Thermodynamics - Eastside Physics

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Oct 27, 2013 (4 years and 14 days ago)

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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.