Thermodynamic Processes Tip Sheet

coralmonkeyMechanics

Oct 27, 2013 (3 years and 9 months ago)

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Thermodynamic States

The ideal gas law relates the pressure (P), volume (V), number of moles (n) and temperature
(T)

of a gas.


PV = nRT


In SI units, R = 8.31 J/(mol*K).
Be especially careful with using SI units exclusively as this
equation is often used
in chemistry with a different set of units.


Not that t
his law
is approximate. It only
applies to cases where the temperature is well above
the condensation point and the volume of the molecules is much less than the volume of the
container.

The term “law”

does not

make it universally applicable or exact.


R is a constant and n is often held constant (a sealed container). This

normally

leaves three
variables in this equation. Do not assume that one of the three is held constant unless you
have evidence. Som
etimes all three change. Using Charles’ law or Boyle’s law (special cases
of the ideal gas law) inappropriately are common ways to make mistakes.

Thermodynamic Processes

Δ
E
th

= Q


W


The first law of thermodynamics is a statement of conservation of energy. A system can
exchange energy wit
h its environment with heat (Q), a microscopic transfer of energy,
or
work (W
, also denoted as W
s
)
, a macroscopic transfer of energy
.
T
his results in changes in its
internal or thermal energy (
E
th
, also denoted as

U in some texts
).
By convention, heat
coming into the system is considered positive and heat going out of the system is considered
negative. Work done by the system is considere
d positive, while work done on the system is
considered negative (
note that
this is the opposite of the classical mechanics sign convention
for work
, thus the minus sign in the first law
).


W = ∫PdV
(± the area under the curve in the PV diagram)
always.

For an isochoric process,
W = 0. For an isobaric process, W = P∫dV = PΔV. For an isothermal process, the ideal gas
law can be used to find P as a function of V and substituted into the equatio
n for work. W =
∫(nRT/V)dV = nRT*ln(V
f
/V
i
).


The definitions of C
p

and C
v

can be used to obtain a formula for heat during isobaric and
isochoric processes. For an isobaric process, dQ = nC
p
dT. If C
p

and n are constant (usually
the case, at least approximat
ely), then Q = nC
p
ΔT. Likewise, for an isochoric process, dQ =
nC
v
dT. If C
v

and n are constant (usually the case, at least approximately), then Q = nC
v
ΔT.


This formula for heat during an

isochoric process can be used with the first law and the
formula for

work to obtain expressions for the change in
thermal

energy for
any

process. For
an isochoric process, W = 0, so
Δ
E
th
= Q = nC
v
ΔT.
T
his
formula

for
Δ
E
th

can
be generalized
to all processes since
thermal

energy is path independent.



Physical Situation

Name

State
Variables

P

V

T

ΔE
th

Q

W

Insulated
Sleeve

Add
weight to
piston

Adiabatic
compression

PV
γ

=
constant;
TV
γ
-
1
=
constant

Up

Down

Up

nC
v
ΔT

> 0

0

-
nC
v
ΔT

< 0

Insulated
Sleeve

Remove
weight
from
piston

Adiabatic
expansion

PV
γ

=
constant;
TV
γ
-
1

=
constant

Down

Up

Down

nC
v
ΔT

< 0

0

-
nC
v
ΔT

> 0

Heat gas

Lock
piston

Isochoric

V fixed; P
α T

Up

Fixed

Up

nC
v
ΔT

> 0

nC
v
ΔT

> 0

0

Cool gas

Lock
piston

Isochoric

V fixed; P
α T

Down

Fixed

Down

nC
v
ΔT

< 0

nC
v
ΔT

< 0

0

Heat gas

Piston
free to
move

Isobaric
expansion

P
fixed; V
α T

Fixed

Up

Up

nC
v
ΔT

> 0

nC
p
ΔT

> 0

PΔV

> 0

Cool gas

Piston
free to
move

Isobaric
compression

P fixed; V
α T

Fixed

Down

Down

nC
v
ΔT

< 0

nC
p
ΔT

< 0

PΔV

< 0

Immerse
gas in
large bath

Add
weight to
piston

Isothermal
compression

T fixed,
PV =
constan
t

Up

Down

Fixed

nC
v
ΔT
=
0

n
R
T*ln(V
f
/V
i
)

< 0

nR
T*ln(V
f
/V
i
)

< 0

Immerse
gas in
large bath

Remove
weight
from
piston

Isothermal
expansion

T fixed,
PV =
constant

Down

Up

Fixed

nC
v
ΔT
= 0

nR
T*ln(V
f
/V
i
)

> 0

nR
T*ln(V
f
/V
i
)

> 0

Unknown

Unknown

No Name

PV =
nRT

?

