9: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

State Functions

12 UUWQU −=−=∆

The internal energy U is a state function: the energy gain (loss) only depends on the initial and final states, and not

on the path.

Even though Q and W depend on the path, ∆U does not!

Page 10: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

This pV–diagram shows two ways to take a system from state a (at lower left) to state c (at upper right):

• via stateb (at upper left), or• via stated (at lower right)

For which path is W > 0?

A. path abc only B. path adc only

C. both path abc and path adc

D. neither path abc nor path adc

E. The answer depends on what the system is made of.

CPS question

Page 11: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

A system can be taken from state a to state b along any of the three paths shown in the pV–diagram.

If state b has greater internal energy than state a, along which path is the absolute value |Q| of the heat transfer the greatest?

A. path 1 B. path 2 C. path 3

D. |Q| is the same for all three paths.

E. not enough information given to decide

CPS question

Page 12: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Thermodynamic Processes:•Adiabatic: no heat transfer (by insulation or by ve ry fast process)

Q=0 →→→→ U2 – U1 = -W

•Isochoric: constant volume process (no work done)

W=0 →→→→ U2 – U1 = Q

•Isobaric: constant pressure process

p=const. →→→→ W = p (V2 – V1)

•Isothermal: constant temperature process (heat may flow but veryslowly so that thermal equilibrium is not disturbed )

∆∆∆∆U=0,Q =-W only for ideal gas. Generally ∆∆∆∆U,Q, W not zero any energy entering as heat must leave as work

Page 13: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Thermodynamic processes

Page 14: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Isolated systems:

fi UU

U

WQ

=→=∆→==

0

0

Cyclic processes

Pi, f

Vinitial state = final state

WQ

U

=→=∆ 0

The internal energyof an isolated systemsremains constant

Adiabatic processes

WU

Q

−=∆→= 0

Expansion: U decreasesCompression: U increasesEnergy exchange between

“heat” and “work”

First Law for Several Types of Processes

Page 15: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

More about cyclic processesP

i, f

V

Work equals the area enclosed by the curves (careful with the sign!!!)

Page 16: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

A. Q > 0, W > 0, and ∆U = 0.

B. Q > 0, W > 0, and ∆U > 0.

C. Q = 0, W > 0, and ∆U < 0.

D. Q = 0, W < 0, and ∆U > 0.

E. Q > 0, W = 0, and ∆U > 0.

CPS question

An ideal gas is taken around the cycle shown in this pV–diagram, from a to b to c and back to a. Process b → c is isothermal.

For this complete cycle,

Page 17: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Important formulas

∆U=Q-W (1st law)

(work during a volume change)

pV=nRT (Ideal gas law)

CV=f/2nR (Equipartition theorem)

∫=2

1

V

V

pdVW

Page 18: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Isochoric process: V = constant

V

P

V1,2

1

2 22 nRTVp =

11 nRTVp =

02

1

==→ ∫VV pdVW

TC

TTCQ

V

V

∆=−= )( 12

Heat

reservoir

During an isochoric process, heat enters (leaves) the system and increases (decreases) the internal energy.

(CV: heat capacity at constant volume)

TCQU

WQU

V ∆==∆→−=∆

Ideal gas: isochoric process

Page 19: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Isobaric process: p = constant

V

P

V1

12

22 nRTpV =

11 TNkpV B=

VpVVppdVWV

V∆=−==→ ∫ )( 12

2

1

V2 TC

TTCQ

p

p

∆=

−= )( 12(CP: heat capacity at constant pressure)

VpTC

WQU

P ∆−∆=−=∆→

During an isobaric expansion process, heat enters the system. Part of the heat is used by the system to do work on the environment; the rest of the heat is used to increase the internal energy.

Heat

Work

reservoir

Ideal gas: isobaric process

Page 20: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Isothermal process: T = constant

V

P

V1

1

2

nRTpV =

1

2ln

2

1

2

1

2

1

V

VnRT

V

dVnRT

dVV

nRTpdVW

V

V

V

V

V

V

=

=

==

∫∫∫

V2

00 =∆⇒=∆ UT

1

2lnV

VnRTWQ ==→

Expansion: heat enters the system all of the heat is used by the system to do work on the environment.

Compression: the work done on the system increases its internal energy, all of the energy leaves the system at the same time as the heat is removed.

