Update to use partial derivative identities

Also fix equations in comments

Author: corykinney

include/cantera/thermo/PengRobinson.h

@@ -208,7 +208,7 @@ class PengRobinson : public MixtureFugacityTP
     //! Returns the isothermal compressibility (\f$\beta_T\f$). Units: 1/Pa.
     /*!
      *  \f[ 
-     *    \beta_T = \frac{1}{v} \left(\frac{\partial v}{\partial p}\right)_T
+     *    \beta_T = -\frac{1}{v} \left(\frac{\partial v}{\partial p}\right)_T
      *  \f]
      */
     virtual double isothermalCompressibility() const;

include/cantera/thermo/RedlichKwongMFTP.h

@@ -176,15 +176,15 @@ class RedlichKwongMFTP : public MixtureFugacityTP
     //! Returns the isothermal compressibility (\f$\beta_T\f$). Units: 1/Pa.
     /*!
      *  \f[ 
-     *    \beta_T = \frac{1}{v} \left(\frac{\partial v}{\partial p}\right)_T
+     *    \beta_T = -\frac{1}{v} \left(\frac{\partial v}{\partial p}\right)_T
      *  \f]
      */
     virtual double isothermalCompressibility() const;
 
     //! Return the volumetric thermal expansion coefficient (\f$\alpha_V\f$). Units: 1/K.
     /*!
      *  \f[ 
-     *    alpha_V = \frac{1}{v} \left(\frac{\partial v}{\partial T}\right)_p
+     *    \alpha_V = \frac{1}{v} \left(\frac{\partial v}{\partial T}\right)_p
      *  \f]
      */
     virtual double thermalExpansionCoeff() const;

src/thermo/PengRobinson.cpp

@@ -632,37 +632,14 @@ double PengRobinson::dpdVCalc(double T, double molarVol, double& presCalc) const
 
 double PengRobinson::isothermalCompressibility() const
 {
-    double P = pressure();
-    double RT_ = RT();
-    double Z = z();
-
-    double A = m_aAlpha_mix * P / (RT_ * RT_);
-    double B = m_b * P / RT_;
-
-    double dAdP = A / P;
-    double dBdP = B / P;
-    double dZdP = ((B - Z) * dAdP + (A - 2 * B - 3 * B * B + 2 * Z + 6 * B * Z - Z * Z) * dBdP) /
-        (3 * Z * Z + 2 * Z * (B - 1) + A - 2 * B - 3 * B * B);
-
-    return 1 / P - dZdP / Z;
+    calculatePressureDerivatives();
+    return -1 / (molarVolume() * m_dpdV);
 }
 
 double PengRobinson::thermalExpansionCoeff() const
 {
-    double P = pressure();
-    double T = temperature();
-    double RT_ = RT();
-    double Z = z();
-
-    double A = m_aAlpha_mix * P / (RT_ * RT_);
-    double B = m_b * P / RT_;
-
-    double dAdT = P / (RT_ * RT_) * (daAlpha_dT() - 2 * m_aAlpha_mix / T);
-    double dBdT = -B / T;
-    double dZdT = ((B - Z) * dAdT + (A - 2 * B - 3 * B * B + 2 * Z + 6 * B * Z - Z * Z) * dBdT) /
-        (3 * Z * Z + 2 * Z * (B - 1) + A - 2 * B - 3 * B * B);
-
-    return 1 / T + dZdT / Z;
+    calculatePressureDerivatives();
+    return -m_dpdT / (molarVolume() * m_dpdV);
 }
 
 void PengRobinson::calculatePressureDerivatives() const

src/thermo/RedlichKwongMFTP.cpp

@@ -709,36 +709,14 @@ doublereal RedlichKwongMFTP::dpdVCalc(doublereal TKelvin, doublereal molarVol, d
 
 double RedlichKwongMFTP::isothermalCompressibility() const
 {
-    double P = pressure();
-    double T = temperature();
-    double RT_ = RT();
-    double Z = z();
-
-    double A = m_a_current * P / (RT_ * RT_ * pow(T, 0.5));
-    double B = m_b_current * P / RT_;
-
-    double dAdP = A / P;
-    double dBdP = B / P;
-    double dZdP = ((B - Z) * dAdP + (A + Z + 2 * B * Z) * dBdP) / (3 * Z * Z - 2 * Z + A - B - B * B);
-
-    return 1 / P - dZdP / Z;
+    pressureDerivatives();
+    return -1 / (molarVolume() * dpdV_);
 }
 
 double RedlichKwongMFTP::thermalExpansionCoeff() const
 {
-    double P = pressure();
-    double T = temperature();
-    double RT_ = RT();
-    double Z = z();
-
-    double A = m_a_current * P / (RT_ * RT_ * pow(T, 0.5));
-    double B = m_b_current * P / RT_;
-
-    double dAdT = -2.5 * A / T + A / m_a_current * da_dt();
-    double dBdT = -B / T;
-    double dZdT = ((B - Z) * dAdT + (A + Z + 2 * B * Z) * dBdT) / (3 * Z * Z - 2 * Z + A - B - B * B);
-
-    return 1 / T + dZdT / Z;
+    pressureDerivatives();
+    return -dpdT_ / (molarVolume() * dpdV_);
 }
 
 void RedlichKwongMFTP::pressureDerivatives() const