?

?

nC
v
ΔT

ΔE
th

+ W

∫PdV = ± area
under curve in
PV diagram

Tip
s
: Know which formulas are specific to a particular process and which are true for any
process.

See the next section for notes on C
p
, C
v

and γ.


Heat Capacities

C
v
, the molar heat capacity at co
nstant volume (zero work), can be estimated using the
number of degrees of freedom multiplied by ½R. For a monatomic gas, there are three
degrees of freedom from the translation of the particles in three dimensions, so C
v

= 3/2*R.
For a diatomic gas, there

are five degrees of freedom from the three directions of translation
and two axes of rotation. The third possible axis does not have a significant rotational kinetic
energy and is therefore insignificant. Therefore, C
v

= 5/2*R. For solids, there are three

degrees of freedom from translation and three from vibration, so C
v

= 3R (Dulong
-
Petit).
All three formulas are theoretical and classical, and generally give reasonable agreement with
empirical evaluations. Deviations from these formulas can be explained
with quantum
mechanics which is beyond the scope of this course.


C
p
, the molar heat capacity at constant pressure, can be calculated for an ideal gas. For an
isobaric process where n and C
p

are constant, Q = nC
p
ΔT. The change in
thermal

energy can
be calculated with the general formula
Δ
E
th

= nC
v
ΔT. W = P∫dV = PΔV by the definition of
work. Using the ideal gas law, PΔV = nRΔT.
Δ
E
th

= Q


W by the first law of
thermodynamics. Combine the above formulas to obtain a
n expression for C
p
.


Δ
E
th

= Q


W

nC
v
ΔT = nC
p
ΔT


nRΔT

C
p

= C
v

+ R


The ratio of heat capacities is denoted by the letter gamma (γ), and is defined by the
following formula:


γ = C
p
/C
v


For a monatomic ideal gas, γ = (3/2*R + R)/(3/2*R) = 5/3. For a diat
omic ideal gas, γ =
(5/2*R + R)/(5/2*R) = 7/5.


Gas

C
v

C
p

γ
=
bxamples
=
monatomic
=
P⼲*o
=
R⼲*o
=
R⼳
=
ee,=ke,⁁r
=
摩atomic
=
R⼲*o
=
T⼲*o
=
T⼵
=
e
2
, N
2
, O
2

Thermodynamic Cycles

A cycle must have a total
Δ
E
th

= 0. Therefore ΣQ = ΣW by the first law. Normally, it is
us
eful to separate the values of Q that are positive from those that are negative. Positive
values represent heat input into the system and negative values represent heat output from
the system.


The total work for a cycle can be calculated graphically with
the area enclosed in the PV
diagram. If the cycle is clockwise, then the device is a heat engine and W > 0. If the cycle is
counter
-
clockwise, then the device is a refrigerator/air conditioner or heat pump and W < 0.


Heat Engines

A heat engine has a chara
cteristic called efficiency, e

(also denoted η)
.


e = ΣW/ΣQ
H


The summation of work includes all processes, regardless of sign. The summation for heat
includes only heat input from the “hot reservoir” which in this case
includes only th
e
processes with positive heat
.


There is a theore
tical upper limit on the efficiency of an engine operating between two
temperature extremes. This is the Carnot efficiency.


e
carnot

= 1


T
c
/T
h


Heat Pumps, Refrigerators, and Air Conditioners

If a cycle has a total work less than zero, then the device
mi
ght be

a refrigerator/air
conditioner or a heat pump.
These devices have heat input from a lower temperature system
and have heat output to a higher temperature system.
The physical construction for these
two devices can be exactly the same, but the use an
d desired outcomes are different. With a
refrigerator
/
air condition
er
, the goal is to transfer heat from the colder system. With a heat
pump, the goal is to transfer heat to the hotter system. With any of these devices, the energy
input is in the form of w
ork
. This work is typically done by a compressor (you pay for the
energy to run this).


