Phase Equilibria: Chemical Potential

• This course consists of 5 sessions of 1.5 hours each
• It will be followed by a section on statistical thermodynamics (same duration), with Florent Goujon
• Focus will be on the phase equilibria of pure substances and ideal mixtures
• The central concept of this part is the notion of chemical potential

Thermodynamics Reminders

Mathematical Requirements

• Requires manipulation of common functions: logarithm, exponential, derivatives (partial), and integrals
• Calculatory aspect should not hinder understanding
• Exercising in mathematical foundations is recommended
• Enumeration concepts will be helpful in statistical thermodynamics

State Variables

• In classical thermodynamics, a system's state is characterized by macroscopic state variables:
• nᵢ: quantity of matter of species i, SI unit: mol, 1 mol = Nₐ entities with Avogadro's number Nₐ = 6.022 140 76 × 10²³ mol⁻¹
• V: volume, SI unit: cubic meter, 1 m³ = 10³ L, 1 mL = 1 cm³
• T: temperature, represents kinetic energy of atoms (their motion), SI unit: Kelvin, 0 °C = 273.15 K
• p: pressure, represents shocks on walls (or passages through a virtual wall) and attractions or repulsions between atoms and the wall (or atoms across a virtual wall), force per unit area, or energy per unit volume, SI unit: Pascal
• 1 Pa = 1 N·m⁻² = 1 J·m⁻³
• 1 bar = 10⁵ Pa
• 1 atm = 1.01325 bar
• These quantities are relatively easy to determine experimentally
• Quantities proportional to system size are extensive, such as nᵢ and V
• They characterize the system globally and are defined even out of equilibrium
• Quantities independent of system size are intensive, such as T and p
• They are defined locally, may not be uniform, and may not make sense out of equilibrium
• The ratio of extensive quantities is intensive
• The density of a system A times larger is ρ' = m'/V' = (λm)/(λV) = m/V = ρ
• At equilibrium, intensive state variables are well-defined and uniform within a single phase
• A homogeneous system consists of a single phase, otherwise it is heterogeneous
• State variables characterize the system in a given configuration, at a given instant
• They do depend neither on the external environment nor on the system's history (past states and transformations)

State Functions

• Mathematical functions (continuous and sufficiently differentiable) whose variables are state variables.
• U: internal energy, SI unit: Joule, 1 cal = 4.184 J, microscopic potential and kinetic energy of the system, it is extensive, its variation is governed by the 1st law of thermodynamics
• H = U + pV: enthalpy, SI unit: Joule, it is extensive, simplifies the 1st law at constant p
• S: entropy, SI unit: Joule per Kelvin, it is extensive, interpreted as the degree of "disorder" in the system (number of possible microstates), its variation is governed by the 2nd law of thermodynamics
• F = U - TS: Helmholtz free energy (also denoted A), SI unit: Joule, it is extensive, its variation represents the maximum recoverable work during a transformation at fixed T and V (and nᵢ)
• G = H - TS = F + pV: Gibbs free energy, SI unit: Joule, it is extensive, its variation represents the maximum recoverable work during a transformation at fixed T and p (and nᵢ)
• These quantities are rather abstract and more difficult to measure experimentally
• State functions can be expressed in terms of state variables, such as F = F(nᵢ, T, V…)
• Like state variables, they do not depend on the system's history or environment
• Their functional form (exact expression, mathematical definition) is not always known
• State functions can generally be inverted (in the sense of an inverse function, y = f(x) ⇔ x = f⁻¹(y)), i.e., to invert the roles of a function and a state variable
• A function becomes a variable and a variable becomes a function
• U can be expressed as a function of S, V, and nᵢ: U = U(S, V, nᵢ)
• The differential of a function expresses how the function varies (for small, so-called infinitesimal variations) when its variables vary
• For a function f(x, y, z), df = (∂f/∂x)y,z dx + (∂f/∂y)x,z dy + (∂f/∂z)x,y dz

