## List of Physics Thermodynamics Formulas, Equations Latex Code

rockingdingo 2023-03-29 #physics #thermodynamics 1 0

List of Physics Thermodynamics Formulas, Equations Latex Code

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

##### 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 ideal gas follows

• #### Thermal Heat Capacity

##### 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}=R

##### Explanation

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

• #### Conservation of Energy

##### 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

• #### Thermodynamics Nernst's law

##### Latex Code
            \lim_{T\rightarrow0}\left(\frac{\partial S}{\partial X}\right)_{T}=0

##### Explanation

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.

• : The total added heat
• : The work done
• : The difference in the internal energy

• #### State functions Maxwell Relations

##### 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}dp


##### Explanation

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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##### Latex Code
            \text{adiabatic processes} \\
W=U_1-U_2 \\
\gamma=C_p/C_V

##### Explanation

Latex code for the Reversible Adiabatic Processes. I will briefly introduce the notations in this formulation.

• #### Isobaric Processes

##### 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.

• #### Throttle Processes

##### 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

• #### The Carnot Cycle

##### 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

• #### Phase Transitions

##### 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.

• #### Thermodynamic Potential

##### 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.

• #### Ideal Mixtures

##### 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
• : one component in a second gives rise to decrease of the freezing point

• #### Thermodynamics Statistical Basis

##### 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
• : one component in a second gives rise to decrease of the freezing point