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Table of thermodynamic equations

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Table of thermodynamic equations

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

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 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/s Defining Equation SI Units Dimension
Thermodynamic beta, Inverse temperature β \beta = 1/k_B T \,\! J−1 [T]2[M]−1[L]−2
Entropy S S = -k_B\sum_i p_i\ln p_i 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 U = \sum_i E_i \! J [M][L]2[T]−2
Enthalpy H H = U+pV\,\! J [M][L]2[T]−2
Partition Function Z dimensionless dimensionless
Gibbs free energy G G = H - TS \,\! J [M][L]2[T]−2
Chemical potential (of component i in a mixture) μi \mu_i = \left (\partial U/\partial N_i \right )_{N_{i \neq j}, S, V } \,\!

(Ni, S, V must all be constant)

J [M][L]2[T]−2
Helmholtz free energy A, F F = U - TS \,\! J [M][L]2[T]−2
Landau potential, Landau Free Energy, Grand potential Ω, ΦG \Omega = U - TS - \mu N\,\! J [M][L]2[T]−2
Massieu Potential, Helmholtz free entropy Φ \Phi = S - U/T \,\! J K−1 [M][L]2[T]−2 [Θ]−1
Planck potential, Gibbs free entropy Ξ \Xi = \Phi - pV/T \,\! 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 C = \partial Q/\partial T\,\! J K −1 [M][L]2[T]−2 [Θ]−1
Heat capacity (isobaric) Cp C_{p} = \partial H/\partial T\,\! J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isobaric) Cmp C_{mp} = \partial^2 Q/\partial m \partial T \,\! J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat capacity (isobaric) Cnp C_{np} = \partial^2 Q/\partial n \partial T \,\! J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Heat capacity (isochoric/volumetric) CV C_{V} = \partial Q/\partial T \,\! J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isochoric) CmV C_{mV} = \partial^2 Q/\partial m \partial T \,\! J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat capacity (isochoric) CnV C_{nV} = \partial^2 Q/\partial n \partial T \,\! J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Specific latent heat L L = \partial Q/ \partial m \,\! J kg−1 [L]2[T]−2
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index γ \gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} \,\! dimensionless dimensionless

Thermal transfer

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Temperature gradient No standard symbol \nabla T \,\! K m−1 [Θ][L]−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P P = \mathrm{d} Q/\mathrm{d} t \,\! W = J s−1 [M] [L]2 [T]−3
Thermal intensity I I = \mathrm{d} P/\mathrm{d} A W m−2 [M] [T]−3
Thermal/heat flux density (vector analogue of thermal intensity above) q Q = \iint \mathbf{q} \cdot \mathrm{d}\mathbf{S}\mathrm{d} t \,\! W m−2 [M] [T]−3

Equations

Phase transitions

Physical situation Equations
Adiabatic transition \Delta H = 0, \quad \Delta U = -\Delta W\,\!
Isothermal transition \Delta U = 0, \quad \Delta W = \Delta H \,\!

For an ideal gas
W=kTN \ln(V_2/V_1)\,\!

Isobaric transition p1 = p2, p = constant

\Delta W = p \Delta V, \quad \Delta Q = \Delta U + p \delta V\,\!

Isochoric transition V1 = V2, V = constant

\Delta W = 0, \quad \Delta Q = \Delta U\,\!

Adiabatic expansion p_1 V_1^{\gamma} = p_2 V_2^{\gamma}\,\!

T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \,\!

Free expansion \Delta U = 0\,\!
Work done by an expanding gas Process

\Delta W = \int_{V_1}^{V_2} p \mathrm{d}V \,\!

Net Work Done in Cyclic Processes
\Delta W = \oint_\mathrm{cycle} p \mathrm{d}V \,\!

Kinetic theory

Ideal gas equations
Physical situation Nomenclature 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
pV = nRT = kTN\,\!

\frac{p_1 V_1}{p_2 V_2} = \frac{n_1 T_1}{n_2 T_2} = \frac{N_1 T_1}{N_2 T_2} \,\!

