Thermodynamics
•
Internal energy of a system can be increased
either by adding energy to the system or by
doing work on the system
•
Remember internal energy of a system is
the sum of the kinetic energies and the
potential energy at the molecular level
•
Adding heat is transferring energy
Work done on a gas
•
Consider a gas at equilibrium, pressure and
temperature is the same throughout.
•
The gas is in a piston with area A.
•
Work done on the gas W =

F
y =

PA
y
•
As A
y = Volume then W =

P
V
•
Note as volume is decreased
V is negative
and work W is positive. If
V is positive
work is done on the environment.
Pressure Diagram
•
The equation above is only valid when the
pressure is constant during the work process
•
Such a process is called an isobaric process.
•
The area under a P vs V graph represents
the work done. The arrow represents +ve or

ve work p
V
First Law of Thermodynamics
•
If a system undergoes a change from its
initial state to a final state, where Q is the
energy transferred to the system by heat and
W is the work done on the system, the
change of internal energy of the system
U = U
f

U
i
= Q + W
•
Q is positive when heat is added to the
system, W is positive when work is done on
the system.
Internal energy of a gas
•
Change in internal energy of an ideal gas is,
U = 3/2nR
T
The molar specific heat of a monatomic gas is
Cv
=
3/2R
Therefore change in internal energy of the ideal
gas is
U = nCv
T
The larger the Cv the more energy needed to
change the temperature.
Degrees of Freedom
•
Each different way a molecule can store
energy is called a degree of freedom
•
Each degree of freedom contributes 1/2R to
the molar specific heat.
•
Monatomic molecules can move in three
directions thus C
v
= 3/2R
•
A diatomic molecule can move three and
tumble two ways thus C
v
= 5/2R
Isobaric Processes
•
Pressure remains constant throughout
•
Expanding gas works on the environment
•
When gas works it losses internal energy
•
Temperature decreases as energy decreases
•
If volume increases and temperature
decreases then thermal energy (heat) must
be added to gas to maintain constant
pressure
Isobaric Processes
•
From
U = Q + W then Q =
U

W
Q =
U

P
V
From P
V = nR
T and
U = 3/2nR
T
then Q = 3/2 nR
T + nR
T = 5/2 nR
T
Another way of expressing heat transfer is
Q = nC
p
T where C
p
= 5/2R
C
p
is the molar heat capacity C
p
= C
v
+ R
Adiabatic Processes
•
In adiabatic processes no energy enters or
leaves the system by heat (insulated System
•
A rapid system is considered adiabatic,
there is no time for heat energy transfer.
•
As Q = 0 then
U = W The work done is
the change in internal energy. Work can
calculated from a PV diagram Pv
y
=constant
•
y
= C
p
/C
v
called adiabatic index
Isovolumetric Processes
•
Sometimes called an isochoric process, it
occurs at a constant volume, shown as a
vertical line in a PV diagram.
•
As volume does not change no work is done
by or on the system, thus W= 0
•
U = Q The change in internal energy of
the system is equal to the transfer of heat to
the system. Q = nC
v
T
Isothermal Processes
•
During the process the temperature of the
system does not change. As U depends on
temperature,
U= 0 as
T = 0 and W =

Q
•
The work done on the system is equal to the
negative thermal energy transferred to the
system.
•
For work done on the environment
W
env
= nRT ln(V
f
/V
i
)
Second Law of Thermodynamics
•
Heat engines take in heat energy and
convert it to other forms of energy,
electrical and mechanical. Ex. Coal burnt,
heat converts water to steam, steam turns a
turbine, turbine drives a generator. In
general a heat engine carries a substance
through a cyclic process.
•
Entropy tendency to greater disorder
Heat Engines cont.
•
The Weng done by a heat engine equal the
net work absorbed by the engine. The initial
and final internal energies are equal.
•
U= 0 therefore Qnet =

