Course map and thermodynamic language

State functions and the four potentials. Classical thermodynamics tracks a system through state functions — quantities fixed by the current state alone, independent of path: $U$, $H$, $S$, $A$ (Helmholtz energy), $G$ (Gibbs energy). Starting from the combined First and Second Law for a reversible process with only $PV$ work, $dU = TdS - PdV$, simple substitution ($H = U+PV$, $A = U-TS$, $G = H-TS$) generates the other three differentials:

where: $T$ is temperature, $S$ entropy, $P$ pressure, $V$ volume. Each potential has a specific pair of natural variables — $U:(S,V)$, $H:(S,P)$, $A:(T,V)$, $G:(T,P)$ — and differentiating a potential with respect to its natural variables directly yields the other conjugate quantity, which is why this particular pairing is chosen rather than any other.

Exactness. Each of these differentials is an exact differential: for a function $f(x,y)$ with $df = M\,dx + N\,dy$, the mixed partial derivatives must agree,

This equality holds automatically whenever $f$ is a genuine state function, because the mixed second partial derivative of any well-behaved function does not depend on the order of differentiation. It is the seed from which the four Maxwell relations (met in a later article) are generated — one exactness condition per potential.

Chemical potential. For systems that exchange matter — open systems, or closed systems undergoing reaction — one more variable per species is needed: the chemical potential $\mu_i$, defined as the partial molar Gibbs energy,

where: $n_i$ is the mole number of species $i$, held constant for all other species $j \ne i$ while $T$ and $P$ are also fixed. Adding this term to $dG$ gives the fundamental equation of the entire unit,

Phase equilibria will require equal $\mu_i$ across phases; reaction equilibria will require $\sum_i \nu_i \mu_i = 0$ at the reaction's stoichiometric coefficients $\nu_i$; and real-gas or real-solution corrections describe exactly how $\mu_i$ deviates from its ideal form. Every one of those later results is this one equation, specialised.

Partial molar quantities in general. Chemical potential is a special case of a broader idea: any extensive property $X$ of a mixture (volume, enthalpy, entropy, Gibbs energy) can be written $X = \sum_i n_i \bar{X}_i$, where $\bar{X}_i = (\partial X/\partial n_i)_{T,P,n_{j\ne i}}$ is the partial molar property of component $i$. $\mu_i$ is simply the partial molar Gibbs energy — the general machinery you will reuse for partial molar volume, partial molar enthalpy, and so on.

Express the Gibbs energy of a two-component ideal-gas mixture and identify $\mu_1, \mu_2$.

Step 1. Write the total Gibbs energy as additive over the two ideal gases (Gibbs' theorem for ideal mixtures), at fixed $T$, total pressure $P$, and mole numbers $n_1, n_2$:

where $p_i = x_iP$ is the partial pressure of species $i$, $x_i = n_i/(n_1+n_2)$ its mole fraction, and $\mu_i^\circ(T)$ the standard chemical potential at reference pressure $p^\circ$.

Step 2. Differentiate with respect to $n_1$ at constant $T, P, n_2$ to extract $\mu_1$. The mixing-entropy contribution from $x_1$ works out cleanly, giving

and, by the identical calculation with the labels swapped, $\mu_2 = \mu_2^\circ(T) + RT\ln(p_2/p^\circ)$.

Step 3. Read the physical content. Both results match the general ideal-gas form $\mu = \mu^\circ + RT\ln(p/p^\circ)$, applied component by component inside the mixture. Crucially, $\mu_i$ depends on the partial pressure $p_i$ of species $i$ alone, not on the total pressure $P$ — each species "feels" only its own concentration in the mixture.

Answer: $\mu_1 = \mu_1^\circ(T) + RT\ln(p_1/p^\circ)$ and $\mu_2 = \mu_2^\circ(T) + RT\ln(p_2/p^\circ)$, the ideal-mixture chemical potentials that later derive Raoult's law and vapour-phase equilibrium.

Competitive stretch

If a third inert ideal gas is added at partial pressure $p_3$ without reacting, $\mu_1$ and $\mu_2$ are unchanged in form — each still depends only on its own $p_i$ — but $x_1$ and $x_2$ (and hence $p_1, p_2$ at fixed total $P$) shift because the mole fractions must now sum to $1$ over three species instead of two. This is exactly why $\mu_i$, not $G$ itself, is the natural bookkeeping variable for multi-component systems.

• Treating $\mu_i$ as the molar Gibbs energy of pure $i$ — it is a partial molar quantity evaluated inside the actual mixture, and $\mu_i \ne G_{m,i}^{\text{pure}}$ except exactly at the reference state.
• Dropping the $T, P$ terms from $dG = -SdT + VdP + \sum_i \mu_i\,dn_i$ too early. At constant $T, P$ this correctly collapses to $dG = \sum_i \mu_i\,dn_i$ — the form used throughout equilibrium analysis — but only after $T, P$ are fixed, not before.
• Confusing extensive and intensive roles: $G$ is extensive (scales with system size), while $\mu_i$ is intensive (independent of how much material is present), exactly like $T$ and $P$.
• Assuming all four potentials share one set of natural variables. Each has its own specific pair — $U:(S,V)$, $H:(S,P)$, $A:(T,V)$, $G:(T,P)$ — and mixing these up breaks the exactness argument that generates the Maxwell relations.

Ask the tutor: if the substitution steps between $U, H, A, G$ or the partial molar quantity notation still feel shaky, ask the AI tutor to walk through two or three more examples — this vocabulary underpins every remaining topic in the thermodynamics unit.