Thermodynamics
The Ideal Gas Law¶
\[
pV=nRT=Nk_BT,p=\dfrac{\rho k_BT}{m},U=\alpha nRT
\]
where:
- \(p\) is the absolute pressure of the gas,
- \(V\) is the volume of the gas,
- \(n\) is the number of moles,
- \(R\) is the ideal gas constant,
- \(k_B\) is the Boltzmann constant,
- \(N_A\) is the Avogadro constant,
- \(T\) is the absolute temperature of the gas,
- \(\alpha\) is the number of degrees of freedom divided by \(2\). (\(\alpha = 3/2\) for monatomic gas, \(\alpha = 5/2\) for diatomic gas).
- \(N\) is the number of particles.
Van der Waals equation¶
\[
(p+\dfrac{a}{V^2})(V-b)=RT,p=\dfrac{RT}{V-b}-\dfrac{a}{V^2}
\]
The First Law of Thermodynamics¶
\[
dU= dW +dQ,dW=-pdV
\]
The Second Law of Thermodynamics¶
\[
dS=\dfrac{dQ}{T}
\]
Adiabatic Process¶
\[pV^\gamma = \text{constant}\]
\[p^{1-\gamma}T^\gamma = \text{constant}\]
\[TV^{\gamma - 1}=\text{constant}\]
where \(\gamma\) is the adiabatic index or heat capacity ratio defined as:
\[
\gamma=\dfrac{C_p}{C_V} = \dfrac{f+2}{f}
\]
- \(C_p,C_V\) is the specific heat for constant pressure, \(C_V\) is the specific heat for constant volume, and \(f\) is the number of degrees of freedom (3 for monoatomic gas, 5 for diatomic gas or a gas of linear molecules).
Derivation for P-V relation:
The adiabatic process is defined as a process where heat transfer is zero, then we can apply the first law of thermodynamics:
\[
dQ=dU-dW=d(\alpha pV)+pdV=(\alpha +1)pdV+\alpha Vdp=0
\]
We can separate the variable:
\[
-(\alpha + 1)\dfrac{dV}V=\alpha\dfrac{dp}p
\]
Integrate on both sides:
\[
\ln(p/p_0)=-\dfrac{\alpha + 1}{\alpha}(V/V_0)
\]
And then exponentiate them, substituting \(\gamma=\dfrac{\alpha + 1}{\alpha}\) .
Similar derivations can be done for Van der Waals equation.