Thermodynamics III Notes

Helmholtz and Gibbs Free Energies

• Two thermodynamic functions: Helmholtz energy (A) and Gibbs free energy (G).

• Used to predict spontaneity of reactions, more convenient than entropy.

Defining Helmholtz and Gibbs Free Energies

• Helmholtz energy: A ≡ U − TS. At constant V and T, \Delta A = \Delta U − T\Delta S \le 0.

• Gibbs free energy: G ≡ H − TS. At constant P and T, \Delta G = \Delta H − T\Delta S \le 0.

• Gibbs free energy ($\Delta G$) is more useful because most reactions occur at constant P.

Maximum Work Done

• Maximum work done for reversible, isothermal change: w_{max} = \Delta A

• \Delta G equals maximum free work at constant T & P.

• \Delta G \le w_{non-expansion}, which is the theoretical limit of free work.

Standard Gibbs Energy

• \Delta G^o = \Delta H^o − T\Delta S^o

• \Delta G_f^o for elements in their most stable form is zero.

• \Delta Gr^o = \sum n \Delta Gf^o(products) − \sum m \Delta G_f^o(reactants), where n and m are stoichiometric coefficients.

Spontaneity and Signs of \Delta H and \Delta S

• \Delta G = \Delta H − T\Delta S. Temperature contribution is through entropy.

• Spontaneity depends on the signs of \Delta H and \Delta S and temperature.

U, H, A, and G

• Internal Energy: dU = TdS − PdV

• Enthalpy: dH = TdS + VdP

• Helmholtz: dA = −SdT − PdV

• Gibbs energy: dG = −SdT + VdP

Effect of P and T on Gibbs Free Energy

• dG = (\frac{\partial G}{\partial T})P dT + (\frac{\partial G}{\partial P})T dP

• \frac{\partial G}{\partial T}_P = -S

• \frac{\partial G}{\partial P}_T = V

Effect of P Change to G for a Pure System

• For ideal gases: G(P) = G^o + nRT \ln \frac{P}{P_o}

Gibbs Free Energy with Composition Changes

• dG = −SdT + VdP + \mu1dn1 + \mu2dn2

Chemical Potential

• \mu = (\frac{\partial G}{\partial n})_{P,T}

• For a pure substance: \mu = G_M

• \mu(P) = \mu^o + RT \ln \frac{P}{P_o}

Chemical Potential in Spontaneous Diffusion

• Molecules move from high to low concentration (or chemical potential).

• Equilibrium is reached when chemical potentials are equal.

Mixing of Two Gases

• \Delta G{mix} = nRT(xA \ln xA + xB \ln x_B)

• \Delta S{mix} = −nR(xA \ln xA + xB \ln x_B)

• Mixing is spontaneous because \Delta G_{mix} < 0

\Delta G for a Non-Equilibrium Mixture

• \Delta Gr = \Delta Gr^o + RT \ln Q_P

\Delta Gr, \Delta Gr^o, and Q

• \Delta G_r < 0: forward reaction spontaneous.

• \Delta G_r > 0: backward reaction spontaneous.

• \Delta G_r = 0: equilibrium.

• \Delta G_r^o = −RT \ln K

Le Chatelier’s Principle

• An equilibrium mixture would shift in a direction that counteracts the change.

Effect of T change to an Equilibrium

• \frac{\partial \ln K_P}{\partial T} = \frac{\Delta H^o}{RT^2}

Effect of P change to an Equilibrium

• KP = Kx(\frac{P}{P_o})^{\Delta n}

Maxwell Equations

• \left(\frac{\partial T}{\partial V}\right)S = -\left(\frac{\partial P}{\partial S}\right)V

• \left(\frac{\partial T}{\partial P}\right)S = \left(\frac{\partial V}{\partial S}\right)P

• \left(\frac{\partial S}{\partial V}\right)T = \left(\frac{\partial P}{\partial T}\right)V

• \left(\frac{\partial S}{\partial P}\right)T = -\left(\frac{\partial V}{\partial T}\right)P