Table of Contents
Thermodynamic States
Thermodynamic Processes
Heat Capacities
Thermodynamic Cycles
Calorimetry
Entropy
Misuses of the Laws of Thermodynamics
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 to use SI units exclusively as this
equation is often used in chemistry with a different set of units.
Note that this 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. Sometimes 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
ΔEth = Q + W
ΔEth = Q – Ws
The first law of thermodynamics is a statement of conservation of energy for the special case
of no changes in potential, kinetic, or chemical energy for the system. A system can
exchange energy with its environment with heat (Q), a microscopic transfer of energy, or
work (W or Ws), a macroscopic transfer of energy. This results in changes in its thermal
energy (Eth).
By convention, heat coming into the system is considered positive and heat going out of the
system is considered negative.
Unfortunately, the first law has two sign conventions for work. For the first equation listed,
a compression is considered positive work on the gas (W > 0) while an expansion is
considered negative work on the gas (W < 0). For the second equation an expansion is
considered positive work done by the gas (Ws > 0) while a compression is considered
negative work done by the gas (Ws < 0).
Ws = ∫PdV (± the area under the curve in the PV diagram) always. For an isochoric process,
Ws = 0. For an isobaric process, Ws = 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 equation for work. W s
= ∫(nRT/V)dV = nRT*ln(Vf/Vi).
The definitions of Cp and Cv can be used to obtain a formula for heat during isobaric and
isochoric processes. For an isobaric process, dQ = nCpdT. If Cp and n are constant (usually
the case, at least approximately), then Q = nCpΔT. Likewise, for an isochoric process, dQ =
nCvdT. If Cv and n are constant (usually the case, at least approximately), then Q = nCvΔT.
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, Ws = 0, so ΔEth= Q = nCvΔT. This formula for ΔEth can be
generalized to all processes since thermal energy is path independent.
Plausible Physical
Situations
Insulated
Add weight
sleeve or
to or push
rapid
down on
process
piston
Insulated
Remove
sleeve or
weight
rapid
from or
process
pull up on
piston
Heat gas
Locked
piston or
rigid
container
Cool gas
Locked
piston or
rigid
container
Heat gas
Piston free
to move,
load
unchanged
Cool gas
Piston free
to move,
load
unchanged
Immerse
Add weight
gas in large to piston
bath
Name
State
Variables
PVγ =
constant;
TVγ-1 =
constant
PVγ =
constant;
TVγ-1 =
constant
P
V
T
ΔEth
Q
Ws
W
Up
Down
Up
nCvΔT
>0
0
-nCvΔT < 0
nCvΔT > 0
Down
Up
Down
nCvΔT
<0
0
-nCvΔT > 0
nCvΔT < 0
Isochoric
V fixed; P α
T
Up
Fixed
Up
nCvΔT
>0
nCvΔT
>0
0
0
Isochoric
V fixed; P α
T
Down
Fixed
Down
nCvΔT
<0
nCvΔT
<0
0
0
Isobaric
expansion
P fixed; V α
T
Fixed
Up
Up
nCvΔT
>0
nCpΔT
>0
PΔV > 0
-PΔV < 0
Isobaric
compression
P fixed; V α
T
Fixed
Down
Down
nCvΔT
<0
nCpΔT
<0
PΔV < 0
-PΔV > 0
Isothermal
compression
T fixed at
temperature
of bath, PV
= constant
T fixed at
temperature
of bath, PV
= constant
PV/T =
constant
Up
Down
Fixed
nCvΔT
=0
nRT*ln(
Vf/Vi)
<0
nRT*ln(Vf/
Vi) < 0
-nRT
*ln(Vf/Vi)
>0
Immerse
gas in large
bath
Remove
weight
from piston
Isothermal
expansion
Down
Up
Fixed
nCvΔT
=0
nRT*ln(
Vf/Vi)
>0
nRT*ln(Vf/
Vi) > 0
-nRT*
ln(Vf/Vi) <
0
Unknown
Unknown
No Name
?
?
?
nCvΔT
ΔEth +
Ws
∫PdV = ±
area under
curve in PV
diagram
-∫PdV = ±
area under
curve in PV
diagram
Adiabatic
compression
Adiabatic
expansion
Tips: Know which formulas are specific to a particular process and which are true for any
process. See the next section for notes on Cp, Cv and γ.
Heat Capacities
Cv, the molar heat capacity at constant 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 Cv = 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, Cv = 5/2*R. For solids, there are three
degrees of freedom from translation and three from vibration, so Cv = 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.
