View source for Physics Formulae/Thermodynamics Formulae

'''Lead Article: [[Tables of Physics Formulae]]'''


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


==Thermodynamics Laws==


{| class="wikitable"
|-

|[[Zeroth Law of Thermodynamics]]
|<math>(T_A = T_B) \land (T_B=T_C) \Rightarrow T_A=T_C\,\!</math> 

(systems in thermal equilibrium)
|-

|First Law of Thermodynamics
|△Q = △U +△W


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

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

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

|-

|Second Law of Thermodynamics
|<math>\Delta S \ge 0\,\!</math>
|-
|Third Law of Thermodynamics
|<math>S = S_\mathrm{structural} + CT\,\!</math>
|-
|}

==Thermodynamic Quantities==

{| class="wikitable"
|-

! Quantity (Common Name/s) !! '''(Common Symbol/s)''' !! '''Defining Equation''' !! '''SI Units''' !! '''Dimension'''
|-

|Number of Molecules
||<math>N\,\!</math>
||
|| dimensionless
|| dimensionless
|-

|Temperature
||<math>T\,\!</math>
||
|| K
|| [Θ]
|-

|Heat Energy
||<math>Q\,\!</math>
||
|| J
|| [M][L]<sup>2</sup>[T]<sup>-2</sup>
|-

|Latent Heat
||<math>Q_L \,\!</math>
||
|| J 
|| [M][L]<sup>2</sup>[T]<sup>-2</sup>
|-


|Entropy
||<math> S \,\!</math>
||
|| J K<sup>-1</sup> 
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-

|Heat Capacity (isobaric)
||<math>C_{p} \,\!</math>
||<math> C_{p} = \frac{\partial Q}{\partial T}\,\!</math>
|| J K <sup>-1</sup>
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-


|Specific Heat Capacity (isobaric)
||<math> C_{mp} \,\!</math>
||<math> C_{mp} = \frac{\partial^2 Q}{\partial m \partial T} \,\!</math>
|| J kg<sup>-1</sup> K<sup>-1</sup> 
|| [L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-

|Molar Specific Heat 

Capacity (isobaric)
||<math>C_{np} \,\!</math>
||<math>C_{np} = \frac{\partial^2 Q}{\partial n \partial T}\,\!</math>
|| J K <sup>-1</sup> mol<sup>-1</sup>
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup> [N]<sup>-1</sup>
|-

|Heat Capacity (isochoric)
||<math> C_{V} \,\!</math>
||<math> C_{V} = \frac{\partial Q}{\partial T} \,\!</math>
|| J K <sup>-1</sup>
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-

|Specific Heat Capacity (isochoric)
||<math> C_{mV} \,\!</math>
||<math> C_{mV} = \frac{\partial^2 Q}{\partial m \partial T} \,\!</math>
|| J kg<sup>-1</sup> K<sup>-1</sup> 
|| [L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-

|Molar Specific Heat 

Capacity (isochoric)
||<math> C_{nV} \,\!</math>
||<math> C_{nV} = \frac{\partial^2 Q}{\partial n \partial T} \,\!</math>
|| J K <sup>-1</sup> mol<sup>-1</sup>
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> [Θ]<sup>-1</sup> [N]<sup>-1</sup>
|-


|'''Internal Energy'''

Sum of all total energies which 

constitute the system

||<math> U \,\!</math>
||<math>U = \sum_i  E_i \!</math>
|| J 
|| [M][L]<sup>2</sup>[T]<sup>-2</sup>
|-


|Enthalpy
||<math> H \,\!</math>
||<math> H = U+pV\,\!</math>
|| J 
|| [M][L]<sup>2</sup>[T]<sup>-2</sup>
|-

|Gibbs Free Energy
||<math> \Delta G \,\!</math>
||<math> \Delta G = \Delta H - T\Delta S \,\!</math>
|| J 
|| [M][L]<sup>2</sup>[T]<sup>-2</sup>
|-

