Thermodynamics
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The classical Carnot heat engine |
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Common thermodynamic equations and quantities in thermodynamics, using mathematical notation, are as follows:
Definitions
Many 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, 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 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, Laplace coefficient | γ | | dimensionless | dimensionless |
Thermal transfer
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 |
Equations
The equations in this article are classified by subject.
Thermodynamic processes
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
Ideal gas
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.
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
- ℓ = 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,
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
The four most common Maxwell's relations are:
Physical situation | Nomenclature | Equations |
Thermodynamic potentials as functions of their natural variables | - = Internal energy
- = Enthalpy
- = Helmholtz free energy
- = Gibbs free energy
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More relations include the following.
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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 | |
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) | |
Thermal transfer
Thermal efficiencies
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
References
- ^ Keenan, Thermodynamics, Wiley, New York, 1947
- ^ 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.
External links
- Thermodynamic equation calculator