Thermodynamic Potentials & Cryogenics

1. Mathematical Scaling of Thermodynamic Functions

A state variable $X$ is extensive if scaling the system size by a positive parameter $\lambda$ scales the variable by $\lambda$: $X(\lambda N, \lambda V) = \lambda X(N, V)$. A state variable $Y$ is intensive if it remains invariant: $Y(\lambda N, \lambda V) = Y(N, V)$.

Let the fundamental relation for internal energy be $U = U(S, V, N)$. Because $S, V, N$ are all extensive:

Differentiating with respect to $\lambda$ via the chain rule:

Setting $\lambda = 1$:

Recalling definitions of temperature $T = (\partial U/\partial S)_{V, N}$, hydrostatic pressure $P = -(\partial U/\partial V)_{S, N}$, and chemical potential $\mu = (\partial U/\partial N)_{S, V}$, we obtain the fundamental Euler Equation:

2. The Gibbs-Duhem Relation

Taking the total differential of the Euler equation:

Subtracting the fundamental differential relation $dU = T dS - P dV + \mu dN$:

This is the Gibbs-Duhem Relation. It demonstrates that the intensive parameters $(T, P, \mu)$ of a single-component system cannot all vary independently; varying any two strictly determines the third.

1. Legendre Transformations

In experimental physics, controlling extensive entropy $S$ is difficult; controlling intensive temperature $T$ using a thermal bath is straightforward. Legendre transformations systematically replace an extensive variable with its conjugate intensive derivative without losing any fundamental thermodynamic information.

2. The Four Potentials and Natural Variables

1. Internal Energy $U(S, V)$:

Natural variables: $(S, V, N)$. Conjugate relations: $T = \left(\frac{\partial U}{\partial S}\right)_{V}$, $P = -\left(\frac{\partial U}{\partial V}\right)_{S}$.

2. Enthalpy $H(S, P) \equiv U + P V$:

Natural variables: $(S, P, N)$. Conjugate relations: $T = \left(\frac{\partial H}{\partial S}\right)_{P}$, $V = \left(\frac{\partial H}{\partial P}\right)_{S}$. Enthalpy represents heat transferred under constant pressure ($dH = \delta Q_P$).

3. Helmholtz Free Energy $F(T, V) \equiv U - T S$:

Natural variables: $(T, V, N)$. Conjugate relations: $S = -\left(\frac{\partial F}{\partial T}\right)_{V}$, $P = -\left(\frac{\partial F}{\partial V}\right)_{T}$. Helmholtz free energy represents the maximum reversible work extractable from a system during an isothermal process: $W_{\max} = -\Delta F$.

4. Gibbs Free Energy $G(T, P) \equiv H - T S = U + P V - T S$:

Natural variables: $(T, P, N)$. Conjugate relations: $S = -\left(\frac{\partial G}{\partial T}\right)_{P}$, $V = \left(\frac{\partial G}{\partial P}\right)_{T}$. From Euler's relation $U = TS - PV + \mu N$, we obtain $G = \mu N$, proving that chemical potential $\mu = G/N$ is simply molar/molecular Gibbs free energy.

1. Thermodynamic Equilibrium Criteria

From the Clausius inequality for a non-isolated system in contact with surroundings at temperature $T_0$ and pressure $P_0$:

Applying this inequality to constrained systems generates specific extremum principles:

1. Isolated System ($U, V$ constant):

2. System at Constant Entropy and Volume ($S, V$ constant):

3. System at Constant Temperature and Volume ($T = T_0, V$ constant):

4. System at Constant Temperature and Pressure ($T = T_0, P = P_0$):

2. Practical Importance of Gibbs Energy Minimum

Most real-world chemical reactions, phase transitions, and biological processes occur at constant atmospheric pressure and ambient temperature. Therefore:

1. Interfacial Work and Surface Free Energy

Creating an interface of area $A$ requires pulling molecules from the bulk liquid against attractive intermolecular cohesive forces to the surface. The work performed to expand surface area by $dA$ at surface tension $\gamma$ is:

Including interfacial work in the fundamental First Law relation:

For the Helmholtz free energy $F = U - TS$:

At constant temperature and volume:

Thus, surface tension is the surface Helmholtz/Gibbs free energy per unit area.

2. Temperature Dependence of Surface Tension & Total Surface Energy

Applying Maxwell's cross-derivative reciprocity to $dF$:

Since surface tension decreases monotonically with temperature ($d\gamma/dT < 0$), surface entropy per unit area $s_A = -d\gamma/dT$ is positive; creating new surface area absorbs heat isothermally ($q_A = T s_A = -T d\gamma/dT$).

The Total Surface Internal Energy Density $u_A = U_A / A$ is:

Because $d\gamma/dT < 0$, total surface energy $u_A$ is strictly greater than surface tension $\gamma$.

1. Thermodynamics of Magnetic Systems

For a paramagnetic material placed inside an external magnetic field $\vec{H}$, work done by the magnetic field generator to alter total sample magnetic dipole moment $\vec{M}$ is:

The fundamental thermodynamic relation becomes:

Neglecting small magnetostrictive volume changes ($dV \approx 0$):

The magnetic Gibbs free energy is defined as $G_{\text{mag}} = U - TS - \mu_0 HM$:

2. Curie's Law and Spin Entropy

For an ideal paramagnetic salt (e.g., cerium magnesium nitrate, gadolinium sulfate), the magnetic dipoles follow Curie's law:

Applying Maxwell's cross-derivative to $dG_{\text{mag}}$:

Applying an external magnetic field forces randomly tumbling atomic spins to align parallel to $\vec{H}$, drastically reducing spin orientation entropy ($S_{\text{spin}}$).

3. Cooling by Adiabatic Demagnetization

Adiabatic demagnetization (proposed by Debye and Giauque in 1926) achieves sub-millikelvin temperatures through a two-stage thermodynamic sequence:

1. Isothermal Magnetization ($A \to B$): The paramagnetic salt is immersed in a liquid helium bath at $T_i \approx 1.0\text{ K}$. A powerful magnetic field ($H_i \sim 2\text{ to } 5\text{ T}$) is applied. The spins align, releasing heat of magnetization $Q = T_i \Delta S_{\text{mag}}$ to the helium bath via helium exchange gas.

2. Adiabatic Demagnetization ($B \to C$): The exchange gas is pumped out to thermally isolate the salt. The external magnetic field is reduced slowly to zero ($H \to 0$).

Because the process is isentropic ($\Delta S_{\text{total}} = 0$):

As $H \to 0$, the magnetic spins disorder thermally, absorbing entropy and heat from the crystal lattice vibrations, causing the lattice temperature to plummet:

where $h_{\text{int}} \sim 0.01\text{ T}$ is the small internal local dipole field of the crystal. By this technique, temperatures down to $10^{-3}\text{ K}$ (electronic demagnetization) and $10^{-6}\text{ K}$ (nuclear demagnetization of copper nuclei) are routinely achieved.