|Helmholtz Free Energy
||<math> A, F \,\!</math>
||<math> A = U - TS \,\!</math>
|| J
|| [M][L]<sup>2</sup>[T]<sup>-2</sup> 
|-

|Specific Latent Heat
||<math>L \,\!</math>
||<math>L = \frac{Q}{m} \,\!</math>
|| J kg<sup>-1</sup>
|| [L]<sup>2</sup>[T]<sup>-2</sup>
|-

|Ratio of Isobaric to 

Isochoric Heat Capacity,

Adiabatic Index
||<math>\gamma \,\!</math>
||<math>\gamma = \frac{C_p}{C_V} = \frac{c_p}{c_V} = \frac{C_{mp}}{C_{mV}} \,\!</math>
|| dimensionless
|| dimensionless
|-

|Linear Coefficient of Thermal Expansion
||<math> \alpha \,\!</math>
||<math> \frac{\partial L}{\partial t} = \alpha L \,\!</math>
|| K<sup>-1</sup>
|| [Θ]<sup>-1</sup>
|-

|Volume Coefficient of Thermal Expansion
||<math> 3 \alpha \,\!</math>
||<math> \frac{\partial V}{\partial t} = 3 \alpha V \,\!</math>
|| K<sup>-1</sup>
|| [Θ]<sup>-1</sup>
|-

|Temperature Gradient
||No standard symbol
||<math> \nabla T \,\!</math>
|| K m<sup>-1</sup>
|| [Θ][L]<sup>-1</sup>
|-

|Thermal Conduction Rate/

Thermal Current
||<math> P \,\!</math>
||<math>P = \frac{\partial Q}{\partial t} \,\!</math>
|| W = J s<sup>-1</sup>
|| [M] [L]<sup>2</sup> [T]<sup>-2</sup>
|-

|Thermal Intensity
||<math> I \,\!</math>
||<math>I = \frac{\partial P}{\partial A}= \frac{\partial^2 P}{\partial A \partial t} \,\!</math>
|| W m<sup>-2</sup>
|| [M] [L]<sup>-1</sup> [T]<sup>-2</sup>
|-

|Thermal Conductivity
||<math>\kappa, K, \lambda \,\!</math>
||<math> \lambda = - \frac{P}{\mathbf{A} \cdot \nabla T } \,\!</math>
|| W m<sup>-1</sup> K<sup>-1</sup>
|| [M] [L] [T]<sup>-2</sup> [Θ]<sup>-1</sup>
|-

|Thermal Resistance
||<math> R \,\!</math>
||<math>R=\frac{\Delta x}{\lambda}\,\!</math>
|| m<sup>2</sup> K W<sup>-1</sup>  
|| [L] [T]<sup>2</sup> [Θ]<sup>1</sup> [M]<sup>-1</sup>
|-

|Emmisivity Coefficient
||<math> \epsilon \,\!</math>
|| Can only be found from experiment

<math> 0 \leqslant \epsilon \leqslant 1\,\!</math>

<math> \epsilon = 0\,\!</math> for perfect reflector

<math> \epsilon = 1\,\!</math> for perfect absorber

(true black body)

|| dimensionless  
|| dimensionless
|-

|}


==Kinetic Theory==


{| class="wikitable"
|-
|Ideal Gas Law
|<math>pV = nRT\,\!</math>

<math>pV = kTN\,\!</math>

<math>\frac{p_1 V_1}{n_1 T_1} = \frac{p_2 V_2}{n_2 T_2} \,\!</math>


<math>\frac{p_1 V_1}{N_1 T_1} = \frac{p_2 V_2}{N_2 T_2} \,\!</math>
|-

|Translational Energy
|<math> \langle E_\mathrm{k} \rangle = \frac{f}{2}kT\,\!</math>
|-

