In thermodynamics, some properties depend only on the current state of a system, not on how it got there. We call these state functions — temperature, pressure, and volume are classic examples. It doesn't matter if you heated a gas quickly or slowly; its final temperature is just its final temperature. Internal energy ($U$) and enthalpy ($H$), which you've already met, also belong to this club. They provide a snapshot of a system's energy content.

Other quantities, like heat ($Q$) and work ($W$), are path functions. The amount of work done or heat transferred depends entirely on the specific process taken between two states. This distinction is crucial. State functions are powerful because their changes are easy to calculate: just subtract the initial value from the final value. This simplicity is why we build our most important frameworks around them.

Internal energy and enthalpy are useful, but they don't tell the whole story. They account for the total energy, but not all of that energy is necessarily available to do useful work. Some of it is inevitably lost as disordered, thermal energy, related to entropy ($S$). To get a clearer picture of a system's potential, we need two new functions that account for this entropic cost.

The Helmholtz free energy ($A$) represents the maximum work that can be extracted from a closed system at a constant temperature ($T$) and volume ($V$). Think of it as the

The second is the Gibbs free energy ($G$). This is arguably the most important potential in chemistry and materials science, as it describes the maximum non-expansion work obtainable from a system at constant temperature ($T$) and pressure ($P$). Most chemical reactions in a lab happen in an open beaker, under constant atmospheric pressure, making Gibbs energy the perfect tool to predict their spontaneity.

For a process at constant temperature and pressure, if the change in Gibbs free energy ($ΔG$) is negative, the process will happen spontaneously. If it's positive, it won't. If it's zero, the system is at equilibrium.

We can combine the first and second laws of thermodynamics into a single, powerful equation. The first law, for a reversible process, can be written in differential form as:

$dU = dQ_{rev} - dW$

From the second law, we know that the change in entropy is defined by reversible heat transfer: $dS = dQ_{rev} / T$, or $dQ_{rev} = TdS$. The work done by the system expanding is $dW = PdV$. Substituting these into the first law gives us the Fundamental Thermodynamic Relation:

This compact equation is the cornerstone of thermodynamic analysis. It contains a huge amount of information, connecting the five key state variables: $U$, $T$, $S$, $P$, and $V$. More importantly, we can use it to derive similar relations for our other potentials. By taking the differentials of the definitions for H, A, and G, we get a full set of fundamental equations:

For Enthalpy ($H = U + PV$): $dH = dU + PdV + VdP = (TdS - PdV) + PdV + VdP$

For Helmholtz Free Energy ($A = U - TS$): $dA = dU - TdS - SdT = (TdS - PdV) - TdS - SdT$

For Gibbs Free Energy ($G = H - TS$): $dG = dH - TdS - SdT = (TdS + VdP) - TdS - SdT$

The true power of these fundamental equations comes from a mathematical property of exact differentials. Since $U, H, A,$ and $G$ are all state functions, their differentials are exact. This means that for a function $f(x,y)$, the order of partial differentiation doesn't matter:

$(\frac{\partial}{\partial y}(\frac{\partial f}{\partial x}))_x = (\frac{\partial}{\partial x}(\frac{\partial f}{\partial y}))_y$

Applying this rule, known as Euler's reciprocity relation, to our four fundamental equations yields a set of surprising and incredibly useful relationships called the Maxwell relations . They connect the partial derivatives of entropy, temperature, pressure, and volume, allowing us to calculate quantities that are difficult to measure (like a change in entropy) from ones that are easy to measure (like changes in pressure and temperature).

Let's derive one from the fundamental relation for internal energy, $dU = TdS - PdV$. From this, we can see that $T = (\frac{\partial U}{\partial S})_V$ and $P = -(\frac{\partial U}{\partial V})_S$. Applying the reciprocity relation:

Substituting our expressions for $T$ and $P$ gives the first Maxwell relation:

Applying the same logic to the other three fundamental equations gives the complete set of four relations.

These relationships form a powerful toolkit for the practical analysis of thermodynamic systems.