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Table of Thermodynamic Equations
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Thermodynamics
The classical Carnot heat engine
Branches
Classical
Statistical
Chemical
Equilibrium / Non-equilibrium
Laws
Zeroth
First
Second
Third
Systems
State
Equation of state
Ideal gas
Real gas
State of matter
Equilibrium
Control volume
Instruments
Processes
Isobaric
Isochoric
Isothermal
Adiabatic
Isentropic
Isenthalpic
Quasistatic
Polytropic
Free expansion
Reversibility
Irreversibility
Endoreversibility
Cycles
Heat engines
Heat pumps
Thermal efficiency
System properties
Note: Conjugate variables in italics
Table of thermodynamic equations
From Wikipedia, the free encyclopedia
For list of mathematical notation used in these equations, see mathematical notation.
Main article: List of thermodynamic properties
This article is summary of common equations and quantities in thermodynamics (seethermodynamic equations for more elaboration). SI units are used for absolutetemperature, not celsius or fahrenheit.
Contents
1 Definitions
1.1 General basic quantities
1.2 General derived quantities
1.3 Thermal properties of matter
1.4 Thermal transfer
2 Equations
2.1 Phase transitions
2.2 Kinetic theory
2.2.1 Ideal gas
2.3 Entropy
2.4 Statistical physics
2.5 Quasi-static and reversible processes
2.6 Thermodynamic potentials
2.7 Maxwell's relations
2.8 Quantum properties
3 Thermal properties of matter
3.1 Thermal transfer
3.2 Thermal efficiencies
4 See also
5 References
Definitions
Main articles: List of thermodynamic properties, Thermodynamic potential, Free entropy,
Defining equation (physical chemistry)
Many of the definitions below are also used in the thermodynamics of chemicalreactions.
General basic quantities
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Property diagrams
Intensive and extensive properties
Functions of state
Temperature / Entropy (introduction)
Pressure / Volume
Chemical potential / Particle number
Vapor quality
Reduced properties
Process functions
Work
Heat
Material properties
Property databases
Specific heat capacity
Compressibility
Thermal expansion
Equations
Carnot's theorem
Clausius theorem
Fundamental relation
Ideal gas law
Maxwell relations
Onsager reciprocal relations
Bridgman's equations
Table of thermodynamic equations
Potentials
Free energy
Free entropy
Internal energy
Enthalpy
Helmholtz free energy
Gibbs free energy
History
Culture
History
General
Heat
Quantity (Common
Name/s)
(Common)
Symbol/sSI Units Dimension
Number of molecules y' ' dimensionless dimensionless
Number of moles n mol [N]
Temperature T K [Θ]
Heat Energy Q, q J [M][L]2[T]−2
Latent Heat QL J [M][L]2[T]−2
General derived quantities
Quantity
(Common
Name/s)
(Common)
Symbol/sDefining Equation SI Units Dimension
Thermodynamic
beta, Inverse
temperature
β J−1 [T]2[M]−1[L]−2
Entropy S J K−1[M][L]2[T]−2
[Θ]−1
Negentropy J J K−1[M][L]2[T]−2
[Θ]−1
Internal Energy U J [M][L]2[T]−2
Enthalpy H J [M][L]2[T]−2
Partition
FunctionZ dimensionless dimensionless
Gibbs free
energyG J [M][L]2[T]−2
Chemical
potential (of
component i ina mixture)
μi (Ni, S, V must all be
constant)
J [M][L]2[T]−2
Helmholtz free
energyA, F J [M][L]2[T]−2
Landau
potential,
Landau Free
Energy, Grand
potential
Ω, ΦG J [M][L]2[T]−2
Massieu
Potential,
Helmholtz free
entropy
Φ J K−1[M][L]2[T]−2
[Θ]−1
Planck
potential, Gibbs
free entropy
Ξ J K−1[M][L]2[T]−2
[Θ]−1
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Entropy
Gas laws
"Perpetual motion" machines
Philosophy
Entropy and time
Entropy and life
Brownian ratchet
Maxwell's demon
Heat death paradox
Loschmidt's paradox
Synergetics
Theories
Caloric theory
Theory of heat
Vis viva ("living force")
Mechanical equivalent of heat
Motive power
Key publications
"An Experimental EnquiryConcerning ... Heat"
"On the Equilibrium ofHeterogeneous Substances"
