Thermal contact
Two systems are in thermal (diathermic) contact, if they can
exchange energy without performing macroscopic work.
This form of energy transfer (random work) is called heat.
Mechanisms of Heat Transfer
1. Thermal Conduction
law of thermal conduction:
dQ
T
 kA 
dt
x

 more precisely :

A
dx
  
dQ
 kA  T 
dt

Mechanisms of Heat Transfer
1. Convection
natural convection:
resulting from differences in density
forced convection:
the substance is forced to move by a
fan or a pump.
The rate of heat transfer is directly related
to the rate of flow of the substance.
dQ = cTdm
Mechanisms of Heat Transfer
1. Radiation
Energy is transmitted in the form of
electromagnetic radiation.
Stefan’s Law
dQ
 AeT4
dt
 = 6  10-8 W/m2K
A – area of the source surface
e – emissivity of the substance
T – temperature of the source
E
B
Zeroth law of thermodynamics
Thermal Equilibrium:
If the systems in diathermic contact do not exchange
energy (on the average), we say that they are in thermal
equilibrium.
If both systems, A and B, are in thermal equilibrium
with a third system, C, then A and B are in thermal
equilibrium with each other.
Temperature
We say that two systems in thermal equilibrium have the same
temperature. (Temperature is a macroscopic scalar quantity
uniquely assigned to the state of the system.)
T  273.16 K  lim P
P3 0 P3
Gas Thermometer
h
T3 = 273.16 K is the temperature at
which water remains in thermal
equilibrium in three phases (solid,
liquid, gas).
The Celsius scale and, in the US, the Fahrenheit scale are often used.
TC  T  273.15 ;
TF  9 TC  32
5
Thermal expansion
For all substances, changing the temperature of a body
while maintaining the same stress in the body causes a
change in the size of the body.
D
dD
l
dl
linear expansion:
dl = ldl
The proportionality coefficient (T) is called
the linear thermal expansion coefficient.
volume expansion:
dV =VdV
The proportionality coefficient (T) is called
the volume thermal expansion coefficient.
Heat Capacity
The differential amount of absorbed heat (dQ), necessary to
change the temperature of the system, is proportional to the
change in temperature (dT) of the system.
dQ  CdT
Tf


 Q   CT dT 
Ti


The proportionality coefficient C is called the heat capacity of the system.
If heat capacity does not depend on temperature:
Q = C  T
specific heat and molar heat capacity
The heat capacity of a system in proportional to the amount of
matter in the system and depends on the material of the system.
If the amount of matter is expressed by mass (m):
C=cm
where c is called specific heat
If the amount of matter is expressed by the number of moles (n),
C = Cm  n
where Cm is called molar heat capacity
temperature dependency of heat capacity
The specific heat of lead
(at atmospheric pressure)
128 (J/kgK)
600 K
The heat capacity of a
system depends on the
thermodynamic process
and the temperature of
the system.
temperature
Latent Heat
At (first type) phase transitions, the amount of absorbed heat
is proportional to the amount of transformed substance
Q = L  m
The proportionality coefficient is called latent heat.
Thermodynamic Process
Macroscopically, the state
of a system of particles is
described
by
uniquely
assigned parameters.
A thermodynamic process
is a sequence of states of the
system. In a thermodynamic
process the state parameters
are functions of time.
T
V
P
internal energy in common processes
• adiabatic process - no heat is transferred
U = W
(dU = -dW)
• isochoric process - constant volume process
U = Q
(dU = dQ)
• cyclical process - the system returns to the initial state
U = 0
• isothermal process - constant temperature
U = Q - W
(dU = dQ - dW)
(for an ideal gas dU = 0)
• isobaric process - constant pressure
U = Q - W
(dU = dQ - dW)
• free expansion - adiabatic with no work done
U = 0
(dU = 0)
Ideal Gas
macroscopic definition:
An ideal gas is one that obeys the equation of state
PV = nRT
P - pressure
V - volume
n - amount (in moles)
R - universal gas constant
T - temperature
microscopic definition:
Except for elastic collisions the particles of an ideal gas
do not interacts - the range of interaction is very short.
Isothermal process in an ideal gas
(Boyle-Mariotte law)
pressure
T1< T2 < T3
The temperature of the system
(an ideal gas) is kept constant.
nRT
V nRT
PV
P
V
volume
Isobaric process in an ideal gas
(Charles and Gay-Lussac law )
volume
P3 < P2< P1
The pressure of the system
is kept constant.
nR
T nRT  T
VPV
P
temperature
Isochoric process in an ideal gas
pressure
V3 < V2< V1
The volume of the system
is kept constant.
nR
T nRT T
PPV
V
temperature
Macroscopic Work
When the volume of the system changes,
the system performs (macroscopic) work.
dW  F  dx