Cp, the molar heat capacity at constant pressure, can be calculated for an ideal gas. For an
isobaric process where n and Cp are constant, Q = nCpΔT. The change in thermal energy can
be calculated with the general formula ΔEth = nCvΔT. Ws = P∫dV = PΔV by the definition
of work. Using the ideal gas law, PΔV = nRΔT. ΔEth = Q – Ws by the first law of
thermodynamics. Combine the above formulas to obtain an expression for Cp.
ΔEth = Q – Ws
nCvΔT = nCpΔT – nRΔT
Cp = Cv + R
The ratio of heat capacities is denoted by the letter gamma (γ), and is defined by the
following formula:
γ = Cp/Cv
For a monatomic ideal gas, γ = (3/2*R + R)/(3/2*R) = 5/3. For a diatomic ideal gas, γ =
(5/2*R + R)/(5/2*R) = 7/5.
Gas
Cv
Cp
γ
Examples
monatomic 3/2*R 5/2*R 5/3 He, Ne, Ar
diatomic
5/2*R 7/2*R 7/5 H2, N2, O2
Thermodynamic Cycles
A cycle must have ΣΔEth = 0. Therefore ΣQ = ΣWs by the first law. Normally, it is useful 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 ΣWs > 0 (ΣW < 0). If
the cycle is counter-clockwise, then the device is likely a refrigerator/air conditioner or heat
pump and ΣWs < 0 (ΣW > 0).
Heat Engines
A heat engine has a characteristic called efficiency, e (also denoted η).
e = ΣWs/ΣQH
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 the
processes with positive heat.
There is a theoretical upper limit on the efficiency of an engine operating between two
temperature extremes. This is the Carnot or maximum efficiency.
ecarnot = 1 – Tc/Th
Heat Pumps, Refrigerators, and Air Conditioners
If a cycle has a total work less than zero, then the device might 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 and desired outcomes are different. With a
refrigerator/air conditioner, 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 work. 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 = ΣQc/|ΣWs|
The summation for heat includes only the heat input from the “cold reservoir” which in this
case includes only the processes with positive heat. The summation of work includes all
processes, regardless of sign.
There is a theoretical upper limit 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.
Kcarnot = Tc/(Th – Tc)
A heat pump also has a characteristic called the coefficient of performance, C.O.P. or K.
K = ΣQh/ΣWs
The summation for heat includes only the heat output to the “hot reservoir” which in this
case includes only the processes with negative heat. The summation of work includes 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 thermodynamics.
Kcarnot = Th/(Th – Tc)
Calorimetry
Calorimetry is an application of conservation of energy. For an isolated system, the total
energy does not change. Individual members of the system with different temperatures may
exchange energy through the mechanism of heat. Any gain in energy by one is a loss by
another. So the starting equation for an isolated system should be the following:
ΣQ = 0
For each member of the system, use one or both of the following equations, depending on
the situation:
Q = mcΔT (temperature change in response to heat)
Q = ±mL (phase change in response to heat)
m = mass
c = specific heat capacity
ΔT = change in temperature
L = heat of fusion or heat of vaporization
± depends on direction of phase change
The first equation is quite possibly the most misused equation in physics. It is not true that
heat always leads to temperature change (see the second equation). It is also not true that the
only mechanism for changing temperature is heat (see adiabatic processes above).
Entropy
Entropy, S, is often characterized 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
A system that is not isolated can decrease its entropy. But any decrease in entropy for one
system must be accompanied by an equal or greater increase in entropy for another system:
ΣΔS ≥ 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
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 willfully 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 couldn’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 law 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 prove that they are different (putting aside objections that
there might be no such thing as “before the Big Bang”). No one has calculated the energy of
the universe “before the Big Bang.” But even if one assumes that the energy is zero “before
the Big Bang”, a calculation of the total energy of the universe based on classical physics
also yields a total energy of zero.i How can that be so with all the stuff moving around
(kinetic energy), light energy, etc.? It turns out that the negative gravitational potential 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 total
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, an 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 things. Hand waving metaphors are no substitute for a proper
calculation of entropy. Second, life forms are not isolated systems. They continuously
exchange energy and particles with their environments, so the second law has nothing to say
about them unless you include their environments in the calculation of entropy. Evolution
has not been proven to violate the second law of thermodynamics.
There are similar claims based on invented probabilities or ill-defined and uncalculated
“information” which should not be confused with science.
i
Tryon, Edward P., 1973. Is the universe a vacuum fluctuation? Nature 246: 396-397