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Chapter 18
Heat, Work, and the First Law of
Thermodynamics
Heat and work
 
dW  F  ds  ( PA)  ds  P( Ads)  PdV
Vf
W   PdV
Vi
Thermodynamic cycle
Heat and work
• Work is done by the system:
Vf
W   PdV
Vi
• Work is done on the system :
Vf
W    PdV
Vi
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
Eint  Eint, f  Eint,i  Q  W
The first law of thermodynamics
• Adiabatic process: no heat transfer between the
system and the environment
Eint  0  W  W
• Isochoric (constant volume) process
Eint  Q  0  Q
• Free expansion:
Eint  0  0  0
• Cyclical process:
Eint  Q  W  0
Q  W
Chapter 18
Problem 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
PV  nRT
P  (nRT ) / V  const / V
• Work (isothermal expansion)
nRT
dV
W   PdV  
Vi
Vi
V
Vf
V f dV
 nRT ln
 nRT 
Vi V
Vi
Vf
W  nRT ln
Vi
Vf
Vf
Work done by an ideal gas at constant
volume and constant pressure
• Isovolumetric process – a process at a constant
volume
Vf
W   PdV  0
Vi
V f Vi
W 0
• Isobaric process – a process at a constant pressure
Vf
Vf
Vi
Vi
W   PdV  P  dV  PV
W  PV
Molar specific heat at constant volume
• Heat related to temperature change:
Q  cV (nm0 N A )T  nCV T
• Internal energy change:
Eint  nCV T  W  nCV T  0  nCV T
3

 nRT 
Eint
3 R T 3
2


CV 

 R

nT
2 T
2
nT
Eint  nCV T
3
CV  R  12.5 J / mol  K
2
Molar specific heat at constant pressure
• Heat related to temperature change:
Q  nCP T
• Internal energy change:
Eint  Q  W  nCP T  PV
nCV T  nCP T  nRT
5
CP  CV  R  R
2
3
CV  R
2
Adiabatic expansion of an ideal gas
PV  nRT
d ( PV )  d (nRT )
PdV  VdP  nRdT
PdV  VdP PdV  VdP

ndT 
C P  CV
R
dEint  dQ  PdV nCV dT  0  PdV
PdV
ndT  
CV
PdV PdV  VdP


CV
C P  CV
Adiabatic expansion of an ideal gas
dP  CP  dV
PdV PdV  VdP
  
0


P  CV  V
CV
C P  CV
 CP 
ln P    ln V  const
 CV 
PiVi   Pf V f
nRT
PV  nRT P 
V
nRT 
V  const
V
TV  1  const
PV
CP
CV

 PV  const
Chapter 18
Problem 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
Eint  0
Ti  T f
PV  nRT
PiVi  Pf V f
Degrees of freedom and molar specific
heat
3
CV  R
2
• Degrees of freedom:
3 translations, 3 rotations, + oscillations
Degrees of freedom and molar specific
heat
f 3
3f
C
CVV  R
2
• Degrees of freedom:
3 translations, 3 rotations, + oscillations
• In polyatomic molecules different
degrees of freedom contribute at
different temperatures
f 5
f 6
Chapter 18
Problem 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