Chemical Potential

• For a pure substance, the chemical potential μ is the molar Gibbs free energy (μ = Ḡ = G/n)
• In a mixture, it corresponds to the partial molar Gibbs free energy (μ̄ᵢ = Ḡᵢ = (∂G/∂nᵢ)p,T,nⱼ≠ᵢ)
• The chemical potential represents the capacity of a system to donate molecules of species i
• It is an intensive quantity expressed in Joules per mole
• By definition of activity, μᵢ(T, p, nⱼ) = μᵢ⁰(T) + RT ln aᵢ
• μᵢ⁰(T) is the standard chemical potential, in a reference state where species i is pure (or the only solute in a solvent), under 1 bar (or at 1 mol·L⁻¹) and without interaction (ideal gas or infinite dilution)
• R = 8.314 462 618 153 24 J·K⁻¹·mol⁻¹ is the ideal gas constant

Ideal Gas

• An ideal gas consists of molecules without interactions
• Real gases tend to behave like ideal gases at low densities (≤ 10 bar) and/or high temperatures
• In an ideal gas at equilibrium, pV = nRT
• If the ideal gas is a mixture, pᵢ = (nᵢ/V)RT = (nᵢ/n)p = xᵢp
• pᵢ is the partial pressure of species i
• It is the pressure that species i would exert alone in volume V at temperature T
• xᵢ = nᵢ/n is the mole fraction of species i, with n = Σᵢ nᵢ
• p = Σᵢ pᵢ
• In an ideal gas, U and H depend only on T (given nᵢ)
• If p and V vary but T remains constant, then U and H are also constant
• For a transformation at fixed T (isothermal), pV is constant
• For a transformation at fixed S (isentropic, particularly adiabatic reversible), pVγ is constant
• γ is the adiabatic index (1.4 for a diatomic ideal gas)

First Law of Thermodynamics

• States that there exists a state function called internal energy, denoted U, such that for any transformation of a closed system
• ΔU = W + Q
• W is the work and Q is the heat (thermal transfer) received by the system during the transformation
• This is the principle of conservation of energy
• W and Q are not state functions as they characterize a transformation
• For a reversible transformation, δW = -pdV (work of pressure forces)
• For a transformation at constant V (isochoric), W = 0 and ΔU = Q
• For a transformation at constant p (isobaric), ΔH = Q

Second Law of Thermodynamics

• States that there exists a state function called entropy, denoted S, such that for any spontaneous transformation of a closed system
• ΔS ≥ Q/T
• This is the principle of evolution: the entropy of an isolated system increases spontaneously
• Entropy is not conserved but is created during an irreversible transformation
• When the entropy can no longer increase, the isolated system reaches a state of equilibrium
• For a reversible transformation, δQ = TdS
• If the transformation is also adiabatic (Q = 0), then it is isentropic (dS = 0)

Partial Molar Quantities

• For an extensive state quantity Y, the contribution of each species in a mixture can be defined via defining the partial molar Y of species i by Ȳᵢ = (∂Y/∂nᵢ)T, p, nⱼ≠ᵢ
• Derivative is taken at constant T, p, and all other quantities of matter
• Y = Σᵢ nᵢȲᵢ can be shown
• By posing Y = Y/n with n = Σᵢ nᵢ, it follows that Ȳ = Σᵢ xᵢȲᵢ
• By considering Y= f(T, p, nⱼ)
• For a system λ times larger Y' = λY = λf(T, p, nⱼ)
• Finally f(T, p, λnⱼ) = λf(T, p, nⱼ)

Gibbs-Duhem Relation

• Calculates the differential of an extensive state quantity Y expressed as a function of T, p, and nᵢ
• dY = (∂Y/∂T)p,nᵢ dT + (∂Y/∂p)T,nᵢ dp + Σᵢ (∂Y/∂nᵢ)T,p,nⱼ≠ᵢ dnᵢ
• At T, p constant, dT = 0 and dp = 0 , leaving dY = Σᵢ Ȳᵢ dnᵢ
• Differentiating Y = Σᵢ nᵢȲᵢ yields dY = Σᵢ Ȳᵢ dnᵢ + nᵢ dȲᵢ
• Identifying these two equations, results in the Gibbs-Duhem equation at fixed T, p
• Σᵢ nᵢ dȲᵢ = 0
• Dividing by n yields Σᵢ xᵢ dȲᵢ = 0
• Choosing, Y = G and Ȳᵢ = μᵢ this equation is often used in practice: Σᵢ nᵢ dμᵢ = 0

