Table of thermodynamic equations 1

Table of thermodynamic equationsThis article is summary of common equations and quantities in thermodynamics (see thermodynamic equations formore elaboration). SI units are used for absolute temperature, not celsius or fahrenheit.

DefinitionsMany of the definitions below are also used in the thermodynamics of chemical reactions.

General basic quantities

Quantity (Common Name/s) (Common) Symbol/s SI Units Dimension

Number of molecules N 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/s

Defining Equation SI Units Dimension

Thermodynamic beta, Inversetemperature

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 Function Z dimensionless dimensionless

Gibbs free energy G J [M][L]2[T]−2

Chemical potential (of component i ina mixture)

μi (Ni, S, V must all beconstant)

J [M][L]2[T]−2

Helmholtz free energy A, F J [M][L]2[T]−2

Landau potential, Landau FreeEnergy

Ω J [M][L]2[T]−2

Grand potential ΦG J [M][L]2[T]−2

Massieu Potential, Helmholtz freeentropy

Φ J K−1 [M][L]2[T]−2

[Θ]−1

Table of thermodynamic equations 2

Planck potential, Gibbs free entropy Ξ J K−1 [M][L]2[T]−2

[Θ]−1

Thermal properties of matter

Quantity (common name/s) (Common)symbol/s

Defining equation SI units Dimension

General heat/thermal capacity C 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

Quantity (common name/s) (Common)symbol/s

Defining equation SI units Dimension

Temperature gradient No standardsymbol

K m−1 [Θ][L]−1

Thermal conduction rate, thermal current, thermal/heat flux, thermalpower transfer

P W = Js−1

[M] [L]2

[T]−2

Thermal intensity I W m−2 [M] [T]−3

Thermal/heat flux density (vector analogue of thermal intensity above) q W m−2 [M] [T]−3

Table of thermodynamic equations 3

EquationsThe equations in this article are classified by subject.

Phase transitions

Physical situation Equations

Adiabatic transition

Isothermal transition

For 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 situation Nomenclature Equations

Ideal gas law • p = pressure• V = volume of container• T = temperature• n = number of moles• N = number of molecules• k = Boltzmann’s constant

Pressure of an ideal gas • m = mass of one molecule• Mm = molar mass

Ideal gas

Table of thermodynamic equations 4

Quantity General Equation IsobaricΔp = 0

IsochoricΔV = 0

IsothermalΔT = 0

Adiabatic

WorkW

HeatCapacity

C

(as for real gas)

(for monatomicideal gas)

(for monatomicideal gas)

InternalEnergy

ΔU

EnthalpyΔH

EntropyΔS

[1]

Constant

Entropy

• , where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phasespace or otherwise called thermodynamic probability.

• , for reversible processes only

Statistical physicsBelow are useful results from the Maxwell-Boltzmann distribution for an ideal gas, and the implications of theEntropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.

Physical situation Nomenclature Equations

Maxwell–Boltzmanndistribution

• 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:

K2 is the Modified Bessel function of the secondkind.

Non-relativistic speeds

Relativistic speeds (Maxwell-Juttner distribution)

Entropy Logarithm of thedensity of states

• Pi = probability of system in microstate i•• Ω = total number of microstates

where:

Entropy change

Entropic force

Table of thermodynamic equations 5

Equipartition theorem • df = degree of freedom Average kinetic energy per degree of freedom

Internal energy

Corollaries of the non-relativistic Maxwell-Boltzmann distribution are below.

Physical situation Nomenclature 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 processesFor 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 potentialsThe 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:

Table of thermodynamic equations 6

Potential Differential

Internal energy

Enthalpy

Helmholtz free energy

Gibbs free energy

Maxwell's relationsThe four most common Maxwell's relations are:

Physical situation Nomenclature Equations

Thermodynamic potentials as functions of their naturalvariables

• = 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

Table of thermodynamic equations 7

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, invarious forms:

Degree of freedom Partition function

Translation

Vibration

Rotation

•• where:• σ = 1 (heteronuclear molecules)• σ = 2 (homonuclear)

Thermal properties of matter

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

Table of thermodynamic equations 8

Derivation of heat capacity (constant volume)

Since

(where δWrev is the work done by the system),

Thermal transfer

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 situation Nomenclature Equations

Thermodynamicengines

• η = efficiency• W = work done by engine• QH = heat energy in higher

temperature reservoir• QC = heat energy in lower

temperature reservoir• TH = temperature of higher temp.

reservoir• TC = temperature of lower temp.

reservoir

Thermodynamic engine:

Carnot engine efficiency:

Refrigeration • K = coefficient of refrigerationperformance Refrigeration performance

Carnot refrigeration performance

Table of thermodynamic equations 9

References[1] Keenan, Thermodynamics, Wiley, New York, 1947[2][2] Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 7

• Atkins, Peter and de Paula, Julio Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 [ISBN0-7167-3539-3].• 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.

Article Sources and Contributors 10

Article Sources and ContributorsTable of thermodynamic equations  Source: http://en.wikipedia.org/w/index.php?oldid=519398450  Contributors: Apple17, Boardhead, Ckk253, Dapeders, Dhollm, EncMstr, Hesacon, Ianalis,Jamoche, Jdpipe, KRMorison, Kbrose, Klaas van Aarsen, Koumz, Maschen, MaxEspinho, Meithan, Memming, Onegumas, PAR, PhilKnight, Pmetzger, Power.corrupts, Protonk, Roastytoast,Sadi Carnot, Saehry, Shorelander, Sumanch, Tetracube, Tommy2010, Utimatu, Vishktl, Werson, Youandme, Z = z² + c, 79 ,قلی زادگان anonymous edits

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