0List of Physics Thermodynamics Formulas, Equations Latex Code
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In this blog, we will introduce most popuplar formulas in thermodynamics, Physics. We will also provide latex code of the equations. Topics include thermodynamics, heat capacity, conservation of energy, nernst's law, maxwell relations, adiabatic processes, isobaric processes, throttle processes, The Carnot Cycle, phase transitions, thermodynamic potential, ideal mixtures, thermodynamics statistical basis, etc.
- 1. Thermodynamics
- Thermodynamics Definitions
- Thermal Heat Capacity
- Thermodynamics Conservation of Energy
- Thermodynamics Nernst's law
- State functions Maxwell relations
- Reversible Adiabatic Processes
- Isobaric Processes
- Throttle Processes
- The Carnot Cycle
- Phase Transitions
- Thermodynamic Potential
- Ideal Mixtures
- Thermodynamics Statistical Basis
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Thermodynamics Definitions
Equation
Latex Code
f(x,y,z)=0 \\ dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{\partial y}\right)_{x}dy \\ \left(\frac{\partial x}{\partial y}\right)_{z}\cdot\left(\frac{\partial y}{\partial z}\right)_{x}\cdot\left(\frac{\partial z}{\partial x}\right)_{y}=-1 \\ \varepsilon^m F(x,y,z)=F(\varepsilon x,\varepsilon y,\varepsilon z) \\ mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}Explanation
Latex code for the Thermodynamics Introduction. I will briefly introduce the notations in this formulation.
: The total differential dz
A homogeneous function of degree m
: The isochoric pressure coefficient
: The isothermal compressibility
: The isobaric volume coefficient
: The adiabatic compressibility
: The ideal gas follows
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Thermal Heat Capacity
Equation
Latex Code
C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0 \\ \displaystyle C_X=T\left(\frac{\partial S}{\partial T}\right)_{X} \\ \displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_{p} \\ \displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_{V} \\ C_{mp}-C_{mV}=RExplanation
Latex code for the Thermodynamics Introduction. I will briefly introduce the notations in this formulation.
: The specific heat at constant at X
: The specific heat at constant pressure
: The specific heat at constant volume
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Conservation of Energy
Equation
Latex Code
Q=\Delta U+W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt} Q=dU+d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W=pdV \\ Q=\Delta H+W_{\rm i}+\Delta E_{\rm kin}+\Delta E_{\rm pot}Explanation
Latex code for the Thermodynamics Introduction. I will briefly introduce the notations in this formulation.
: The total added heat
: The work done
: The difference in the internal energy
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Thermodynamics Nernst's law
Equation
Latex Code
\lim_{T\rightarrow0}\left(\frac{\partial S}{\partial X}\right)_{T}=0Explanation
Latex code for the Thermodynamics Nernst's law. I will briefly introduce the notations in this formulation. Absolute zero temperature cannot be reached by cooling through a finite number of steps.
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State functions Maxwell Relations
Equation
Latex Code
\left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial p}{\partial S}\right)_{V}~,~~\left(\frac{\partial T}{\partial p}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{p}~,~~ \left(\frac{\partial p}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}~,~~\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T} \\ TdS=C_VdT+T\left(\frac{\partial p}{\partial T}\right)_{V}dV~~\mbox{and}~~TdS=C_pdT-T\left(\frac{\partial V}{\partial T}\right)_{p}dpExplanation
Latex code for the State functions Maxwell Relations. I will briefly introduce the notations in this formulation. Absolute zero temperature cannot be reached by cooling through a finite number of steps.
: Internal energy,
: Enthalpy ,
: Free energy,
: Gibbs free energy,
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Reversible Adiabatic Processes
Equation
Latex Code
\text{adiabatic processes} \\ W=U_1-U_2 \\ \text{reversible adiabatic processes} \\ \gamma=C_p/C_VExplanation
Latex code for the Reversible Adiabatic Processes. I will briefly introduce the notations in this formulation.
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Isobaric Processes
Equation
Latex Code
\text{Isobaric Processes} \\ H_2-H_1=\int_1^2 C_pdT \\ \text{Reversible isobaric process} \\ H_2-H_1=Q_{\rm rev}Explanation
Latex code for the Isobaric Processes. I will briefly introduce the notations in this formulation.
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Throttle Processes
Equation
Latex Code
\alpha_H=\left(\frac{\partial T}{\partial p}\right)_{H}=\frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_{p}-V\right]Explanation
Latex code for the Throttle Processes. The throttle process is used in refrigerators for example. I will briefly introduce the notations in this formulation.
: conserved quantity
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The Carnot Cycle
Equation
Latex Code
\eta=1-\frac{|Q_2|}{|Q_1|}=1-\frac{T_2}{T_1}:=\eta_{\rm C} \\ \xi=\frac{|Q_2|}{W}=\frac{|Q_2|}{|Q_1|-|Q_2|}=\frac{T_2}{T_1-T_2}Explanation
Latex code for the Carnot Cycle. The throttle process is used in refrigerators for example. I will briefly introduce the notations in this formulation.
: efficiency for a Carnot cycle
: the cold factor when process is applied in reverse order and the system performs a work -W
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Phase Transitions
Equation
Latex Code
\Delta S_m=S_m^\alpha - S_m^\beta=\frac{r_{\beta\alpha}}{T_0} \\ S_m=\left(\frac{\partial G_m}{\partial T}\right)_{p} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}} \\ r_{\beta\alpha}=r_{\alpha\beta} \\ r_{\beta\alpha}=r_{\gamma\alpha}-r_{\gamma\beta} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}}Explanation
Latex code for the Carnot Cycle. Phase transitions are isothermal and isobaric. I will briefly introduce the notations in this formulation.
: ideal gas one finds for the vapor line at some distance from the critical point.
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Thermodynamic Potential
Equation
Latex Code
dG=-SdT+Vdp+\sum_i\mu_idn_i \\ \displaystyle\mu=\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_j} \\ V=\sum_{i=1}^c n_i\left(\frac{\partial V}{\partial n_i}\right)_{n_j,p,T}:=\sum_{i=1}^c n_i V_i \\ \begin{aligned} V_m&=&\sum_i x_i V_i\\ 0&=&\sum_i x_i dV_i\end{aligned}Explanation
Latex code for the Thermodynamic Potential. I will briefly introduce the notations in this formulation.
: thermodynamic potential.
: partial volume of component i.
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Ideal Mixtures
Equation
Latex Code
U_{\rm mixture}=\sum_i n_i U^0_i \\ H_{\rm mixture}=\sum_i n_i H^0_i \\ S_{\rm mixture}=n\sum_i x_i S^0_i+\Delta S_{\rm mix} \\ \Delta S_{\rm mix}=-nR\sum\limits_i x_i\ln(x_i)Explanation
Latex code for the Ideal Mixtures. I will briefly introduce the notations in this formulation.
: one component in a second gives rise to an increase in the boiling point
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Thermodynamics Statistical Basis
Equation
Latex Code
P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)Explanation
Latex code for the Thermodynamics Statistical Basis. I will briefly introduce the notations in this formulation.
: number of possibilities
: number of particles
: number of possible energy levels
: g-fold degeneracy
: The occupation numbers in equilibrium(with the maximum value for P)
: State sum Z is a normalization constant
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