12.3 General relations for du, dh, ds, cv, and cp

Internal energy, enthalpy, and entropy changes

General relations for thermodynamic property changes

These three equations are called the Gibbs equations (or $T\,ds$ equations). They come directly from combining the first and second laws, and they apply to any simple compressible substance undergoing any process.

Change in specific internal energy:

$du = T\,ds - P\,dv$

This says that a differential change in internal energy comes from two contributions: heat interaction ($T\,ds$) and boundary work ($P\,dv$).

Change in specific enthalpy:

$dh = T\,ds + v\,dP$

Since enthalpy is defined as $h = u + Pv$, you can derive this by differentiating that definition and substituting the $du$ relation above.

Change in specific entropy:

Rearranging the first Gibbs equation gives:

$ds = \frac{du + P\,dv}{T}$

Or equivalently, from the enthalpy form:

$ds = \frac{dh - v\,dP}{T}$

These two entropy expressions are extremely useful because they let you compute $\Delta s$ using measurable quantities ($T$, $P$, $v$, $c_v$, $c_p$) rather than needing to measure heat transfer directly.

Derivation and validity of general relations

The Gibbs equations are derived by combining:

1. The first law for a closed system: $\delta q - \delta w = du$

2. The second law for a reversible process: $\delta q_{rev} = T\,ds$

3. Boundary work for a simple compressible system: $\delta w_{rev} = P\,dv$

Substituting steps 2 and 3 into step 1 gives $du = T\,ds - P\,dv$.

A critical point: even though the derivation uses a reversible process, the resulting equations involve only properties (state variables). Since properties depend only on the state and not the path, these relations are valid for every process, reversible or irreversible, as long as the substance is a simple compressible system.

Specific heat capacities and thermodynamic properties

Specific heat capacities as partial derivatives

The specific heats are formally defined as partial derivatives:

$c_v = \left(\frac{\partial u}{\partial T}\right)_v$

This tells you how internal energy changes with temperature when volume is held fixed. Physically, it's the energy you need to add per degree of temperature rise in a constant-volume process.

$c_p = \left(\frac{\partial h}{\partial T}\right)_P$

This tells you how enthalpy changes with temperature when pressure is held fixed. In a constant-pressure process, $c_p$ gives the energy input per degree of temperature rise.

For most real substances, $c_p > c_v$ because at constant pressure the substance expands as it heats up, so extra energy goes into $P\,dv$ work.

Relationship between specific heat capacities

For an ideal gas, the relationship simplifies to:

$c_p - c_v = R$

where $R$ is the specific gas constant ($R = R_u / M$, with $R_u = 8.314 \text{ kJ/(kmol·K)}$ and $M$ the molar mass).

For a general substance, the more complete relation is:

$c_p - c_v = -T \frac{\left(\dfrac{\partial P}{\partial T}\right)_v^2}{\left(\dfrac{\partial P}{\partial v}\right)_T}$

This general form reduces to $R$ for an ideal gas, but it also works for real gases, liquids, and solids. Notice that $\left(\partial P / \partial v\right)_T$ is always negative for stable substances, so $c_p - c_v$ is always positive.

The specific heats can also be linked to the thermodynamic potentials:

• Helmholtz free energy: $f = u - Ts$, and $c_v = -T\left(\dfrac{\partial^2 f}{\partial T^2}\right)_v$
• Gibbs free energy: $g = h - Ts$, and $c_p = -T\left(\dfrac{\partial^2 g}{\partial T^2}\right)_P$

Calculating thermodynamic property changes

Applying general relations to calculate changes

To find finite changes in $u$, $h$, or $s$, you integrate the differential relations between the initial and final states.

For an ideal gas, internal energy and enthalpy depend only on temperature, which simplifies things considerably:

$\Delta u = \int_{T_1}^{T_2} c_v \, dT$

$\Delta h = \int_{T_1}^{T_2} c_p \, dT$

If $c_v$ and $c_p$ are approximately constant over the temperature range, these reduce to:

$\Delta u = c_v \, \Delta T$

$\Delta h = c_p \, \Delta T$

For real substances, $u$ and $h$ also depend on pressure or volume, so you need departure functions or property tables. The general relations tell you exactly what additional terms appear. For example, writing $u = u(T, v)$:

$du = c_v \, dT + \left[T\left(\frac{\partial P}{\partial T}\right)_v - P\right] dv$

The bracketed term is zero for an ideal gas but nonzero for real substances.

Calculating entropy changes

General $ds$ equations for any substance:

$ds = \frac{c_v}{T}\,dT + \left(\frac{\partial P}{\partial T}\right)_v dv$

$ds = \frac{c_p}{T}\,dT - \left(\frac{\partial v}{\partial T}\right)_P dP$

These are the workhorses for computing entropy changes. Pick the one that matches the independent variables you know ($T, v$ or $T, P$).

For an ideal gas (with constant specific heats), integrating gives:

$\Delta s = c_v \ln\frac{T_2}{T_1} + R \ln\frac{v_2}{v_1}$

$\Delta s = c_p \ln\frac{T_2}{T_1} - R \ln\frac{P_2}{P_1}$

For an isothermal process ($T_1 = T_2$), the first equation reduces to:

$\Delta s = R \ln\frac{v_2}{v_1}$

For a reversible adiabatic (isentropic) process, $ds = 0$, which leads to the familiar relations for an ideal gas:

$Pv^{\gamma} = \text{constant}, \quad Tv^{\gamma - 1} = \text{constant}, \quad TP^{(1-\gamma)/\gamma} = \text{constant}$

where $\gamma = c_p / c_v$ is the specific heat ratio.

Relationships between thermodynamic properties

Interdependence of thermodynamic properties

The general relations reveal that thermodynamic properties are tightly coupled. The Maxwell relations formalize this by equating certain cross-partial derivatives. They come from the fact that the Gibbs equations involve exact differentials, so the order of mixed second partial derivatives doesn't matter.

The four Maxwell relations are:

The cyclic relation (also called the triple product rule) is another useful identity:

$\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1$

This holds for any three properties $x$, $y$, $z$ and is handy for converting one partial derivative into others that are easier to evaluate.

Analyzing thermodynamic processes

The general relations simplify in predictable ways depending on what's held constant:

• Isothermal (constant $T$): $dT = 0$, so entropy changes depend only on volume or pressure changes.
• Isobaric (constant $P$): $dP = 0$, so $dh = c_p\,dT$ and $ds = (c_p / T)\,dT$.
• Isochoric (constant $v$): $dv = 0$, so $du = c_v\,dT$ and $ds = (c_v / T)\,dT$.
• Isentropic (constant $s$): $ds = 0$, linking $T$, $P$, and $v$ changes through $c_v$, $c_p$, and $\gamma$.

Engineers rely on these relations constantly. Designing a gas turbine, sizing a refrigeration compressor, or predicting how a real gas behaves at high pressure all require you to compute $\Delta u$, $\Delta h$, and $\Delta s$ from measurable data. The general relations are what make that possible.