Chapter 18

Heat, Work, and the First Law of Thermodynamics

Heat and work

Thermodynamic cycle

sdFdW

dsPA )( )(AdsP PdV

f

i

V

VPdVW

Heat and work

• Work is done by the system:

• Work is done on the system :

f

i

V

VPdVW

f

i

V

VPdVW

The first law of thermodynamics

• Work and heat are path-dependent quantities

• Quantity Q + W = ΔEint (change of internal energy)

is path-independent

• 1st law of thermodynamics: the internal energy of a system increases if heat is added to the system or work is done on the system

WQEEE if int,int,int

The first law of thermodynamics

• Adiabatic process: no heat transfer between the system and the environment

• Isochoric (constant volume) process

• Free expansion:

• Cyclical process:

WWE 0int

QQE 0int

000int E

0int WQE

WQ

Chapter 18Problem 19

In a certain automobile engine, 17% of the total energy released in burning gasoline ends up as mechanical work. What’s the engine’s mechanical power output if its heat output is 68 kW?

Work done by an ideal gas at constant temperature

• Isothermal process – a process at a constant temperature

• Work (isothermal expansion)

nRTPV VconstVnRTP //)(

f

i

V

VPdVW f

i

V

VdV

V

nRT

f

i

V

V V

dVnRT

i

f

V

VnRT ln

i

f

V

VnRTW ln

Work done by an ideal gas at constant volume and constant pressure

• Isovolumetric process – a process at a constant volume

• Isobaric process – a process at a constant pressure

0if

f

i VV

V

VPdVW

VPW

0W

f

i

V

VPdVW f

i

V

VdVP VP

Molar specific heat at constant volume

• Heat related to temperature change:

• Internal energy change:

TNnmcQ AV )( 0 TnCV

WTnCE V int TnCTnC VV 0

Tn

ECV

int

Tn

nRT

23

T

TR

2

3R

2

3

KmolJRCV / 5.122

3TnCE V int

Molar specific heat at constant pressure

• Heat related to temperature change:

• Internal energy change:

TnCQ P

WQE int

RCC VP

VPTnCP

TnRTnCTnC PV

RCV 2

3R

2

5

Adiabatic expansion of an ideal gas

PdVdTnCV 0PdVdQdE int

nRTPV )()( nRTdPVd nRdTVdPPdV

R

VdPPdVndT

VC

PdVndT

VP CC

VdPPdV

VPV CC

VdPPdV

C

PdV

Adiabatic expansion of an ideal gas

VPV CC

VdPPdV

C

PdV

0

V

dV

C

C

P

dP

V

P

constVC

CP

V

P

lnln constPVPV V

P

C

C

ffii VPVP

nRTPV V

nRTP

constVV

nRT

constTV 1

Chapter 18Problem 24

How much work does it take to compress 2.5 mol of an ideal gas to half its original volume while maintaining a constant 300 K temperature?

Free expansion of an ideal gas

0int E

fi TT

nRTPV

ffii VPVP

Degrees of freedom and molar specific heat

• Degrees of freedom:3 translations, 3 rotations, + oscillations

RCV 2

3

Degrees of freedom and molar specific heat

• Degrees of freedom:3 translations, 3 rotations, + oscillations

• In polyatomic molecules different degrees of freedom contribute at different temperatures

Rf

CV 2

3f

5f

6f

RCV 2

3

Chapter 18Problem 26

A gas mixture contains 2.5 mol of O2 and 3.0 mol of Ar. What are this mixture’s molar specific heats at constant volume and at constant pressure?

Questions?

Answers to the even-numbered problems

Chapter 18

Problem 22

1.2 kJ