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Table of thermodynamic equations

DefinitionsEdit

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

General basic quantitiesEdit

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 quantitiesEdit

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Thermodynamic beta, Inverse temperature β   J−1 [T]2[M]−1[L]−2
Thermodynamic temperature τ  

   

J [M] [L]2 [T]−2
Entropy S  

  ,  

J K−1 [M][L]2[T]−2 [Θ]−1
Pressure P  

 

Pa M L−1T−2
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 in a mixture)

μi  

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

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

Thermal properties of matterEdit

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 transferEdit

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Temperature gradient No standard symbol   K m−1 [Θ][L]−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P   W = J s−1 [M] [L]2 [T]−3
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

EquationsEdit

The equations in this article are classified by subject.

Thermodynamic processesEdit

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 theoryEdit

Ideal gas equations
Physical situation Nomenclature Equations
Ideal gas law
 

 

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

Ideal gasEdit

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          

EntropyEdit

  •  , 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 physicsEdit

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 processesEdit

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 potentialsEdit

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 relationsEdit

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 propertiesEdit

  •  
  •  
  •   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  

Thermal properties of matterEdit

Coefficients Equation
Joule-Thomson coefficient  
Compressibility (constant temperature)  
Coefficient of thermal expansion (constant pressure)  
Heat capacity (constant pressure)  
Heat capacity (constant volume)  

Thermal transferEdit

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 efficienciesEdit

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 alsoEdit

ReferencesEdit

  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. American Physical Society (APS). 3 (4): 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.

External linksEdit