First Derivatives of State Functions

• Considering U as a function of S, V, and nᵢ, its differential is dU = (∂U/∂S)V,nᵢ dS + (∂U/∂V)S,nᵢ dV + Σᵢ (∂U/∂nᵢ)S,V,nⱼ≠ᵢ dnᵢ
• The 1st law states that for a closed system and constant amounts of matter, dU = δW + δQ
• Choosing a reversible path (when dnᵢ = 0) yields dU = T dS − pdV and identifying T = (∂U/∂S)V,nᵢ and p = −(∂U/∂V)S,nᵢ
• From the definition of H = U + pV, dH = dU + pdV + V dp = TdS + V dp is deduced
• In the same way
• dF = -S dT – pdV
• dG = −SdT + V dp
• By considering F as a function of T, V, and nᵢ and G as a function of T, p, and nᵢ, these examples are identified (non-exhaustive list)
• S = −(∂F/∂T)V,nᵢ
• V = (∂G/∂p)T,nᵢ
• Considering the possible variations of nᵢ with respect to G, dG = (∂G/∂T)p,nᵢ dT + (∂G/∂p)T,nᵢ +Σᵢ ∂G/∂nᵢ dnᵢ
• This implies that dG = −SdT + V dp + Σᵢ μᵢ dnᵢ
• Inverting the previous steps results in the entire series of thermodynamic identities
• dH = TdS + V dp + Σᵢ μᵢ dnᵢ
• dF = -SdT – pdV + Σᵢ μᵢ dnᵢ
• dU = T dS - pdV +Σᵢ μᵢ dnᵢ
• Thus, one can also define the chemical potential by
• μᵢ = (∂U/∂nᵢ)S,V,nⱼ≠ᵢ
• μᵢ = (∂F/∂nᵢ)T,V,nⱼ≠ᵢ
• μᵢ = (∂H/∂nᵢ)S,p,nⱼ≠ᵢ
• The chemical potential μᵢ cannot simply be stated as Ūᵢ, as Ūᵢ is the derivative at T, p constants and not S, V

Second Derivatives of State Functions

• A math theorem states that the order of partial derivatives can be inverted if they are well-defined, for a function f(x, y)
• Applying this equation in U(S, V, nᵢ ) results in (∂/∂S (∂U/∂V))s,nᵢ = (∂/∂V (∂U/∂S))v,nᵢ
• ((∂p/∂S)v,nᵢ =− (∂T/∂V)s,nᵢ
• With H, F, and G (Maxwell relations):
• (∂V/∂T)s,nᵢ = (∂S/∂p)T,nᵢ
• (∂p/∂S)v,nᵢ = (∂T/∂p)V,nᵢ
• (∂S/∂p)T,nᵢ = (−∂V/∂T)p,nᵢ

Gibbs-Helmholtz (GH) Relation

• Derivatives G/T with respect to T at fixed p and nᵢ dT ∂G/T/∂T = 1/T((∂G/∂T) − G/T
• Recognized to be H = G + TS, which can be translated as (∂G/T/∂T)p,nᵢ = −H/T²
• It can be further derived with respect to nᵢ at fixed T and p(∂μᵢ⁰/T/∂T)p,nⱼ = −Hᵢ⁰/T²*

Heat capacities

• Heat capacities express the relationship between the heat received and the temperature increase (excluding changes of state)
• Defined as Cv = (∂H/∂T)v,nᵢ and Cp = (∂H/∂T)p,nᵢ
• They are extensive quantities expressed in Joules per Kelvin and often used in molar versions
• Cv = nCv = mcv
• Cp = nCp = mcp
• For liquid water Cp ≈ 1 cal K⁻¹ g⁻¹
• For a reversible transformation at fixed V and nᵢ, dU = δQ = Cv dT = T dS
• Cv is then equal to T(∂S/∂T)v,nᵢ
• Similarly, for a reversible transformation at fixed p and nᵢ, dH = δQ = Cp dT = T dS
• Cp is then equal to T(∂S/∂T)p,nᵢ