Table of Thermodynamic Equations - Wikipedia, The Free Encyclopedia

Thermodynamics

The classical Carnot heat engine

Branches

Classical

Statistical

Chemical

Equilibrium / Non-equilibrium

Laws

Zeroth

First

Second

Third

Systems

State

Equation of state

Ideal gas

Real gas

State of matter

Equilibrium

Control volume

Instruments

Processes

Isobaric

Isochoric

Isothermal

Adiabatic

Isentropic

Isenthalpic

Quasistatic

Polytropic

Free expansion

Reversibility

Irreversibility

Endoreversibility

Cycles

Heat engines

Heat pumps

Thermal efficiency

System properties

Note: Conjugate variables in italics

Table of thermodynamic equations

From Wikipedia, the free encyclopedia

For list of mathematical notation used in these equations, see mathematical notation.

Main article: List of thermodynamic properties

This article is summary of common equations and quantities in thermodynamics (seethermodynamic equations for more elaboration). SI units are used for absolutetemperature, not celsius or fahrenheit.

Contents

1 Definitions

1.1 General basic quantities

1.2 General derived quantities

1.3 Thermal properties of matter

1.4 Thermal transfer

2 Equations

2.1 Phase transitions

2.2 Kinetic theory

2.2.1 Ideal gas

2.3 Entropy

2.4 Statistical physics

2.5 Quasi-static and reversible processes

2.6 Thermodynamic potentials

2.7 Maxwell's relations

2.8 Quantum properties

3 Thermal properties of matter

3.1 Thermal transfer

3.2 Thermal efficiencies

4 See also

5 References

Definitions

Main articles: List of thermodynamic properties, Thermodynamic potential, Free entropy,

Defining equation (physical chemistry)

Many of the definitions below are also used in the thermodynamics of chemicalreactions.

General basic quantities

Table of thermodynamic equations - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Table_of_thermodynamic_equations

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Property diagrams

Intensive and extensive properties

Functions of state

Temperature / Entropy (introduction)

Pressure / Volume

Chemical potential / Particle number

Vapor quality

Reduced properties

Process functions

Work

Heat

Material properties

Property databases

Specific heat capacity

Compressibility

Thermal expansion

Equations

Carnot's theorem

Clausius theorem

Fundamental relation

Ideal gas law

Maxwell relations

Onsager reciprocal relations

Bridgman's equations

Table of thermodynamic equations

Potentials

Free energy

Free entropy

Internal energy

Enthalpy

Helmholtz free energy

Gibbs free energy

History

Culture

History

General

Heat

Quantity (Common

Name/s)

(Common)

Symbol/sSI Units Dimension

Number of molecules y' ' 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/sDefining Equation SI Units Dimension

Thermodynamic

beta, Inverse

temperature

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

FunctionZ dimensionless dimensionless

Gibbs free

energyG J [M][L]2[T]−2

Chemical

potential (of

component i ina mixture)

μi (Ni, S, V must all be

constant)

J [M][L]2[T]−2

Helmholtz free

energyA, 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


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Entropy

Gas laws

"Perpetual motion" machines

Philosophy

Entropy and time

Entropy and life

Brownian ratchet

Maxwell's demon

Heat death paradox

Loschmidt's paradox

Synergetics

Theories

Caloric theory

Theory of heat

Vis viva ("living force")

Mechanical equivalent of heat

Motive power

Key publications

"An Experimental EnquiryConcerning ... Heat"

"On the Equilibrium ofHeterogeneous Substances"

"Reflections on theMotive Power of Fire"

Timelines

Thermodynamics

Heat engines

Art

Education

Maxwell's thermodynamic surface

Entropy as energy dispersal

Scientists

Bernoulli

Carnot

Clapeyron

Clausius

Carathéodory

Duhem

Gibbs

von Helmholtz

Joule

Maxwell

von Mayer

Onsager

Rankine

Smeaton

Stahl

Thompson

Thomson

Waterston

Book:Thermodynamics

Thermal properties of matter

Main Articles: Heat capacity, Thermal expansion

Quantity (common

name/s)

(Common)

symbol/sDefining equation SI units Dimension

General heat/thermal

capacityC J K −1

[M][L]2[T]−2

[Θ]−1

Heat capacity

(isobaric)Cp J K −1

[M][L]2[T]−2

[Θ]−1


(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


(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

Main article: Thermal conductivity


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action=edit)

Quantity

(common

name/s)

(Common)

symbol/sDefining equation SI units Dimension

Temperature

gradient

No standard

symbolK m−1 [Θ][L]−1

Thermal

conduction

rate, thermal

current,

thermal/heat

flux, thermal

power

transfer

PW = J

s−1

[M] [L]2

[T]−3

Thermal

intensityI 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.

Phase transitions


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Physical

situationEquations

Adiabatic

transition

Isothermal

transitionFor 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

situationNomenclature Equations

Ideal gas law

p = pressure

V = volume of container

T = temperature

n = number of moles

R = Gas constant

N = number of molecules

k = Boltzmann's constant

Pressure of

an ideal gas

m = mass of one molecule

Mm = molar mass

Ideal gas


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Quantity General EquationIsobaric

Δp = 0

Isochoric

ΔV = 0

Isothermal

ΔT = 0

Adiabatic

Work

W

Heat

Capacity

C

(as for real gas)

(for monatomic ideal

gas);

(for diaatomic ideal gas)

(for monatomic

ideal gas);

(for diatomicideal gas)

Internal

Energy

ΔU

Enthalpy

ΔH

Entropy

ΔS

[1]

Constant

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 isvalid for atoms or molecules constituting ideal gases.


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Physical


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:

</div">

K2 is the Modified Bessel function

of the second kind.

Non-relativistic speeds

Relativistic speeds (Maxwell-Jüttnerdistribution)

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

theoremdf = degree of freedom

Average kinetic energy per degree of

freedom

Internal energy

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


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Physical


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 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

Main article: Thermodynamic potentials

See also: Maxwell relations

The following energies are called the thermodynamic potentials,

Name Symbol Formula Natural variables

Internal 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


Gibbs free energy

Maxwell's relations

The four most common Maxwell's relations are:


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Physical


Thermodynamic

potentials as

functions of

their natural

variables

= 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

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 where:

σ = 1 (heteronuclear molecules)

σ = 2 (homonuclear)

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)

Derivation of heat capacity (constant pressure)

Since

Derivation of heat capacity (constant volume)

Since

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

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 = 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


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:

RefrigerationK = coefficient of

refrigeration performance

Refrigeration performance

Carnot refrigeration performance

See also

Antoine equation Gibbs–Helmholtz equation


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Bejan number

Bowen ratio

Bridgman's equations

Clausius–Clapeyron relation

Departure functions

Duhem–Margules equation

Ehrenfest equations

Gibbs' phase rule

Kopp's law

Kopp–Neumann law

Noro–Frenkel law of corresponding states

Onsager reciprocal relations

Stefan number

Triple product rule

References

^ Keenan, Thermodynamics, Wiley, New York, 19471.

^ Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0 19 855148 72.

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., 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.

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