"Reflections on theMotive Power of Fire"
Timelines
Thermodynamics
Heat engines
Art
Education
Maxwell's thermodynamic surface
Entropy as energy dispersal
Scientists
Bernoulli
Carnot
Clapeyron
Clausius
Carathéodory
Duhem
Gibbs
von Helmholtz
Joule
Maxwell
von Mayer
Onsager
Rankine
Smeaton
Stahl
Thompson
Thomson
Waterston
Book:Thermodynamics
Thermal properties of matter
Main Articles: Heat capacity, Thermal expansion
Quantity (common
name/s)
(Common)
symbol/sDefining equation SI units Dimension
General heat/thermal
capacityC J K −1
[M][L]2[T]−2
[Θ]−1
Heat capacity
(isobaric)Cp J K −1
[M][L]2[T]−2
[Θ]−1
Specific heat capacity
(isobaric)Cmp J kg−1 K−1
[L]2[T]−2
[Θ]−1
Molar specific heat
capacity (isobaric)Cnp J K −1 mol−1
[M][L]2[T]−2
[Θ]−1 [N]−1
Heat capacity
(isochoric/volumetric)CV J K −1
[M][L]2[T]−2
[Θ]−1
Specific heat capacity
(isochoric)CmV J kg−1 K−1
[L]2[T]−2
[Θ]−1
Molar specific heat
capacity (isochoric)CnV J K −1 mol−1
[M][L]2[T]−2
[Θ]−1 [N]−1
Specific latent heat L J kg−1 [L]2[T]−2
Ratio of isobaric to
isochoric heat
capacity, heat
capacity ratio,
adiabatic index
γ dimensionless dimensionless
Thermal transfer
Main article: Thermal conductivity
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e (https://en.wikipedia.org/w/index.php?title=Template:Thermodynamics&
action=edit)
Quantity
(common
name/s)
(Common)
symbol/sDefining equation SI units Dimension
Temperature
gradient
No standard
symbolK m−1 [Θ][L]−1
Thermal
conduction
rate, thermal
current,
thermal/heat
flux, thermal
power
transfer
PW = J
s−1
[M] [L]2
[T]−3
Thermal
intensityI W m−2 [M] [T]−3
Thermal/heat
flux density
(vector
analogue of
thermal
intensity
above)
q W m−2 [M] [T]−3
Equations
The equations in this article are classified by subject.
Phase transitions
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Physical
situationEquations
Adiabatic
transition
Isothermal
transitionFor an ideal gas
Isobaric
transition
p1 = p2, p = constant
Isochoric
transition
V1 = V2, V = constant
Adiabatic
expansion
Free
expansion
Work done
by an
expanding
gas
Process
Net Work Done in Cyclic Processes
Kinetic theory
Ideal gas equations
Physical
situationNomenclature Equations
Ideal gas law
p = pressure
V = volume of container
T = temperature
n = number of moles
R = Gas constant
N = number of molecules
k = Boltzmann's constant
Pressure of
an ideal gas
m = mass of one molecule
Mm = molar mass
Ideal gas
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Quantity General EquationIsobaric
Δp = 0
Isochoric
ΔV = 0
Isothermal
ΔT = 0
Adiabatic
Work
W
Heat
Capacity
C
(as for real gas)
(for monatomic ideal
gas);
(for diaatomic ideal gas)
(for monatomic
ideal gas);
(for diatomicideal gas)
Internal
Energy
ΔU
Enthalpy
ΔH
Entropy
ΔS
[1]
Constant
Entropy
, where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called
thermodynamic probability.
, for reversible processes only
Statistical physics
Below are useful results from the Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution isvalid for atoms or molecules constituting ideal gases.
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Physical
situationNomenclature Equations
Maxwell–
Boltzmann
distribution
v = velocity of
atom/molecule,
m = mass of each molecule
(all molecules are identical in
kinetic theory),
γ(p) = Lorentz factor as
function of momentum (see
below)
Ratio of thermal to rest
mass-energy of each
molecule:
</div">
K2 is the Modified Bessel function
of the second kind.
Non-relativistic speeds
Relativistic speeds (Maxwell-Jüttnerdistribution)
Entropy
Logarithm of
the density
of states
Pi = probability of system in
microstate i
Ω = total number of
microstates
where:
Entropy
change
Entropic
force
Equipartition
theoremdf = degree of freedom
Average kinetic energy per degree of
freedom
Internal energy
Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.
