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First law of thermodynamics
Thermal systems
Classical mechanics
Thermal system
• Thermal systems:
- deals with many individual objects
- conceptually different from mechanical systems
- don’t know the position, velocity, and energy of
any molecules or atoms or objects
- can’t perform any calculation on them
• Sacrifice microscopic knowledge of the system,
using macroscopic parameters instead
- volume (V)
- temperature (T)
- pressure (P)
- number of particles (N)
- energy (E), etc.
• Macroscopic systems with many individual objects:
- processes are often irreversible
- arrow of time does exist
- energy conservation is not enough to describe
the thermal states
Thermodynamic systems
Isolated systems can exchange
neither energy nor matter with the
environment.
reservoir
reservoir
Heat
Heat
Work
Work
Closed systems exchange energy
but not matter with the environment.
Open systems can exchange
both matter and energy with
the environment.
Idea gas model
Lattice model for solid state materials
The ideal gas model
• all the particles are identical
•
•
•
•
the particles number N is huge
the particles can be treated as point masses
the particles do not interact with each other
the particles obey Newton’s laws of motion, but their motion is random
• collisions between the particles are elastic
The ideal gas
equation of state:
PV  NkBT
kB = 1.38  10-23 J/K
Internal energy
• The internal energy of a system of N particles,
U, is all the energy of the system that is
associated with its microscopic components
when view from a reference frame at rest
with respect to the object.
• Internal energy includes:
- kinetic energy of translation, rotation, and
vibration of particles
- potential energy within the particles
- potential energy between particles
• Internal energy is a state function – it depends
only on the values of macroparameters (the
state of a system)
For a non-ideal gas:
U  U (V , T )
For an ideal gas (no interactions):
U  U (T )
f
 Nk BT
2
Monatomic:
f 3
Diatomic:
f 5
Heat
Heat and work are both defined to describe energy transfer across a
system boundary.
• Heat (Q): the transfer of energy across the boundary of a
system due to a temperature difference between the system and
its surroundings.
- Q > 0: temperature increases; heating process
- Q < 0: temperature decreases; cooling process
- Q  CT (C: heat capacity)
• Heat transfer mechanisms
- conduction: exchange of kinetic energy between
microscopic particles (molecules, atoms, and
electrons) through collisions
- convection: energy transfer by the movement of a
heated substance such as air
- radiation: energy transfer in the form of electromagnetic
waves
heat
• Work (W): any other kind of energy transfer across boundary
Quasi-static processes
Quasi-static (quasi-equilibrium) processes:
• Sufficiently slow processes, and any intermediate state
can be considered as at thermal equilibrium. The macro
parameters are well-defined for all intermediate states.
• The state of a system that participates in a quasiequilibrium process can be described with the same
number of macro parameters as for a system in
equilibrium.
• Examples of quasi-static processes:
- isothermal: T = constant
- isovolumetric: V = constant
- isobaric: P = constant
- adiabatic: Q = 0
Work done during volume changes
Quasi-static process
at each infinitesimal
movement
dH
V  A H
H
dV  A  dH
dW  F  dH
dW  PdV
 ( P  A)  dH

