Mathematical Structure: The Fundamental Relations

The previous lesson introduced entropy and the second law, from which we derived the fundamental relation $dU=T\,dS-P\,dV$ for a closed system. We will now explore what this relation tells us about the variables that describe a thermodynamic system and how they depend on one another. Here we consider a simple system: a pure substance (a single chemical component) that is homogeneous (the same local macroscopic properties at every point), with pressure work as its only form of work. Throughout the lesson, we will also show how this description extends to mixtures, which contain several chemical species.

1. Understanding the fundamental relation

Before interpreting this relation, let us extend it to open systems so that we can deal directly with cases in which the amount of matter can vary.

1.1. Extension to open systems

In Lesson 6, we considered a locally reversible path between two nearby equilibrium states of a closed system to obtain $\delta Q_{\mathrm{rev}}=T\,dS$ and $\delta W_{\mathrm{rev}}=-P\,dV$, and then

using the first law. In an open system, the number of particles $N$ can vary. The differential of the energy then contains an additional term:

Notice that we treat $N$ as a continuous variable, although strictly speaking it is an integer. This approximation is appropriate for macroscopic systems, for which $N\gg1$.

Despite its name, the coefficient $\mu$ plays a role even in the absence of chemical reactions. To understand what it measures, let us hold $S$ and $V$ fixed: this leaves $dU=\mu\,dN$. The chemical potential therefore measures the change in internal energy per particle added, at constant entropy and volume. It has the dimensions of energy.

We can build an intuition by analogy with temperature: just as heat flows spontaneously from a hot medium to a cold one, particles of a species tend, at a common temperature, to move from the medium where their chemical potential is higher to the one where it is lower, provided the partition allows them to pass. Equality of chemical potentials is therefore a condition for equilibrium with respect to matter exchange, just as equality of temperatures is a condition for thermal equilibrium. We will prove this explicitly in Section 6.

1.2. Fundamental variables and equations of state

We now have many thermodynamic variables: $U$, $S$, $V$, $N$, $T$, $P$ and $\mu$. This proliferation often confuses students. A natural question arises: can these variables be chosen independently? If not, which ones must be specified to determine the others? We have already seen that some are linked by an equation of state, as in the case of an ideal gas. But how many equations of state does a simple system have, and how many genuinely independent variables remain?

The fundamental relation answers this crucial question. Recall the definition of the differential of a function $f(x,y,z)$:

Comparing this with the relation $dU=T\,dS-P\,dV+\mu\,dN$ leads to two consequences.

It is essential to understand the role of the different variables in $dU=T\,dS-P\,dV+\mu\,dN$. The presence of $T$, $P$ and $\mu$ does not mean that they must be added to $S$, $V$ and $N$ as three further independent variables. They are the coefficients of the differential, and are themselves functions of $S$, $V$ and $N$: $T=T(S,V,N)$, and similarly for $P$ and $\mu$. The nontrivial content of the fundamental relation is that these three functions are derivatives of the same function $U(S,V,N)$. Their variations are therefore already accounted for through those of $S$, $V$ and $N$; they do not add three independent variables to the description.

The above shows that the seven parameters we have introduced are not all independent. For a given simple system, three extensive parameters, $S$, $V$ and $N$, suffice to determine the equilibrium state: the function $U(S,V,N)$ gives the internal energy, and its three derivatives give the temperature, pressure and chemical potential.

We must therefore distinguish specifying the parameters $S, V, N$, which fixes the state of the system, from specifying the function $U(S,V,N)$, which describes the system itself. This function is called the system's fundamental equation in the energy representation. The differential relation (1) is satisfied by all simple systems, but the function $U$ differs from one system to another: it does not have the same expression for an ideal gas, a real gas, a liquid or a solid.

1.3. Example of a monatomic ideal gas

Consider a monatomic ideal gas, for which we already know the expression

This gives $U$ as a function of $T$ and $N$. To write it in the form $U(S,V,N)$, we still need to express $T$ as a function of $S$, $V$ and $N$. Until this dependence is known, we cannot directly calculate the last two equations of state using the derivatives of $U$ above. In particular, it would be a mistake to conclude that the pressure is zero because $V$ does not appear explicitly in the expression for $U$: at fixed $S$ and $N$, $T$ depends on volume. Similarly, calculating $\mu$ requires accounting for the dependence of $T$ on $N$ at fixed $S$ and $V$. The equations of state actually give

We will calculate these partial derivatives in Section 5: the first recovers the equation of state $PV=Nk_BT$, while the second gives the chemical potential of the ideal gas.