Pressure of an ideal gas
• m = mass of one molecule
• Mm = molar mass
p = \frac{Nm \langle v^2 \rangle}{3V} = \frac{nM_m \langle v^2 \rangle}{3V} = \frac{1}{3}\rho \langle v^2 \rangle \,\!

Ideal gas

Quantity General Equation Isobaric
Δp = 0
Isochoric
ΔV = 0
Isothermal
ΔT = 0
Q=0
Work
W
\delta W = p dV\; p\Delta V\; 0\; nRT\ln\frac{V_2}{V_1}\; \frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma} = C_V \left(T_1 - T_2 \right)
Heat Capacity
C
(as for real gas) C_p = \frac{5}{2}nR\;
(for monatomic ideal gas)
C_V = \frac{3}{2}nR \;
(for monatomic ideal gas)
Internal Energy
ΔU
\Delta U = C_v \Delta T\; Q + W\;

Q_p - p\Delta V\;
Q\;

C_V\left ( T_2-T_1 \right )\;
0\;

Q=-W\;
W\;

C_V\left ( T_2-T_1 \right )\;
Enthalpy
ΔH
H=U+pV\; C_p\left ( T_2-T_1 \right )\; Q_V+V\Delta p\; 0\; C_p\left ( T_2-T_1 \right )\;
Entropy
ΔS
\Delta S = C_v \ln{T_2 \over T_1} + nR \ln{V_2 \over V_1}
\Delta S = C_p \ln{T_2 \over T_1} - nR \ln{p_2 \over p_1}[1]
C_p\ln\frac{T_2}{T_1}\; C_V\ln\frac{T_2}{T_1}\; nR\ln\frac{V_2}{V_1}\;
\frac{Q}{T}\;
C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;
Constant \; \frac{V}{T}\; \frac{p}{T}\; p V\; p V^\gamma\;

Entropy

• dS = \frac{\delta Q}{T} , 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:\theta = k_B T/mc^2 \,\!

K2 is the Modified Bessel function of the second kind.

Non-relativistic speeds

P\left ( v \right )=4\pi\left ( \frac{m}{2\pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2 k_B T} \,\!

Relativistic speeds (Maxwell-Jüttner distribution)
f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} e^{-\gamma(p)/\theta}

Entropy Logarithm of the density of states
• Pi = probability of system in microstate i
• Ω = total number of microstates
S = - k_B\sum_i P_i \ln P_i = k_\mathrm{B}\ln \Omega\,\!

where:
P_i = 1/\Omega\,\!

Entropy change \Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T} \,\!

\Delta S = k_B N \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1} \,\!

Entropic force \mathbf{F}_\mathrm{S} = -T \nabla S \,\!
Equipartition theorem
• df = degree of freedom
Average kinetic energy per degree of freedom

\langle E_\mathrm{k} \rangle = \frac{1}{2}kT\,\!

Internal energy U = d_f \langle E_\mathrm{k} \rangle = \frac{d_f}{2}kT\,\!

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

Physical situation Nomenclature Equations
Mean speed \langle v \rangle = \sqrt{\frac{8 k_B T}{\pi m}}\,\!
Root mean square speed v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3k_B T}{m}} \,\!
Modal speed v_\mathrm{mode} = \sqrt{\frac{2k_B T}{m}}\,\!
Mean free path
• σ = Effective cross-section
• n = Volume density of number of target particles
• = Mean free path
\ell = 1/\sqrt{2} n \sigma \,\!

Quasi-static and reversible processes

For quasi-static and reversible processes, the first law of thermodynamics is:

dU=\delta Q - \delta W

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 U \int ( T dS - p dV + \sum_i \mu_i dN_i ) S, V, \{N_i\}
Helmholtz free energy F U-TS T, V, \{N_i\}
Enthalpy H U+pV S, p, \{N_i\}
Gibbs free energy G U+pV-TS T, p, \{N_i\}
Landau Potential (Grand potential) \Omega, \Phi_{G} U - T S -\sum_i\,\mu_i N_i T, V, \{\mu_i\}

and the corresponding fundamental thermodynamic relations or "master equations"[2] are:

Potential Differential
Internal energy dU\left(S,V,{n_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i
Enthalpy dH\left(S,p,n_{i}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}
Helmholtz free energy dF\left(T,V,n_{i}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}
Gibbs free energy dG\left(T,p,n_{i}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}

Maxwell's relations

The four most common Maxwell's relations are:

Physical situation Nomenclature Equations
Thermodynamic potentials as functions of their natural variables
\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V}

\left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H }{\partial S \partial P}

+\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V = - \frac{\partial^2 F }{\partial T \partial V}

-\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P = \frac{\partial^2 G }{\partial T \partial P}

More relations include the following.

 \left ( {\partial S\over \partial U} \right )_{V,N} = { 1\over T } \left ( {\partial S\over \partial V} \right )_{N,U} = { p\over T } \left ( {\partial S\over \partial N} \right )_{V,U} = - { \mu \over T } \left ( {\partial T\over \partial S} \right )_V = { T \over C_V } \left ( {\partial T\over \partial S} \right )_P = { T \over C_P } -\left ( {\partial p\over \partial V} \right )_T = { 1 \over {VK_T} }

Other differential equations are:

Name H U G
Gibbs–Helmholtz equation H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p U = -T^2\left(\frac{\partial \left(F/T\right)}{\partial T}\right)_V G = -V^2\left(\frac{\partial \left(F/V\right)}{\partial V}\right)_T
\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_P \left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P

Quantum properties

• U = N k_B T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V ~
• S = \frac{U}{T} + N * ~ S = \frac{U}{T} + N k_B \ln Z - N k \ln N + Nk ~ 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 Z_t = \frac{(2 \pi m k_B T)^\frac{3}{2} V}{h^3}
Vibration Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_B T}}
Rotation Z_r = \frac{2 I k_B T}{\sigma (\frac{h}{2 \pi})^2}

Thermal properties of matter

Coefficients Equation
Joule-Thomson coefficient \mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H
Compressibility (constant temperature) K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N}
Coefficient of thermal expansion (constant pressure) \alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p
Heat capacity (constant pressure) C_p = \left ( {\partial Q_{rev} \over \partial T} \right )_p = \left ( {\partial U \over \partial T} \right )_p + p \left ( {\partial V \over \partial T} \right )_p = \left ( {\partial H \over \partial T} \right )_p = T \left ( {\partial S \over \partial T} \right )_p
Heat capacity (constant volume) C_V = \left ( {\partial Q_{rev} \over \partial T} \right )_V = \left ( {\partial U \over \partial T} \right )_V = T \left ( {\partial S \over \partial T} \right )_V

Thermal transfer

Physical situation Nomenclature Equations
Net intensity emission/absorption
• Texternal = external temperature (outside of system)
• Tsystem = internal temperature (inside system)
• ε = emmisivity
I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) \,\!
Internal energy of a substance
• CV = isovolumetric heat capacity of substance
• ΔT = temperature change of substance
\Delta U = N C_V \Delta T\,\!
Meyer's equation
• Cp = isobaric heat capacity
• CV = isovolumetric heat capacity
• n = number of moles
C_p - C_V = nR \,\!
Effective thermal conductivities
• λi = thermal conductivity of substance i
• λnet = equivalent thermal conductivity
Series

\lambda_\mathrm{net} = \sum_j \lambda_j \,\!

Parallel \frac{1}{\lambda}_\mathrm{net} = \sum_j \left ( \frac{1}{\lambda}_j \right ) \,\!

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:

\eta = \left |\frac{W}{Q_H} \right|\,\!

Carnot engine efficiency:
\eta_c = 1 - \left | \frac{Q_L}{Q_H} \right | = 1-\frac{T_L}{T_H}\,\!

Refrigeration
• K = coefficient of refrigeration performance
Refrigeration performance

K = \left | \frac{Q_L}{W} \right | \,\!

Carnot refrigeration performance K_C = \frac{|Q_L|}{|Q_H|-|Q_L|} = \frac{T_L}{T_H-T_L}\,\!

References

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., 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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