 P  ( A  dH )
Vf
W 
PdV
 P  dV
V

i
Work done by the
gas as its volume
changes from Vi to Vf
Work done during volume changes (cont.)
dW  PdV
• dV > 0: the work done on the gas is negative
• dV < 0: the work done on the gas is positive
In thermodynamics, positive work represents a transfer of energy out of the
system, and negative work represents a transfer of energy into the system.
P
Vf
W   PdV
Vi
P  P(V , T )
Pi
i
P-V diagram
f
The work done by a gas in the expansion
is the area under the curve connecting
the initial and final states
Pf
Vi
Vf
V
Work and heat are not state functions
a
c
b
a. isovolumetric
b. isobaric
W  Pf (V f  Vi )
a. isobaric
b. isovolumetric
W  Pi (V f  Vi )
isothermal
Vf
W   PdV
Vi
• Because the work done by a system depends on the initial and final states and
on the path followed by the systems between the states, it is not a state function.
• Energy transfer by heat also depends on the initial, final, and intermediate states
of the system, it is not a state function either.
• When heat enters a system, will it increase the system’s internal energy?
• When work is done on a system, will it increase the system’s internal energy?
It depends on the path!
The first law of thermodynamics
• Two ways to exchange energy between a system
and its surroundings (reservoir):
heat and work
• Such exchanges only modify the internal energy of
the system
• The first law of thermodynamics: conservation of energy
U  Q  W
reservoir
Heat
Work
Q > 0: energy enters the system
Q < 0: energy leaves the system
W > 0: work done on the system is negative;
energy leaves the system
W < 0: work done on the system is positive;
energy enters the system
• For infinitesimal processes:
dU  dQ  dW  dQ  pdV
Several examples
Isolated systems:
P
i, f
Q W  0
 U  0
 Ui  U f
The internal energy
of an isolated systems
remains constant
Adiabatic processes
Cyclic processes
Insulating
wall
V
initial state = final state
U  0
 Q W
Energy exchange between
“heat” and “work”
Q0
 U  W
Expansion: U decreases
Compression: U increases
Idea gas isovolumetric process
P
2
PV  NkBT2
Isovolumetric process: V = constant
V2
 W   PdV  0
V1
1
PV  NkBT1
V1,2
reservoir
Heat
V
Q  CV (T2  T1 )
 CV T
(CV: heat capacity
at constant volume)
U  Q  W
 U  Q  CV T
During an isovolumetric process, heat enters
(leaves) the system and increases (decreases)
the internal energy.
Idea gas isobaric process
P
PV  NkBT2
V2
 W   PdV  P(V2  V1 )  PV
2
1
Isobaric process: P = constant
V1
PV  NkBT1
V1
V2
V
Q  C p (T2  T1 )
 C p T
(CP: heat capacity
at constant pressure)
 U  Q  W
reservoir
 C P T  PV
Heat
Work
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.
Idea gas isothermal process
P
1
PV  Nk BT
2
Isothermal process: T = constant
U  0
V2
V2
V1
V1
W   PdV  
V1
V2
V
During an isothermal expansion process,
heat enters the system and all of the heat
is used by the system to do work on the
environment.
During an isothermal compression process,
energy enters the system by the work done
on the system, but all of the energy leaves
the system at the same time as the heat
is removed.
NkBT
dV
V
dV
 NkBT 
V1 V
V2
 NkBT ln
V1
V2
V2
 Q  W  Nk BT ln
V1
Idea gas adiabatic process
P
2
P  P(V , T )
1
V2
V1
Idea gas:
f
U  NkBT
2
f
 dU  NkB dT
2
Adiabatic process:
dU  dQ  dW
  PdV
Adiabatic process: Q = 0
V2
V2
V1
V1
W   PdV   P(V , T )dV
PV  NkBT
V
 d ( PV )  d ( NkBT )
 PdV  VdP  NkB dT
PdV  VdP
f
NkB dT   PdV
2
2
 NkB dT   PdV
f
2
  PdV
f
Idea gas adiabatic process
P
2
P  P(V , T )
PdV  VdP  
 VdP  (1 
1
V2
V1
let   (1 
V
2
PdV
f
2
) PdV  0
f
2
) , and divided by PV
f
dP
dV

0
P
V
V dV
dP
P1 P   V1 V  0
P
V
 ln   ln  0
P1
V1
P
PV 
 ln
0

P1V1

 PV   P1V1  constant
Idea gas adiabatic process

P
PV   constant
2
PV   P1V1  constant
V2
W   PdV  P1V1
V1
1
V2
V1


V
 W  P1V1
V2

V1
dV
V
1
1
1
(   )
(  1) V2 V1
  constant
For monatomic gas,
f 3
2
  1   1.7
3
Idea gas adiabatic process

P
PV   constant
2
P1V1  NkBT1
1
V2
V1
PV   P1V1  constant
V
P2V2  NkBT2

P1V1  P2V2

V2 1 T1
  1 
V1
T2
or
 1
T1 V1

P1 V1 T1

P2 V2 T2

P V
 1  2
P2 V1
 T2 V2
 1
 constant
Idea gas adiabatic process
P

PV  constant
2

PV   P1V1  constant
 1
T1 V1
1
V2
V1
 T2 V2
 1
 constant
PV  NkBT
V
1
1
1
W  P1V1
(   )
(  1) V2 V1

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.
Summary
• Internal energy, heat, and work:
- internal energy is the energy of the system; a state function
- heat and work are two ways to exchange energy between the system
and the environment. They are not state functions and depend on the path
• The first law of thermodynamics connects the internal energy with heat and
work: U  Q  W
Quasi-static
process
Character
isovolumetric V = constant
isobaric
P = constant
isothermal
T = constant
adiabatic
Q0
U
Q
W
U  Q
Q  CV T
W 0
U  Q  W
Q  CP T
W  PV
U  0
Q W
U  W
Q0
W  NkBT ln

W  P1V1
V2
V1
1
1
1
(   )
(  1) V2 V1