W = Weng
•
where Qnet = Q
h

Q
c
•
The work done by an engine for a cyclic
process is the area enclosed by the curve of
PV diagram
Thermal Efficiency
•
The thermal efficiency of a heat engine e is
defined as the work done by the engine,
W
eng
, divided by the energy absorbed
through one cycle.
•
E = W
eng
/
Q
h
= (Q
h

Q
c
)/Q
h
= 1

Q
c
/Q
h
•
The values of Q are absolute values
Refrigerators and Heat Pumps
•
Heat engines operate in reverse, energy is
injected into the engine. Work is done by
the system where heat is removed from a
cool reservoir to a hot reservoir. A
compressor reduces the volume of a gas
increasing its temperature, the gas later
expands requiring heat to do so thus
drawing heat from the cool refrigerator
Coefficients of Performance
•
Coefficient of performance for a refrigerator
is equal to the magnitude of extracted
energy divided by the work performed.
•
COP
(cooling) =
Q
c
/W the larger the ratio
the better the performance
•
For a heat pump operating in heat mode
COP
(heating)
= Q
h
/W
Second Law of Thermodynamics
•
No heat engine operating in a cycle can
absorb energy from a reservoir and use it
entirely for the performance of an equal
amount of work. (e<1) e = W
eng
/ Q
h
•
Some energy Q
c
is always lost to the
environment.
The Carnot Engine
•
The Carnot Cycle is the operation of an ideal
reversible cycle, using two energy reservoirs.
•
Carnot’s theorem states that no real engine
operating between two energy reservoirs can be
more efficient than a Carnot engine operating
between the same two reservoirs.
•
The Carnot engine is only theoretical and would
have to run infinitely slowly, thus having zero
power output.
Carnot cont.
•
The Carnot cycle contains an ideal gas in a
thermally nonconductive cylinder with a
moveable piston at one end. The cycle
passes through four stages, two isothermal
and two adiabatic. During the adiabatic
stages the gas temperatures range between
T
c
and T
h.
All cycles are reversible.
Stage 1 Isothermal Expansion
•
The base of the cylinder consists of a hot
energy reservoir at T
h
.
•
Gas in the cylinder absorbs heat energy Q
h
from a reservoir, thus doing work by raising
the piston. The gas expands isothermally at
temperature T
h.
•
W =

Q
Stage 2 Adiabatic Expansion
•
Base of the cylinder is replaced with a
thermally insulated base.
•
The gas continues to expand, this time
adiabatically. ( no energy enters or leaves
the system by heat.
•
The expanding gas does work on the piston,
raising it further while the gas temperature
decreases from T
h
to T
c
. W =
Δ
U
Stage 3 Isothermal Compression
•
The cylinder base is replaced with a cold
reservoir at T
c
.
•
The gas is compressed at the temperature of
T
c
and during this time expels energy to the
reservoir, Q
c.
•
Work is done on the gas.
Stage 4 Adiabatic Compression
•
The base is again replaced with a thermally
non

conducting wall.
•
The gas is compressed adiabatically
increasing its temperature to T
h
thus doing
work on the gas.
•
This 4 stage cycle is constantly repeated.
•
E
c
= 1
–
T
c
/T
h
T is in Kelvin
•
This is used to rate engine efficiency
Entropy S
•
Entropy is the state of disorder(randomness)
•
The change in entropy
Δ
S = Q
r
/T
Where Q
r
is the energy absorbed or expelled
during a reversible process. T is constant in
Kelvin. r means reversible.
•
A change in entropy occurs between two
equilibrium states. The path taken is not
important
Entropy cont.
•
If the laws of nature are allowed to operate
without interference it is more likely to have
a disorderly arrangement than an orderly
one. Using probability Boltzmann found
that S = k
B
lnW where k
B
is
Boltzmann’s constant and W is a number
proportional to the probability of a specific
occurrence. The second law states what is
most likely not what will happen.
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