1.4. The Maxwell relations

The Maxwell relations are thermodynamic identities, valid regardless of the substance under consideration. They follow from the equality of mixed partial derivatives: if the second derivatives of $U$ are continuous, Schwarz's theorem allows the order of differentiation to be interchanged.

Let us apply it to the variables $S$ and $V$, at fixed $N$:

The left-hand side means that we first differentiate $U$ with respect to $S$, at constant $V$ and $N$, then differentiate the resulting function with respect to $V$, at constant $S$ and $N$. According to the equations of state (2), $\drp{U}{S}{V,N}=T$. This side therefore equals

On the right, we first differentiate $U$ with respect to $V$, at constant $S$ and $N$, then differentiate the resulting function with respect to $S$, at constant $V$ and $N$. Since $\drp{U}{V}{S,N}=-P$, we obtain

Equating the two sides thus gives the first Maxwell relation:

Proceeding in the same way with the pairs $(S,N)$ and $(V,N)$ yields two further relations. The three Maxwell relations in the energy representation are therefore

These relations follow directly from the fundamental relation when $U$ is at least twice continuously differentiable. They constitute a strong prediction of thermodynamics that is independent of the model: any substance described by such a fundamental function must satisfy them (under these regularity assumptions on $U$).

Conversely, if the functions $T$, $P$ and $\mu$ satisfy the Maxwell relations (and are at least $C^1$), then a function $U$ exists locally such that $dU=T\,dS-P\,dV+\mu\,dN$. We can therefore reconstruct the fundamental function from the equations of state by integration, up to an additive constant.

This reverse approach is useful in practice: when studying a substance in the laboratory, we do not initially know its fundamental function $U(S,V,N)$. Measuring entropy itself is not straightforward; we will return to this in the next lesson. What experiments provide are usually relations between measurable quantities, such as pressure, volume and temperature. As we saw in the historical lesson, the ideal gas equation $PV=Nk_BT$ was first established experimentally. These relations must then be integrated to reconstruct the fundamental equation of the substance under study. We will carry out this reconstruction for the ideal gas in Section 5, thereby finding its fundamental equation $U(S,V,N)$.

2. The entropy representation

In the energy representation, $U$ is the dependent variable: we express it as a function of $S$, $V$ and $N$. We can also choose entropy as the dependent variable, provided $U(S,V,N)$ can be inverted to give $S(U,V,N)$ at fixed $V$ and $N$. In what follows, we will work in domains where this inversion is possible, without repeating the assumption each time.

Solving the fundamental relation for $dS$, with $T\ne0$, gives

This is the fundamental relation in the entropy representation. By the same reasoning as before, the natural variables of $S$ are therefore $U$, $V$ and $N$:

Identifying the coefficients of the differential again gives three equations of state

Be careful: these are not three equations of state independent of those obtained in the energy representation. They express the same information, simply using a different choice of variables. The function $S(U,V,N)$ is called the fundamental equation in the entropy representation. It contains the same information as $U(S,V,N)$.

Equality of the mixed derivatives of $S$ gives the Maxwell relations associated with this representation:

These relations are not independent of those obtained in the energy representation; they are simply the ones suited to this choice of variables.

3. The role of extensivity

We still know only two explicit equations of state for the ideal gas. This is not enough to reconstruct the entire fundamental equation by integration, as proposed in Section 1.4. However, we can use an additional property: extensivity. It allows us to apply Euler's theorem to $U$ or $S$ and derive a new nontrivial relation, the Gibbs-Duhem relation.

We study it here in general before applying it to the ideal gas.

3.1. Extensivity and Euler's theorem

If $f$ is differentiable, Euler's theorem states that

We discussed the extensivity of $U$ and $S$ in Lessons 4 and 6. This property applies to macroscopic systems in which surface effects and long-range interactions can be neglected. We assume this is the case here. We then have:

That is, $U$ and $S$ are homogeneous functions of degree one. Applying Euler's theorem to $U(S,V,N)$ with $k=1$ therefore gives:

Substituting the equations of state (2) then gives the Euler relation:

3.2. The Gibbs-Duhem relation

The Euler relation leads to an important thermodynamic identity for extensive systems. Differentiating it gives

Regrouping terms on the left, we obtain:

The fundamental relation shows that the left-hand side vanishes. This leaves the relation

called the Gibbs-Duhem relation. It shows that the three intensive variables $T$, $P$ and $\mu$ cannot vary independently in a homogeneous phase consisting of a single species.

