Thermodynamics: Thermodynamic Relations

Thermodynamics becomes powerful when you can move confidently between properties that are hard to measure and properties that are easy to measure. Thermodynamic relations are the mathematical tools that make this possible. They connect temperature, pressure, volume, entropy, enthalpy, and other state variables in ways that allow advanced analysis of real systems, from steam power plants to refrigeration cycles and chemical processes.

This article focuses on three pillars of practical thermodynamic relations: the Gibbs equations, Maxwell relations, and the disciplined use of property tables.

Why thermodynamic relations matter

Most thermodynamic problems are not solved by measuring everything directly. Entropy is the classic example. It is central to the second law, but it is not measured with a sensor in the way temperature or pressure is. Similarly, internal energy and Helmholtz free energy are rarely measured directly. Instead, engineers and scientists use relations that express changes in these quantities in terms of measurable variables such as $T$, $P$, and $v$.

Two ideas make this work:

• State functions and exact differentials: For a simple compressible system in equilibrium, properties like $u$, $h$, $s$, and $g$ depend only on the state. Their differentials are exact.
• Equality of mixed partial derivatives: If a function is well-behaved, then $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$. This mathematical fact underpins the Maxwell relations.

The Gibbs equations: a bridge to entropy relations

For a simple compressible system, the combined first and second laws yield fundamental differential relations. Two especially useful forms are the Gibbs equations:

$$Tds=du+Pdv$$

$$Tds=dh-vdP$$

These are not “new laws” but rearrangements of the fundamental definitions of internal energy $u$, enthalpy $h=u+Pv$, and the second-law relation $Tds=\delta q_{rev}$. Their practical value is that they express entropy changes in terms of changes in $u$, $v$, $h$, and $P$.

How the Gibbs equations are used

1. Relating entropy changes to measurable variables

If you can express $du$ or $dh$ in terms of $T$, $P$, and $v$, you can integrate to obtain $s(T,P)$ or $s(T,v)$. For ideal gases, this leads to familiar results such as entropy changes expressed through logarithms of temperature and pressure ratios.

1. Deriving property relations for real substances

For real fluids, $u(T,v)$ or $h(T,P)$ may not have simple analytic forms. The Gibbs equations still provide the structure needed to compute entropy changes using tabulated data, equations of state, or measured caloric properties.

1. Building “Tds relations”

Thermodynamic analysis often needs $ds$ in terms of $dT$ and $dP$ (or $dT$ and $dv$). Starting from the Gibbs equations and substituting suitable expressions for $du$ and $dh$ gives forms that are convenient for integration along a process path.

Maxwell relations: extracting information from potentials

Maxwell relations come from thermodynamic potentials and the symmetry of second derivatives. For a simple compressible system, the key potentials and their differentials are:

• Internal energy: $du=Tds-Pdv$
• Enthalpy: $dh=Tds+vdP$
• Helmholtz free energy: $da=-sdT-Pdv$ where $a=u-Ts$
• Gibbs free energy: $dg=-sdT+vdP$ where $g=h-Ts$

Each differential identifies “natural variables.” For example, $g$ naturally depends on $(T,P)$, while $a$ depends on $(T,v)$. Applying equality of mixed partials yields the four classic Maxwell relations:

$${\left(\frac{\partial T}{\partial v}\right)}_s=-{\left(\frac{\partial P}{\partial s}\right)}_v$$

$${\left(\frac{\partial T}{\partial P}\right)}_s={\left(\frac{\partial v}{\partial s}\right)}_P$$

$${\left(\frac{\partial s}{\partial v}\right)}_T={\left(\frac{\partial P}{\partial T}\right)}_v$$

$${\left(\frac{\partial s}{\partial P}\right)}_T=-{\left(\frac{\partial v}{\partial T}\right)}_P$$

Why Maxwell relations are practical (not just elegant)

Maxwell relations allow you to replace derivatives involving entropy with derivatives involving pressure, volume, and temperature. That is exactly what you want in engineering work.

For example, consider $${\left(\frac{\partial s}{\partial P}\right)}_T=-{\left(\frac{\partial v}{\partial T}\right)}_P.$$ The left side involves entropy, often unknown. The right side involves how specific volume changes with temperature at constant pressure, which can be obtained from an equation of state, a $P$-$v$-$T$ chart, or property tables.

In fluid property calculations, Maxwell relations are also used to derive measurable-property expressions for:

• Thermal expansion coefficient: $\alpha =\frac{1}{v}{\left(\frac{\partial v}{\partial T}\right)}_P$
• Isothermal compressibility: ${\kappa }_T=-\frac{1}{v}{\left(\frac{\partial v}{\partial P}\right)}_T$

Those coefficients appear in many derived relations for entropy and enthalpy departures of real fluids from ideal behavior.

A concrete workflow example

Suppose you need an entropy change for a compressed liquid over a pressure change at approximately constant temperature. Direct entropy data may be sparse. Using $$ds={\left(\frac{\partial s}{\partial P}\right)}_{T}dP=-{\left(\frac{\partial v}{\partial T}\right)}_{P}dP,$$ you can estimate $ds$ from how $v$ varies with $T$ at the pressure of interest. Even when the approximation is rough, it is grounded in exact thermodynamics.

Property tables: the practical backbone of thermodynamic analysis

Thermodynamic relations are only as useful as your ability to evaluate the properties and derivatives they contain. In much of applied thermodynamics, that evaluation comes from property tables.

What property tables provide

For common working fluids such as water, refrigerants, and combustion gases (under simplifying assumptions), property tables provide:

• Saturated properties as functions of $T$ or $P$ (including $v_f$, $v_g$, $u_f$, $u_g$, $h_f$, $h_g$, $s_f$, $s_g$)
• Superheated vapor properties indexed by $T$ and $P$
• Compressed liquid properties (often approximated or provided at selected states)

These tables embody experimental measurements and high-fidelity equations of state. They are the reason many cycle calculations can be done accurately without deriving a full analytic model.

Using tables with thermodynamic relations

Property tables are often used alongside relations in two ways:

1. Direct state determination

In Rankine cycle analysis, for instance, you identify the state using measured or given $P$ and $T$ (or quality $x$), then read $h$ and $s$ from tables.

1. Checking consistency and computing changes

Even if you can look up entropy directly, the Gibbs equations and Maxwell relations help you interpret trends and validate assumptions. For example:

• If a process is modeled as isentropic, table-based values of $s$ should remain constant within rounding.
• If a throttling process is modeled, $h$ should remain constant, and tables confirm the resulting exit quality or superheat level.

Interpolation and caution near phase boundaries

Real work often requires interpolation between tabulated points. Linear interpolation is common and often sufficient over small intervals, but it should be used carefully near the saturation dome where properties can change rapidly and nonlinearly with small changes in $T$ or $P$. In two-phase regions, quality-based relations are typically the correct tool:

• $h=h_f+x{h}_{fg}$
• $s=s_f+x{s}_{fg}$
• $v=v_f+x{v}_{fg}$

Putting it together: advanced analysis without losing the physics

Maxwell relations, Gibbs equations, and property tables are complementary:

• The Gibbs equations give you the fundamental structure that ties entropy to energy, pressure, and volume.
• Maxwell relations let you transform “hard” derivatives involving entropy into “accessible” derivatives involving $P$, $v$, and $T$.
• Property tables provide the numerical grounding needed for real substances and real operating conditions.

In practice, advanced thermodynamic analysis is less about memorizing formulas and more about choosing the right relation for the information you have. When you keep the natural variables of each potential in mind and use tables intelligently, thermodynamic relations stop being abstract identities and become a reliable toolkit for solving real engineering problems.