Thermodynamics III Notes

Helmholtz and Gibbs Free Energies

• Spontaneity of reactions can be predicted using thermodynamic functions, specifically Helmholtz energy (A) and Gibbs free energy (G).

• Helmholtz energy is defined as A ≡ U − TS, and Gibbs free energy is defined as G ≡ H − TS.

\DeltaA and \DeltaG

• At constant volume and temperature, \Delta A = \Delta U - T\Delta S. At constant pressure and temperature, \Delta G = \Delta H - T\Delta S.

• \Delta G is more useful because most reactions occur at constant P rather than constant V.

• \Delta A ≤ 0 (spontaneous/reversible) and \Delta G ≤ 0 (spontaneous/reversible).

\DeltaA is the Maximum Work Done

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

What is “free” in Gibbs Free Energy?

• \Delta G equals the maximum free work that can be done by a system at constant T & P.

• \Delta G ≤ w_{non-expansion}, where w is the sum of expansion work and non-expansion work.

Standard Gibbs Energy of Formation (\Delta Gf^o) and Reaction (\Delta Gr^o)

• \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.

How the Signs of \DeltaH and \DeltaS Affect Spontaneity?

• \Delta G = \Delta H − T\Delta S, spontaneity depends on the signs of \Delta H and \Delta S, and temperature.

• Melting of ice is T dependent because \Delta H > 0 and \Delta S > 0.

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 changes on Gibbs Free Energy

• dG = (\frac{\partial G}{\partial T})P dT + (\frac{\partial G}{\partial P})T dP, where \left(\frac{\partial G}{\partial T}\right)P = -S and \left(\frac{\partial G}{\partial P}\right)T = V.

Effect of P Change to G for a Pure System

• At constant T: dG = VdP. For ideal gases: G(P) = G^o + nRT \ln(\frac{P}{P_o}).

Gibbs Free Energy when Composition Changes

• dG = -SdT + VdP + \mu1dn1 + \mu2dn2, where \mu is chemical potential.

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

Chemical Potential (\mu)

• For a pure substance: \mu = G_M (molar Gibbs free energy).

• \mu(P) = \mu^o + RT \ln(\frac{P}{P_o}), where \mu^o is the standard chemical potential.

Chemical Potential in Spontaneous Diffusion

• Molecules move from high to low chemical potential until equilibrium (\muI = \mu{II} or PI = P{II}).

Mixing of two Gases at Constant T and P

• \Delta G{mix} = nRT(xA \ln xA + xB \ln xB), where xA and x_B are mole fractions.

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

\DeltaG for a non-Equilibrium or Reaction Mixture

• \Delta Gr = \Delta Gr^o + RT \ln Q_P, where Qp is the reaction quotient of pressure.

\DeltaGr, \DeltaGor and Q

• \Delta Gr < 0 (forward reaction spontaneous), \Delta Gr > 0 (backward reaction spontaneous), \Delta G_r = 0 (equilibrium).

• \Delta G_r^o = -RT \ln K.

Why does a non-Equilibrium Mixture Tend to Equilibrate?

• Non-equilibrium mixtures minimize Gibbs free energy to reach equilibrium.

Change of S and G with time for a Spontaneous Process

• Spontaneous direction: delocalization of energy, increase in S, decrease in G.

Le Chatelier’s principle

• An equilibrium mixture shifts to counteract any change.

Effect of T change to an Equilibrium

• \frac{\partial \ln KP}{\partial T} = \frac{\Delta Hr^o}{RT^2}. Decreasing T favors exothermic reactions, increasing T favors endothermic reactions.

Effect of P change to an Equilibrium

• KP = Kx (\frac{P}{P_o})^{\Delta n}, where \Delta n is the change in the number of moles of gas species.

The values of \DeltaGo and KP are dependent on T

• \frac{\partial}{\partial T} (\frac{G}{T})P = -\frac{H}{T^2}. van’t Hoff equation: \ln KP(T2) = \ln KP(T1) - \frac{\Delta Hr^o(T1)}{R} (\frac{1}{T2} - \frac{1}{T_1}).

Maxwell Equations of U, H, A and G

• \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

Expression for Internal Pressure (\pi_T)

• \piT = \left(\frac{\partial U}{\partial V}\right)T = T \left(\frac{\partial P}{\partial T}\right)_V - P = 0 for ideal gases.