4. Intensive degrees of freedom

We have described a simple system using three extensive variables, $S$, $V$ and $N$. When the system is extensive, these three variables reduce to two intensive variables for describing its intensive state. Indeed, extensivity immediately gives

so the energy per particle $u=U/N$ is a function of only two intensive variables, $s=S/N$ and $v=V/N$.

Remember that describing the complete state of the system also requires specifying its size, for example by giving $N$, or another suitable extensive quantity. The two intensive degrees of freedom must therefore not be confused with the three extensive variables needed to describe the complete state.

We will extend this count to systems with several components and several phases in Lesson 10, where we will establish Gibbs' phase rule.

5. The fundamental equation of the ideal gas

5.1. The Sackur-Tetrode equation

The equations for a monatomic ideal gas are $PV=Nk_BT$ and $U=\frac{3}{2}Nk_BT$, where $k_B$ is Boltzmann's constant, related to the ideal gas constant $R$ by $k_B=R/N_A$, with $N_A$ being Avogadro's number. These two equations and the Gibbs-Duhem relation allow us to find the fundamental equation of the ideal gas.

It is easier to find $S$ than $U$. We therefore start from the first equation of state in the entropy representation:

Integrating with respect to $U$ at fixed $V$ and $N$, using an arbitrary reference energy $U_0>0$, gives

where $F$ is an as yet unknown function. The argument of the logarithm is thus dimensionless; $U_0$ will be combined with the other constants at the end. The second equation of state then gives

Substituting (9) and integrating with respect to $V$ gives

where $V_0>0$ is likewise an arbitrary reference volume, and therefore

It remains to determine $Z(N)$ using the Gibbs-Duhem relation. The calculation is given in the proof below. Up to a constant $A$, we obtain

where $A$ is a dimensional constant, independent of $U$, $V$ and $N$. Up to the value of $A$, this is the Sackur-Tetrode formula, derived independently by Otto Sackur and Hugo Tetrode in 1912¹.

5.2. Recovering the equations of state

Let us check that the fundamental equation $S=S(U,V,N)$ given in (10) recovers the two equations of state from which we started. Differentiating with respect to $U$ and $V$, we find

and

We do indeed recover $U=3Nk_BT/2$ and $PV=Nk_BT$. All that remains is to calculate the chemical potential $\mu=-T\drp{S}{N}{U,V}$. Evaluating the derivative gives (verify this):

6. Equilibrium and stationarity of entropy

In Lesson 6, in the paragraph “Entropy and equilibrium”, we assumed that the entropy of an isolated composite system is stationary at equilibrium: its first-order variation vanishes when the system's free parameters are varied slightly while respecting the imposed constraints. Let us repeat this calculation for two subsystems that can also exchange particles. Each is assumed to be in internal equilibrium, and contributions from their interface are neglected. The composite system is isolated: its energy $U$, volume $V$ and particle number $N$ are fixed. The constraints therefore read

The parameters of each subsystem may nevertheless vary, provided their changes compensate one another:

The partition may impose additional constraints: if it is fixed, $dV_1=0$; if it is impermeable, $dN_1=0$. We must therefore specify the allowed exchanges to determine which parameters remain free.

Expressing the parameters of the second subsystem using the constraints makes the total entropy a function of $U_1$, $V_1$ and $N_1$:

The fundamental relation gives

If the three free parameters $U_1$, $V_1$ and $N_1$ can vary independently in both directions near equilibrium, stationarity requires the coefficient of each variation $dU_1$, $dV_1$ and $dN_1$ to vanish. We obtain in turn:

Equality of chemical potentials thus complements the thermal and mechanical equilibrium conditions already encountered. These conditions apply only to allowed exchanges: an impermeable partition, for example, imposes $dN_1=0$ and therefore does not require the last condition at equilibrium.

For a permeable but fixed partition, note that at a common temperature, $T_1=T_2=T$, we simply have

which shows that if $\mu_2>\mu_1$, a transfer of particles from 2 to 1 ($dN_1>0$) increases entropy and therefore occurs spontaneously. As stated earlier, at a uniform temperature, matter tends to move from regions of higher chemical potential to regions of lower chemical potential, until the chemical potentials are equal at equilibrium.

7. References