Physics Formulae/Thermodynamics Formulae

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This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Thermodynamics.


Thermodynamics Laws

Zeroth Law of Thermodynamics (T_A = T_B) \and (T_B=T_C) \Rightarrow T_A=T_C\,\!

(systems in thermal equilibrium)

First Law of Thermodynamics \Delta U = \Delta Q + \Delta W \,\!

Internal energy increase \Delta U > 0 \,\!, decrease \Delta U < 0 \,\!

Heat energy transferred to system \Delta Q > 0 \,\!, from system \Delta Q < 0 \,\!

Work done transferred to system \Delta W > 0 \,\! by system \Delta W < 0 \,\!

Second Law of Thermodynamics \Delta S \ge 0\,\!
Third Law of Thermodynamics S = S_\mathrm{structural} + CT\,\!

Thermodynamic Quantities

Quantity (Common Name/s) (Common Symbol/s) Defining Equation SI Units Dimension
Number of Molecules N\,\! dimensionless dimensionless
Temperature T\,\! K [Θ]
Heat Energy Q\,\! J [M][L]2[T]-2
Latent Heat Q_L \,\! J [M][L]2[T]-2
Entropy S \,\! J K-1 [M][L]2[T]-2 [Θ]-1
Heat Capacity (isobaric) C_{p} \,\! C_{p} = \frac{\partial Q}{\partial T}\,\! J K -1 [M][L]2[T]-2 [Θ]-1
Specific Heat Capacity (isobaric) C_{mp} \,\! C_{mp} = \frac{\partial^2 Q}{\partial m \partial T} \,\! J kg-1 K-1 [L]2[T]-2 [Θ]-1
Molar Specific Heat

Capacity (isobaric)

 C_{np} \,\! C_{np} = \frac{\partial^2 Q}{\partial n \partial T}\,\! J K -1 mol-1 [M][L]2[T]-2 [Θ]-1 [N]-1
Heat Capacity (isochoric) C_{V} \,\! C_{V} = \frac{\partial Q}{\partial T} \,\! J K -1 [M][L]2[T]-2 [Θ]-1
Specific Heat Capacity (isochoric) C_{mV} \,\! C_{mV} = \frac{\partial^2 Q}{\partial m \partial T} \,\! J kg-1 K-1 [L]2[T]-2 [Θ]-1
Molar Specific Heat

Capacity (isochoric)

 C_{nV} \,\! C_{nV} = \frac{\partial^2 Q}{\partial n \partial T} \,\! J K -1 mol-1 [M][L]2[T]-2 [Θ]-1 [N]-1
Internal Energy

Sum of all total energies which

constitute the system

 U \,\! U = \sum_i E_i \! J [M][L]2[T]-2
Enthalpy H \,\! H = U+pV\,\! J [M][L]2[T]-2
Gibbs Free Energy \Delta G \,\! \Delta G = \Delta H - T\Delta S \,\! J [M][L]2[T]-2
Helmholtz Free Energy A, F \,\! A = U - TS \,\! J [M][L]2[T]-2
Specific Latent Heat L \,\! L = \frac{Q}{m} \,\! J kg-1 [L]2[T]-2
Ratio of Isobaric to

Isochoric Heat Capacity,

Adiabatic Index

 \gamma \,\! \gamma = \frac{C_p}{C_V} = \frac{c_p}{c_V} = \frac{C_{mp}}{C_{mV}} \,\! dimensionless dimensionless
Linear Coefficient of Thermal Expansion \alpha \,\! \frac{\partial L}{\partial t} = \alpha L \,\! K-1 [Θ]-1
Volume Coefficient of Thermal Expansion 3 \alpha \,\! \frac{\partial V}{\partial t} = 3 \alpha V \,\! K-1 [Θ]-1
Temperature Gradient No standard symbol \nabla T \,\! K m-1 [Θ][L]-1
Thermal Conduction Rate/

Thermal Current

 P \,\! P = \frac{\partial Q}{\partial t} \,\! W = J s-1 [M] [L]2 [T]-2
Thermal Intensity I \,\! I = \frac{\partial P}{\partial A}= \frac{\partial^2 P}{\partial A \partial t} \,\! W m-2 [M] [L]-1 [T]-2
Thermal Conductivity \kappa, K, \lambda \,\! \lambda = - \frac{P}{\mathbf{A} \cdot \nabla T } \,\! W m-1 K-1 [M] [L] [T]-2 [Θ]-1
Thermal Resistance R \,\! R=\frac{\Delta x}{\lambda}\,\! m2 K W-1 [L] [T]2 [Θ]1 [M]-1
Emmisivity Coefficient \epsilon \,\! Can only be found from experiment

0 \leqslant \epsilon \leqslant 1\,\!

\epsilon = 0\,\! for perfect reflector

\epsilon = 1\,\! for perfect absorber

(true black body)

 dimensionless dimensionless

Kinetic Theory

Ideal Gas Law pV = nRT\,\!

pV = kTN\,\!

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


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

Translational Energy \langle E_\mathrm{k} \rangle = \frac{f}{2}kT\,\!
Internal Energy U = \frac{f}{2}NkT\,\!

Thermal Transitions

Adiabatic \Delta Q = 0 \,\!

\Delta U = W\,\!

Work by an Expanding Gas Process

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


Net Work Done in Cyclic Processes

\Delta W = \oint_\mathrm{cycle} p \mathrm{d}V \,\!

Isobaric Transition \Delta U = Q\,\!
Cyclic Process Q + W = 0\,\!
Work, Isochoric W=0\,\!
work, Isobaric W=p\Delta V\,\!
Work, Isothermal W=kTN \ln(V_2/V_1)\,\!
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\,\!

Statistical Physics


Below are usefull results from the Maxell-Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity.

Degrees of Freedom f\,\!
Maxwell-Boltzmann Distribution,

Mean Speed

 \langle v \rangle = \sqrt{\frac{8kT}{\pi m}}\,\!
Maxwell-Boltzmann Distribution

Mode-Speed

 v_\mathrm{mode} = \sqrt{\frac{kT}{2m}}\,\!
Root Mean Square Speed v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{kT}{3m}} \,\!
Mean Free Path \langle x_\mathrm{free} \rangle = \frac{ \langle v \rangle}{\sqrt{2} \pi d^2 N }\,\!?
Maxwell–Boltzmann Distribution P(v)=4\pi\left ( \frac{m}{2\pi kT} \right )^{3/2} v^2 e^{-mv^2/2kT} \,\!
Multiplicity of Configurations W = \frac{N!}{n_1 ! n_2 !}\,\!
Microstate in one half of the box n_1, n_2\,\!
Boltzmann's Entropy Equation S = k \ln W \,\!
Irreversibility \,\!
Entropy S = - k \sum_i P_i \ln P_i \!\,\!
Entropy Change \Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T} \,\!

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

Entropic Force F_S = -T \nabla S \,\!

Thermal Transfer

Stefan-Boltzmann Law I = \sigma \epsilon T ^4 \,\!
Net Intensity Emmision/Absorbtion I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) \,\!
Internal Energy of a Substance \Delta U = N C_V \Delta T\,\!
Work done by an Expanding Ideal Gas \mathrm{d} W = p \mathrm{d} V = N k \mathrm{d}T \,\!
Meyer's Equation C_p - C_V = nR \,\!

Thermal Efficiencies

Engine Efficiency \epsilon = |W|/|Q_H|\,\!
Carnot Engine Efficiency \epsilon_c = (|Q_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H\,\!
Refrigeration Performance K = |Q_L|/|W|\,\!
Carnot Refrigeration Performance K_C = |Q_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L)\,\!
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