Abstract
This chapter presents the procedures to derive the equations for the changes in energy functions and entropy of a system as the temperature, pressure, and volume of the system vary as well as the changes in enthalpy, entropy, and chemical potential or Gibbs free energy of a system due to phase transitions and chemical reactions at a given temperature and pressure in terms of experimentally measured or theoretically computed properties such as heat capacity, compressibility, and thermal expansion coefficient.
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Notes
- 1.
\(\left( {\frac{\partial x}{{\partial y}}} \right)_{z} = - \frac{{\left( {\frac{\partial z}{{\partial y}}} \right)_{x} }}{{\left( {\frac{\partial z}{{\partial x}}} \right)_{y} }}\)
- 2.
\(\mathop \int\limits_{{T_{\text{o}} }}^{T} \left( {\mathop \int \limits_{{T_{\text{o}} }}^{T} \frac{{c_{v} }}{T}{\text{d}}T} \right){\text{d}}T = T\mathop \int \limits_{{T_{\text{o}} }}^{T} \frac{{c_{v} }}{T}{\text{d}}T - \mathop \int \limits_{{T_{\text{o}} }}^{T} c_{v} {\text{d}}T\)
- 3.
\(\mathop \int \limits_{{T_{\text{o}} }}^{T} \left( {\mathop \int \limits_{{T_{\text{o}} }}^{T} \frac{{c_{p} }}{T}{\text{d}}T} \right){\text{d}}T = T\mathop \int \limits_{{T_{\text{o}} }}^{T} \frac{{c_{p} }}{T}{\text{d}}T - \mathop \int \limits_{{T_{\text{o}} }}^{T} c_{p} {\text{d}}T\)
- 4.
\({\text{e}}^{x} \approx 1 + x\;{\text{for}}\;x \ll 1\)
- 5.
\(\int {\ln x{\text{d}}x} = x\ln x - x\)
- 6.
\(\int {x{\text{e}}^{x} {\text{d}}x} = \left( {x - 1} \right){\text{e}}^{x} ;\int {x{\text{e}}^{ax} {\text{d}}x} = \frac{{{\text{e}}^{ax} }}{{a^{2} }}\left( {ax - 1} \right)\)
- 7.
For example, for a cubic to tetragonal transformation, \(\varepsilon_{11}^{\text{o}} = \frac{{a_\mathrm{t} - a_\mathrm{c} }}{{a_\mathrm{c} }}\), \(\varepsilon_{22}^{\text{o}} = \frac{{a_\mathrm{t} - a_\mathrm{c} }}{{a_\mathrm{c} }}\), \(\varepsilon_{33}^{\text{o}} = \frac{{c_\mathrm{t} - a_\mathrm{c} }}{{a_\mathrm{c} }}\), \(\varepsilon_{12}^{\text{o}} = \varepsilon_{21}^{\text{o}} = \varepsilon_{13}^{\text{o}} = \varepsilon_{31}^{\text{o}} = \varepsilon_{23}^{\text{o}} = \varepsilon_{32}^{\text{o}} = 0\), where \(a_\mathrm{t}\) and \(c_\mathrm{t}\) are the lattice parameters of the tetragonal phase measured at the stress-free condition, and \(a_\mathrm{c}\) is the lattice parameter of the cubic phase measured at the stress-free condition.
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Chen, LQ. (2022). Thermodynamic Calculations of Materials Processes . In: Thermodynamic Equilibrium and Stability of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-13-8691-6_8
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DOI: https://doi.org/10.1007/978-981-13-8691-6_8
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