Ideal gas: isothermal process

Page 21: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Heat capacities of an Ideal gasConsider an isobaric process p=constant

TCQ pp ∆=From the 1st Law of Thermodynamics:

pdVdUdTCdQ pp +==but

dTCdU V=pdVdTCdTC Vp +=⇒

nRdTpdVVdppdVnRTpV ==+⇒=From the Ideal gas law:

nRCCnRdTdTCdTC VpVp +=⇒+=⇒

Page 22: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Heat capacities of an Ideal gas

nRf

C

nRCC

V

Vp

2=

+=nR

fCp 2

2+=⇒

f

fnR

fnR

f

C

C

V

p 2

2/

2

2 +=+=⇒

f = #degrees of freedom

For a monoatomic gas f=3

67.13/5;2

5 ===⇒V

pp C

CnRC

Page 23: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Molar heat capacities of various gases at (25 C)

1.058.763.2927.3636.12H2S

1.028.513.4128.3936.90N2O

1.028.453.3928.1736.62CO2

Polyatomic

1.008.402.4920.7429.04CO

1.018.392.5220.9829.37O2

1.018.382.4620.4428.82H2

1.008.322.5020.8029.12N2

Diatomic

0.998.271.5112.5220.79Xe

1.008.341.5012.4520.79Kr

1.008.341.5012.4520.79Ar

0.988.111.5212.6820.79Ne

0.998.271.5112.5220.79He

Monoatomic

(Cp-Cv)/RCp-CvCv/RCvCpGas

Page 24: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

V

P

V1

1

2

V2

Adiabatic process: Q = 0),( TVpp = ∫∫ == 2

1

2

1

),(V

V

V

VdVTVppdVW

TNkpV B=

nRdTVdppdV

nRTdpVd

=+→=→ )()(

pdVRdTf

n

pdVdTC

pdVdWdU

V

−=→

−=→−=−=

2

pdVf

VdppdV2−=+

f is the # of degrees of freedom

Ideal gas: adiabatic process

Page 25: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

V

P

V1

1

2

V2

),( TVpp =

0)2

1(

2

=++→

−=+

pdVf

Vdp

pdVf

VdppdV

0=+V

dV

p

dp γ

constant0ln

0lnln

0

11

11

11

11

==→=→

=+→

=+ ∫∫

γγγ

γ

γ

γ

VppVVp

pV

V

V

p

p

V

dV

p

dp V

V

p

p

let , and dividing by pV)2

1(f

+=γ

Ideal gas: adiabatic process (contd)

Page 26: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

V

P

V1

1

2

V2

constant=γpV

)(1

1

)(

)(

2211

2211

21

VpVp

VpVpR

C

TTnC

TnCUW

V

V

V

−−

=

−=

−=∆−=∆−=

γ

−−

=→ −− 12

11

11

11

)1(

1γγ

γ

γ VVVpW

constant11 == γγ VppV

Ideal gas: adiabatic process (contd)

Using:

Page 27: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

V

P

V1

1

2

V2

constant=γpV

γγ2211 VpVp =

222

111

nRTVp

nRTVp

==

2

1

2

1

2

1

T

T

V

V

p

p =→

γ

γ

1

2

2

1

V

V

p

p =→

2

11

1

12

T

T

V

V =→ −

γ

γ

or

constant11 == γγ VppV

constant122

111 == −− γγ VTVT

Ideal gas: adiabatic process (contd)

Page 28: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

V

P

V1

1

2

V2

constant=γpV constant11 == γγ VppV

constant122

111 == −− γγ VTVT

nRTpV =

During an adiabatic expansion process, the reduction of the internal energy is used by the system to do work on the environment.

During an adiabatic compression process, the environment does work on the system and increases the internal energy.

Ideal gas: adiabatic process (contd)

−−

= −− 12

11

11 11

)1( γγ

γ

γ VV

VpW

Page 29: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

When an ideal gas is allowed to expand isothermally from volume V1 to a larger volume V2, the gas does an amount of work equal to W12.

If the same ideal gas is allowed to expand adiabatically from volume V1 to a larger volume V2, the gas does an amount of work that is

A. equal to W12.

B. less than W12.

C. greater than W12.

D. either A., B., or C., depending on the ratio of V2 to V1.

CPS question

Page 30: Thermodynamic systems - Northeastern ITS · Quasi-static processes : near equilibrium Initial state, final state, intermediate state: p, V & T well defined Sufficiently slow processes

Quasi-static process

Character U∆ WQ

adiabatic 0=Q WU −=∆

isothermal T = constant 0=∆U

isochoric

isobaric

V = constant

p = constant

QU =∆ TCQ V ∆= 0=W

VpW ∆=WQU −=∆ TCQ P∆=

1

2lnV

VnRTW =WQ =

)11

()1(

11

11

2

11 −− −−

= γγγ

γ VVVpW0=Q

Summary


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