(temperature dependence of (K)): [ \fracd\ln KdT = \frac\Delta H^\circRT^2 ] MIT 5.60 trick: Integrate this assuming (\Delta H^\circ) constant for small (T) range, or use (\Delta H^\circ(T) = \Delta H^\circ_298 + \int_298^T \Delta C_p , dT). 8. Connection to Statistical Mechanics (MIT 5.62 Bridge) The partition function (Q(N,V,T)) relates to Helmholtz free energy: [ A = -kT \ln Q ] Then: [ P = -\left(\frac\partial A\partial V\right) T,N = kT \left(\frac\partial \ln Q\partial V\right) T,N ] [ U = kT^2 \left(\frac\partial \ln Q\partial T\right)_V,N ] [ S = k \ln Q + \fracUT ]
Equilibrium condition: [ \sum_i \nu_i \mu_i = 0 ] chemical thermodynamics mit
(const (T,P)): [ \sum_i N_i d\mu_i = 0 ] This is crucial for checking consistency of experimental data. 7. Chemical Equilibrium Consider a reaction (\sum_i \nu_i \mathcalM_i = 0) (with (\nu_i) > 0 for products, < 0 for reactants). (temperature dependence of (K)): [ \fracd\ln KdT =
For an ideal gas mixture: [ \mu_i(T,P) = \mu_i^\circ(T) + RT \ln\left(\fracP_iP^\circ\right) ] where (P_i = y_i P) (partial pressure). Two others from (dU) and (dH)
Two others from (dU) and (dH). These are for converting unmeasurable quantities (entropy change) into measurable ones (volume, pressure, temperature). 5. Chemical Potential & Phase Equilibria The chemical potential of species (i): [ \mu_i = \left(\frac\partial G\partial N_i\right) T,P,N j\neq i ] Phase Equilibrium Condition (MIT Classic Derivation) For two phases (\alpha) and (\beta) in contact: [ T^\alpha = T^\beta,\quad P^\alpha = P^\beta,\quad \mu_i^\alpha = \mu_i^\beta ] Clausius-Clapeyron Equation [ \fracdPdT = \frac\Delta H_\textvapT \Delta V ] Used for calculating vapor pressure vs. temperature. 6. Mixtures & Partial Molar Quantities Partial molar Gibbs free energy = chemical potential (\mu_i).
For non-ideal systems: [ \mu_i = \mu_i^\circ + RT \ln a_i ] where (a_i) = activity, and activity coefficient (\gamma_i = a_i / x_i) (for Raoult’s law basis).
From (dG = -SdT + VdP): [ -\left(\frac\partial S\partial P\right)_T = \left(\frac\partial V\partial T\right)_P ]