P PdV
A  dx  PdV
dW
dx
(integral form
W 
 PdV )
process
Work depends on the thermodynamic process!
P
First law of thermodynamics
a
b
V
For each thermodynamic process, the
difference between heat delivered to the
system and the work done by the system
depends only on the initial and the final state
of the system.
T
There is such a function state U, called internal energy, that
dU = dQ - dW
where dQ is the heat delivered to the system and dW is the
work performed by the system.
Comment: On the microscopic scale, the internal energy of a
system is the total mechanical energy of the system.
Internal energy of an ideal gas
The internal energy of an ideal gas results from the
translational kinetic energy, rotational kinetic energy, and
vibrational energy of the molecules constituting the system.
According to the kinetic theory of gases, the internal energy of
an ideal gas is a function of temperature only
U = nCVT
proof.
From the first law of thermodynamics
dU = dQ = nCVdT
With the reference for the internal energy at 0 K:
T
U  0   nCV dT  nCV T
0
Entropy
For any cyclical quasi-static process
P
a
b
V
T
dQ r
0

process T
(The change in entropy from the
initial state to the final state does not
depend on the process.)
We can introduce a function S, called entropy, that is
a function of state of the system.
definition of entropy
macroscopic:
The change in entropy between two equilibrium states is given by the
heat transferred, dQr, in a quasi-static process leading from the
initial to the final state divided by the absolute temperature, T, of the
system
dQ r
dS 
T
microscopic:
If the number of possible configurations for a considered state
of a system is W (statistical sum), the entropy S of the system in
this state is
S  kB ln W
where kB is a physical constant (Boltzmann’s constant).
Second law of thermodynamics
In any thermodynamic process that
proceeds from one equilibrium state to
another, the total entropy of the system
and its environment (the Universe)
cannot decrease.
consequences of the second law of thermodynamics
It is impossible to construct a heat
engine which when operating in a cycle,
produces no other effect than the
absorption of thermal energy from a
reservoir and the performance of an
equal amount of work
If it was
With
the possible
heat sink
Th
Qh
W
engine
Qc
 QhQh Qc
Suniv
0
S.  
0
ShSh
Seng


S
Seng  c   0 
Tc
Th Th
engine efficiency
e
W
Q h  Qc
T
Qc
 1

1 c
Q h
Qh
Th
Q h
Tc
consequences of the second law of thermodynamics
It is impossible to transfer heat from one body to another
body at a higher temperature with no other consequences
in the universe.
T1
T2
The change in entropy:
 dQ dQ

0  dS  dS1  dS2 
T1
T2
from which
dQ
T1  T2
Reversible and irreversible processes
If, during a thermodynamic process, the entropy of the
Universe is not changed, the process is reversible.
If during a thermodynamic process the entropy of the
Universe is changed (increased), the process is
irreversible.
Gas Constant
(molar heat capacity of an ideal gas)
in an isobaric process:
nCVdT = dU = dQ - dW = nCPdT - nRdT
PV = nRT
PdV = nRdT
Molar heat capacity of an ideal gas in an isobaric process is
related to the molar heal capacity in an isochoric process by
CP = CV + R
Adiabatic process in an ideal gas
nCVdT = dU = -dW = -PdV
(no heat)
(for ideal gas)
PdV + VdP = nRdT
(eliminating temperature)
R
PdV
CV
CV  R dV
dP


CV
V
P
PdV  VdP  
 ln

Pi Vi


Pf Vf
Vf
P
  ln f
Vi
Pi
 1
Ti Vi

 1
Tf Vf
The Carnot cycle
P
Tc
e 1
Th
Qh
A
B
Tc
Qhc
Tch
Th
D
C
adiabatic
isothermal
expantion
compression
Qc
V
The work W done by the gas equals the net heat delivered to
the gas in the cycle
W = Qh - Qc
four-stroke combustion engine
(Otto cycle)
The Otto cycle represents operation of a common gasoline engine. The
cycle includes two isobaric, two isochoric and two adiabatic processes.
e  1
C
P
1. intake
2. compression
1
 V1 / V2  1
Qh
3. work
B
4. exhaust
O
D
W
Qc
A
V1
V2
V
Refrigerators
A refrigerator is a device that moves
heat from a system at a lower temperature
to the system with higher temperature.
The effectiveness of a refrigerator is
described in terms of the coefficient of
performance
COP 
Th
Qh
W
engine
Qc
Tc
Q c
W
The highest possible coefficient of performance is that of a
refrigerator whose working substance is carried through a reverse
Carnot cycle
COPC 
Tc
Th  Tc