Table of thermodynamic equations

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Table of thermodynamic equations

Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:

Definitions

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Perspective

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

General basic quantities

More information Quantity (common name/s), (Common) symbol/s ...
Quantity (common name/s) (Common) symbol/s SI unit Dimension
Number of molecules N 1 1
Amount of substance n mol N
Temperature T K Θ
Heat Energy Q, q J ML2T−2
Latent heat QL J ML2T−2
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General derived quantities

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Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension
Thermodynamic beta, inverse temperature β J−1 T2M−1L−2
Thermodynamic temperature τ

J ML2T−2
Entropy S

,

J⋅K−1 ML2T−2Θ−1
Pressure P

Pa ML−1T−2
Internal Energy U J ML2T−2
Enthalpy H J ML2T−2
Partition Function Z 1 1
Gibbs free energy G J ML2T−2
Chemical potential (of component i in a mixture) μi

, where is not proportional to because depends on pressure. , where is proportional to (as long as the molar ratio composition of the system remains the same) because depends only on temperature and pressure and composition.

J ML2T−2
Helmholtz free energy A, F J ML2T−2
Landau potential, Landau free energy, Grand potential Ω, ΦG J ML2T−2
Massieu potential, Helmholtz free entropy Φ J⋅K−1 ML2T−2Θ−1
Planck potential, Gibbs free entropy Ξ J⋅K−1 ML2T−2Θ−1
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Thermal properties of matter

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Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension
General heat/thermal capacity C J⋅K−1 ML2T−2Θ−1
Heat capacity (isobaric) Cp J⋅K−1 ML2T−2Θ−1
Specific heat capacity (isobaric) Cmp J⋅kg−1⋅K−1 L2T−2Θ−1
Molar specific heat capacity (isobaric) Cnp J⋅K−1⋅mol−1 ML2T−2Θ−1N−1
Heat capacity (isochoric/volumetric) CV J⋅K−1 ML2T−2Θ−1
Specific heat capacity (isochoric) CmV J⋅kg−1⋅K−1 L2T−2Θ−1
Molar specific heat capacity (isochoric) CnV J⋅K⋅−1 mol−1 ML2T−2Θ−1N−1
Specific latent heat L J⋅kg−1 L2T−2
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient γ 1 1
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Thermal transfer

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Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension
Temperature gradient No standard symbol K⋅m−1 ΘL−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P W ML2T−3
Thermal intensity I W⋅m−2 MT−3
Thermal/heat flux density (vector analogue of thermal intensity above) q W⋅m−2 MT−3
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Equations

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The equations in this article are classified by subject.

Thermodynamic processes

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Physical situation Equations
Isentropic process (adiabatic and reversible)

For an ideal gas


Isothermal process

For an ideal gas

Isobaric process p1 = p2, p = constant

Isochoric process V1 = V2, V = constant

Free expansion
Work done by an expanding gas Process

Net work done in cyclic processes

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Kinetic theory

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Ideal gas equations
Physical situation Nomenclature Equations
Ideal gas law

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

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Quantity General Equation Isobaric
Δp = 0
Isochoric
ΔV = 0
Isothermal
ΔT = 0
Adiabatic
Work
W

Heat Capacity
C
(as for real gas)
(for monatomic ideal gas)


(for diatomic ideal gas)


(for monatomic ideal gas)


(for diatomic ideal gas)

Internal Energy
ΔU






Enthalpy
ΔH
Entropy
Δs

[1]

Constant
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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 is valid for atoms or molecules constituting ideal gases.

More information Ratio of thermal to rest mass-energy of each molecule: ...
Physical situation Nomenclature 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:

K2 is the modified Bessel function of the second kind.

Non-relativistic speeds

Relativistic speeds (Maxwell–Jüttner distribution)

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

Internal energy

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

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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
  • = mean free path
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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

The following energies are called the thermodynamic potentials,

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Name Symbol Formula Natural variables
Internal energy
Helmholtz free energy
Enthalpy
Gibbs free energy
Landau potential, or
grand potential
,
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and the corresponding fundamental thermodynamic relations or "master equations"[2] are:

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Potential Differential
Internal energy
Enthalpy
Helmholtz free energy
Gibbs free energy
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Maxwell's relations

The four most common Maxwell's relations are:

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Physical situation Nomenclature Equations
Thermodynamic potentials as functions of their natural variables

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More relations include the following.

Other differential equations are:

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Name H U G
Gibbs–Helmholtz equation
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Quantum properties

  • Indistinguishable Particles

where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

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Degree of freedom Partition function
Translation
Vibration
Rotation
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Thermal properties of matter

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More information , ...
Coefficients Equation
Joule-Thomson coefficient
Compressibility (constant temperature)
Coefficient of thermal expansion (constant pressure)
Heat capacity (constant pressure)
Heat capacity (constant volume)
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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 = emissivity
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 = amount of substance
Effective thermal conductivities
  • λi = thermal conductivity of substance i
  • λnet = equivalent thermal conductivity
Series

Parallel

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Thermal efficiencies

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Physical situation Nomenclature 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:

Refrigeration K = coefficient of refrigeration performance Refrigeration performance

Carnot refrigeration performance

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See also

References

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