Table of thermodynamic equations

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

Definitions

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

General basic quantities

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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

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

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

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

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
 

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
 

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          

Entropy

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  •  , 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

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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.

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  

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
  • = mean free path
 

Quasi-static and reversible processes

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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

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The following energies are called the thermodynamic potentials,

Name Symbol Formula Natural variables
Internal energy      
Helmholtz free energy      
Enthalpy      
Gibbs free energy      
Landau potential, or
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

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The four most common Maxwell's relations are:

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

 

 

 

More relations include the following.

     
   
 

Other differential equations are:

Name H U G
Gibbs–Helmholtz equation      
   

Quantum properties

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  •  
  •   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:

Degree of freedom Partition function
Translation  
Vibration  
Rotation  

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)  

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  

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  

See also

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References

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  1. ^ Keenan, Thermodynamics, Wiley, New York, 1947
  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 ISBN 0-7167-3539-3.
    • Chapters 1–10, Part 1: "Equilibrium".
  • Bridgman, P. W. (1 March 1914). "A Complete Collection of Thermodynamic Formulas". Physical Review. 3 (4). American Physical Society (APS): 273–281. doi:10.1103/physrev.3.273. ISSN 0031-899X.
  • 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 edition, New York: John Wiley & Sons.
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