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on_phenomenological_thermodynamics_._note [2015/11/07 19:48] nikolaj |
on_phenomenological_thermodynamics_._note [2015/11/07 19:49] nikolaj |
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The fact that pressure and volume are then reciprocal, $p=\frac{c^ST}{V}$, leads to the occurrence of all the logarithms that are so typical for phenomenological thermodynamics (see below). E.g. in computing change of energies, we often must integrate external variables against internal ones. | The fact that pressure and volume are then reciprocal, $p=\frac{c^ST}{V}$, leads to the occurrence of all the logarithms that are so typical for phenomenological thermodynamics (see below). E.g. in computing change of energies, we often must integrate external variables against internal ones. | ||
- | $(\Delta G)_{T,\{N_i\}}:=\int_{p_2}^{p_1}V(T,p)\,{\mathrm d}p=c^S T\ln(\frac{p_2}{p_1})\implies p_2 = p_1\cdot{\mathrm e}^{\frac{(\Delta G)_{T,\{N_i\}}}{c^S T}}$. | + | $(\Delta G)_{T,\{N_i\}}:=\int_{p_2}^{p_1}V(T,p)\,{\mathrm d}p=c^S T\ln(\frac{p_2}{p_1})$ |
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+ | $\implies p_2 = p_1\cdot\exp\left(\dfrac{(\Delta G)_{T,\{N_i\}}}{c^S T}\right)$. | ||
This sort of "concentration varies with $\exp(-E/k_BT)$" equation also pops up often in chemistry and electronics (Goldmann equation, Nernst equation, diode current-voltage-characteristics,...) | This sort of "concentration varies with $\exp(-E/k_BT)$" equation also pops up often in chemistry and electronics (Goldmann equation, Nernst equation, diode current-voltage-characteristics,...) |