A refrigerator/air conditioner has a characteristic called the coefficient of performance,
C.O.P. or K.


K = ΣQ
c
/|ΣW|


The summation for heat includes only the heat input from the “cold reservoir”
which in this
case
includes only th
e
processes with positive heat
.

The summation of work includes all
processes, regardless of sign.


There is a theoretical upper l
imit on the coefficient of performance for a refrigerator/air
conditioner operating between two temperature extremes. This is the Carnot coefficient of
performance

and is based on the second law of thermodynamics
.


K
carnot

= T
c
/(T
h



T
c
)


A heat pump also
has a characteristic called the coefficient of performance, C.O.P. or K.


K = ΣQ
h
/ΣW


The summation for heat includes only the heat output to the “hot reservoir”
which in this
case
includes only th
e
processes with negative heat
.

The summation of work inclu
des all
processes, regardless of sign.


There is a theoretical upper limit on the coefficient of performance for a
heat pump
operating between two temperature extremes. This is the Carnot coefficient of performance

and is based on the second law of thermod
ynamics
.


K
carnot

= T
h
/(T
h



T
c
)

Entropy

Entropy, S, is often charact
erized as a measure of disorder, though this is a loose definition.
The second law of thermodynamics states that for an isolated system (no energy or matter
exchanged with external agents
), the total entropy of the system cannot decrease:


ΔS ≥ 0


There are two general methods for calculating entropy. The first is useful when there are
exchanges of energy in the form of heat. The second is useful when there is mixing of
particles:


ΔS = ∫(
dQ/T) ≈ Q/T

S = k*ln(W)


S = entropy

Q = heat

T = temperature

k = Boltzmann’s constant = 1.38E
-
23 J/K

W = number of
possible
microscopic states consistent with
the macroscopic state

Misuses of the Laws of Thermodynamics

Some non
-
scientists

and a few willfu
lly ignorant “scientists” claim
that the laws of
thermodynamics falsify both cosmic and biological evolution.

These
claims

are not
supported by
evidence
.


The
Big Bang
Does Not Violate the First Law

One
claim

unsupported by
evidence

is that the
Big Bang

co
uldn’t possibly be correct
because it violates the first law of thermodynamics. The claim is that there is obviously a lot
of energy now and there couldn’t be any energy before the
Big Bang
. The total energy
apparently
increased thus violating the first la
w of thermodynamics.


For this alleged violation to be true, we must know the total energy of the universe both
before and after the
Big Bang

and show that they are different (putting aside objections that
there
might be

no such thing as “before the
Big Ba
ng
”). No
on
e
has calculated the energy of
the universe
“before the
Big Bang
.”
But e
ven
if
one assumes that the energy is zero “before
the
Big Bang
”, a
calculation

of the total energ
y of the universe based on classical physics also
yields a total energy of
zero!

This was shown in

a paper written by E. Tryon in a 1973 artic
le
in the journal Nature (and not
refuted to date
).
How can that be so with all the stuff
moving
around (kinetic energy),
light energy, etc.? It turns out that the negative
gravitational
po
tential energy balances out
the positive energy
and the net sum

is zero.


Even if we assume that there is such a thing as “before the Big Bang” and that the t
otal
energy

of the universe

before the
Big Bang

is zero
, there is
no
proven
violation of the first

law of thermodynamics.


Evolution
Does Not Violate the Second Law

Another
claim

unsupported by
evidence

is that the second law of thermodynamics prohibits
the evolution of chemicals to
simple

organisms

(abiogenesis)

or of
simple

organisms to more
complex
organisms
,
a
n apparent decrease in entropy.


There are two
important

objections to the "life violates the second law"
claim
. First, there is
no
calculation

that shows that complex life forms are
always
lower entropy than less
complex forms or non
-
living th
ings.
Hand waving metaphors are no substitute for a
proper
calculation

of entropy
.
Second, life forms are not closed system
s
. They continuously
exchange energy and particles with the
ir

environment
s
, so the second law has nothing to say
ab
out them unless yo
u include their
environment
s

in the
calculation

of entropy.

Evolution
has not been proven to violate the second law of thermodynamics.


There are similar claims based on ill
-
defined and uncalculated “information” which should
not be confused with science.