|Internal Energy
|<math>U = \frac{f}{2}NkT\,\!</math>
|-

|}


==Thermal Transitions==

{| class="wikitable"
|-
|Adiabatic
|<math>\Delta Q = 0 \,\!</math>

<math>\Delta U = W\,\!</math>
|-

|Work by an Expanding Gas 
|Process

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


Net Work Done in Cyclic Processes

<math> \Delta W = \oint_\mathrm{cycle} p \mathrm{d}V \,\!</math>
|-

|Isobaric Transition
|<math>\Delta U = Q\,\!</math>
|-

|Cyclic Process
|<math> Q + W = 0\,\!</math>
|-

|Work, Isochoric
|<math>W=0\,\!</math>
|-

|work, Isobaric
|<math>W=p\Delta V\,\!</math>
|-

|Work, Isothermal
|<math>W=kTN \ln(V_2/V_1)\,\!</math>
|-

|Adiabatic Expansion
|<math>p_1 V_1^{\gamma} = p_2 V_2^{\gamma}\,\!</math>

<math>T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \,\!</math>
|-

|Free Expansion
|<math>\Delta U = 0\,\!</math>
|-


|}


==Statistical Physics==


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


{| class="wikitable"
|-
|Degrees of Freedom
|<math>f\,\!</math>
|-

|Maxwell-Boltzmann Distribution, 

Mean Speed
|<math> \langle v \rangle = \sqrt{\frac{8kT}{\pi m}}\,\!</math>
|-
|Maxwell-Boltzmann Distribution
Mode-Speed
|<math> v_\mathrm{mode} = \sqrt{\frac{kT}{2m}}\,\!</math>
|-

|Root Mean Square Speed
|<math> v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{kT}{3m}} \,\!</math>
|-
|Mean Free Path
|<math> \langle x_\mathrm{free} \rangle = \frac{ \langle v \rangle}{\sqrt{2} \pi d^2 N }\,\!</math>?
|-
|Maxwell–Boltzmann Distribution
|<math>P(v)=4\pi\left ( \frac{m}{2\pi kT} \right )^{3/2} v^2 e^{-mv^2/2kT} \,\!</math>
|-

|Multiplicity of Configurations
|<math>W = \frac{N!}{n_1 ! n_2 !}\,\!</math>
|-

|Microstate in one half of the box
|<math>n_1, n_2\,\!</math>
|-

|Boltzmann's Entropy Equation
|<math>S = k \ln W \,\!</math>
|-

|Irreversibility
|<math>\,\!</math>
|-

|Entropy
|<math>S = - k \sum_i P_i \ln P_i \!\,\!</math>
|-
|Entropy Change
|<math>\Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T} \,\!</math>

<math>\Delta S = kN \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1} \,\!</math>
|-

|Entropic Force
|<math>F_S = -T \nabla S \,\!</math>
|-

|}


==Thermal Transfer==


{| class="wikitable"
|-
|Stefan-Boltzmann Law
|<math>I = \sigma \epsilon T ^4 \,\!</math>
|-

|Net Intensity Emmision/Absorbtion

|<math> I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) \,\!</math>
|-

|Internal Energy of a Substance
|<math>\Delta U = N C_V \Delta T\,\!</math>
|-

|Work done by an Expanding Ideal Gas
|<math>\mathrm{d} W = p \mathrm{d} V = N k \mathrm{d}T \,\!</math>
|-

|Meyer's Equation
|<math> C_p - C_V = nR \,\!</math>
|-
|}


==Thermal Efficiencies==

{| class="wikitable"
|-
|Engine Efficiency
|<math>\epsilon = |W|/|Q_H|\,\!</math>
|-
|Carnot Engine Efficiency
|<math>\epsilon_c = (|Q_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H\,\!</math>
|-
|Refrigeration Performance
|<math>K = |Q_L|/|W|\,\!</math>
|-
|Carnot Refrigeration Performance
|<math>K_C = |Q_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L)\,\!</math>
|-
|}

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