Materials Science and Engineering/Equations

< Materials Science and Engineering

Notation

Conjugate variables
   p Pressure V Volume
   T Temperature S Entropy
   μ Chemical potential N Particle number
Thermodynamic potentials
   U Internal energy A Helmholtz free energy
   H Enthalpy G Gibbs free energy
Material properties
   ρ Density
   CV   Heat capacity (constant volume)
   Cp   Heat capacity (constant pressure)
   \beta_T Isothermal compressibility
   \beta_S Adiabatic compressibility
   \alpha Coefficient of thermal expansion
Other conventional variables
   δw infinitesimal amount of Mechanical Work
   δq infinitesimal amount of Heat
Constants
   kB Boltzmann constant
   R Ideal gas constant

Equations

Laws of Thermodynamics

First Law of Thermodynamics:

dU=\delta Q-\delta W\,

Second Law of Thermodynamics:

\int \frac{\delta Q}{T} \ge 0

Fundamental Equations

The Fundamental Equation:

dU \le TdS-pdV+\sum_{i=1}^p\mu_idN_i

The equation may be seen as a particular case of the chain rule:

dU= \left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}}dS+ \left(\frac{\partial U}{\partial V}\right)_{S,\{N_i\}}dV+ \sum_i\left(\frac{\partial U}{\partial N_i}\right)_{S,V,\{N_{j \ne i}\}}dN_i

from which the following identifications can be made:

\left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}}=T
\left(\frac{\partial U}{\partial V}\right)_{S,\{N_i\}}=-p
\left(\frac{\partial U}{\partial N_i}\right)_{S,V,\{N_{j \ne i}\}}=\mu_i

These equations are known as "equations of state" with respect to the internal energy.

Thermodynamic Potentials

Thermodynamic Potentials:

Name Formula Natural variables
Internal energy U\, ~~~~~S,V,\{N_i\}\,
Helmholtz free energy A=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\}\,

For the above four potentials, the fundamental equations are expressed as:

dU\left(S,V,{N_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i
dA\left(T,V,N_{i}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}
dH\left(S,p,N_{i}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}
dG\left(T,p,N_{i}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}

Euler Integrals:

Because all of natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that

U=TS-pV+\sum_i \mu_i N_i\,

Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:

A= -pV+\sum_i \mu_i N_i\,
H=TS +\sum_i \mu_i N_i\,
G= \sum_i \mu_i N_i\,

Note that the Euler integrals are sometimes also referred to as fundamental equations.

Gibbs Duhem Relationship:

Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that:

0=SdT-Vdp+\sum_iN_id\mu_i\,

which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Josiah Gibbs and Pierre Duhem.

Materials Properties

Compressibility: At constant temperature or constant entropy

~ \beta_{T \text{ or } S} = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N \text{ or } S,N}

Heat Capacity at Constant Pressure:

~ C_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 at Constant Volume:

~ C_V=\left ( {\partial U\over \partial T} \right )_V = T \left ( {\partial S\over \partial T} \right )_V ~

Coefficient of Thermal Expansion:

\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p

Maxwell Relations:

Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables. They follow directly from the fact that the order of differentiation does not matter when taking the second derivative. The four most common Maxwell relations are:

~ \left ( {\partial T\over \partial V} \right )_{S,N} = -\left ( {\partial p\over \partial S} \right )_{V,N} ~ ~ \left ( {\partial T\over \partial p} \right )_{S,n} = \left ( {\partial V\over \partial S} \right )_{p,N} ~
~ \left ( {\partial T\over \partial V} \right )_{p,N} = -\left ( {\partial p\over \partial S} \right )_{T,N} ~ ~ \left ( {\partial T\over \partial p} \right )_{V,N} = \left ( {\partial V\over \partial S} \right )_{T,N} ~

Processes

Incremental Processes:

~ dU = T\,dS-p\,dV + \mu\,dN ~

~ dA = -S\,dT-p\,dV + \mu\,dN ~

~ dG = -S\,dT+V\,dp + \mu\,dN = \mu\,dN +N\,d\mu ~

~ dH = T\,dS+V\,dp + \mu\,dN ~ Equation Table for an Ideal Gas (PV^m=constant):

Constant Pressure Constant Volume Isothermal Adiabatic
Variable \Delta p=0\; \Delta V=0\; \Delta T=0\; q=0\;
m\; 0\; \infty\; 1\; \gamma=\frac {C_p}{C_V} \;
Work
\begin{matrix}w=-\int_{V_1}^{V_2} pdV \end{matrix}		 -p\left ( V_2-V_1 \right )\; 0\; -nRT\ln\frac{V_2}{V_1}\; C_V\left ( T_2-T_1 \right )\;
Heat Capacity, C\; C_p = (5/2)nR\; C_V = (3/2)nR \; C_p\; or C_V\; C_p\; or C_V\;
Internal Energy, \Delta U = 3/2 *nR\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, \Delta H\;
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
\begin{matrix}\Delta S=-\int_{T_1}^{T_2} \frac {C}{T}dT \end{matrix}		 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}\;		 0\;
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