diff --git a/doc/source/derivs/derivs.ipynb b/doc/source/derivs/derivs.ipynb index c621e71dd5e10dbadb7d9705ab86375542a39462..a2015a904564a8e106710bef0bb1e54d56aba7e0 100644 --- a/doc/source/derivs/derivs.ipynb +++ b/doc/source/derivs/derivs.ipynb @@ -24,7 +24,42 @@ "\n", "$$ p^r = \\rho R T \\Lambda^r_{01}$$\n", "\n", - "Similar definitions apply for all the other residual thermodynamic properties" + "Similar definitions apply for all the other thermodynamic properties, with the tot superscript indicating it is the sum of the residual and ideal-gas (not included in teqp) contributions:\n", + "\n", + "$$\n", + "\\frac{p}{\\rho R T} = 1 + \\Lambda^r_{01}\n", + "$$\n", + "Internal energy ($u= a+Ts$):\n", + "$$\n", + "\t\\frac{u}{RT} = \\Lambda^{\\rm tot}_{10}\n", + "$$\n", + "Enthalpy ($h= u+p/\\rho$):\n", + "$$\n", + "\\frac{h}{RT} = 1+\\Lambda^r_{01} + \\Lambda^{\\rm tot}_{10}\n", + "$$\n", + "Entropy ($s\\equiv -(\\partial a/\\partial T)_v$):\n", + "$$\n", + "\\frac{s}{R} = \\Lambda^{\\rm tot}_{10}-\\Lambda^{\\rm tot}_{00}\n", + "$$\n", + "Gibbs energy ($g= h-Ts$):\n", + "$$\n", + "\t\\frac{g}{RT} = 1+\\Lambda^r_{01}+\\Lambda^{\\rm tot}_{00}\n", + "$$\n", + "Derivatives of pressure:\n", + "$$\n", + "\t\\left(\\frac{\\partial p}{\\partial \\rho}\\right)_{T} = RT\\left(1+2\\Lambda^r_{01}+\\Lambda^r_{02}\\right)\n", + "$$\n", + "$$\n", + "\t\\left(\\frac{\\partial p}{\\partial T}\\right)_{\\rho} = R\\rho\\left(1+\\Lambda^r_{01}-\\Lambda^r_{11}\\right)\n", + "$$\n", + "Isochoric specific heat ($c_v\\equiv (\\partial u/\\partial T)_v$):\n", + "$$\n", + "\\frac{c_v}{R} = -\\Lambda^{\\rm tot}_{20}\n", + "$$\n", + "Isobaric specific heat ($c_p\\equiv (\\partial h/\\partial T)_p$; see Eq. 3.56 from Span \\cite{Span-BOOK-2000} for the derivation):\n", + "$$\n", + "\\frac{c_p}{R} = -\\Lambda^{\\rm tot}_{20}+\\frac{(1+\\Lambda^r_{01}-\\Lambda^r_{11})^2}{1+2\\Lambda^r_{01}+\\Lambda^r_{02}}\n", + "$$" ] }, {