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Physical
situationNomenclature Equations
Mean speed
Root mean
square speed
Modal speed
Mean free
path
σ = Effective cross-section
n = Volume density of
number of target particles
ℓ = Mean free path
Quasi-static and reversible processes
For quasi-static and reversible processes, the first law of thermodynamics is:
where δQ is the heat supplied to the system and δW is the work done by the system.
Thermodynamic potentials
Main article: Thermodynamic potentials
See also: Maxwell relations
The following energies are called the thermodynamic potentials,
Name Symbol Formula Natural variables
Internal energy
Helmholtz free energy
Enthalpy
Gibbs free energy
Landau Potential (Grand potential) ,
and the corresponding fundamental thermodynamic relations or "master equations"[2] are:
Potential Differential
Internal energy
Enthalpy
Helmholtz free energy
Gibbs free energy
Maxwell's relations
The four most common Maxwell's relations are:
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Physical
situationNomenclature Equations
Thermodynamic
potentials as
functions of
their natural
variables
= Internal energy
= Enthalpy
= Helmholtz free
energy
= Gibbs free
energy
More relations include the following.
Other differential equations are:
Name H U G
Gibbs–Helmholtz equation
Quantum properties
Indistinguishable Particles
where N is number of particles, h is Planck's constant, I is moment of inertia, and Z is the partition function, in various forms:
Degree of freedom Partition function
Translation
Vibration
Rotation where:
σ = 1 (heteronuclear molecules)
σ = 2 (homonuclear)
Thermal properties of matter
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Coefficients Equation
Joule-Thomson coefficient
Compressibility (constant temperature)
Coefficient of thermal expansion (constant pressure)
Heat capacity (constant pressure)
Heat capacity (constant volume)
Derivation of heat capacity (constant pressure)
Since
Derivation of heat capacity (constant volume)
Since
(where δWrev is the work done by the system),
Thermal transfer
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Physical situation Nomenclature Equations
Net intensity
emission/absorption
Texternal = external
temperature (outside of
system)
Tsystem = internal
temperature (inside system)
ε = emmisivity
Internal energy of a
substance
CV = isovolumetric heat
capacity of substance
ΔT = temperature change of
substance
Meyer's equation
Cp = isobaric heat capacity
CV = isovolumetric heat
capacity
n = number of moles
Effective thermal
conductivities
λi = thermal conductivity of
substance i
λnet = equivalent thermal
conductivity
Series
Parallel
Thermal efficiencies
Physical
situationNomenclature Equations
Thermodynamic
engines
η = efficiency
W = work done by engine
QH = heat energy in higher
temperature reservoir
QL = heat energy in lower
temperature reservoir
TH = temperature of higher
temp. reservoir
TL = temperature of lower
temp. reservoir
Thermodynamic engine:
Carnot engine efficiency:
RefrigerationK = coefficient of
refrigeration performance
Refrigeration performance
Carnot refrigeration performance
See also
Antoine equation Gibbs–Helmholtz equation
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Bejan number
Bowen ratio
Bridgman's equations
Clausius–Clapeyron relation
Departure functions
Duhem–Margules equation
Ehrenfest equations
Gibbs' phase rule
Kopp's law
Kopp–Neumann law
Noro–Frenkel law of corresponding states
Onsager reciprocal relations
Stefan number
Triple product rule
References
^ Keenan, Thermodynamics, Wiley, New York, 19471.
^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 72.
Atkins, Peter and de Paula, Julio Physical Chemistry, 7th edition,
W.H. Freeman and Company, 2002 [ISBN 0-7167-3539-3].
Chapters 1 - 10, Part 1: Equilibrium.
Bridgman, P.W., Phys. Rev., 3, 273 (1914).
Landsberg, Peter T. Thermodynamics and Statistical Mechanics.
New York: Dover Publications, Inc., 1990. (reprinted from Oxford
University Press, 1978).
Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition,
McGraw-Hill Book Company, New York, 1961.
Reichl, L.E., "A Modern Course in Statistical Physics", 2nd edition,
New York: John Wiley & Sons, 1998.
Schroeder, Daniel V. Thermal Physics. San Francisco: Addison
Wesley Longman, 2000 [ISBN 0-201-38027-7].
Silbey, Robert J., et al. Physical Chemistry. 4th ed. New Jersey:
Wiley, 2004.
Callen, Herbert B. (1985). "Thermodynamics and an Introduction
to Themostatistics", 2nd Ed., New York: John Wiley & Sons.
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