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.
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