6.6: Equations of state
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Equations of state are laws that relate changes in density to changes in other thermodynamic variables. Each equation of state is an approximation. In a given situation, we want to choose the simplest approximation consistent with the level of accuracy we need. Here we will list several possibilities grouped into four categories of increasing complexity and (one hopes) accuracy.
6.6.1 Homogeneous fluid
The simplest approximation we can make is to assume that density is uniform. We refer to such a fluid as homogeneous. In this case, the equation of state is simply
\[\rho=\rho_{0}, \nonumber \]
and our set of equations is closed.
6.6.2 Barotropic fluid
A slightly more general choice is to assume that density varies only due to changes in pressure:
\[\rho=\rho(p). \nonumber \]
A fluid for which this is true is called barotropic. It can be shown that the variation of density with pressure supports the propagation of sound waves, and
\[\left(\frac{\partial \rho}{\partial p}\right)_{T}=\frac{1}{c^{2}}\label{eqn:} \]
where \(c\) is the speed of sound. The subscript \(T\) reminds us that the partial derivative is to be evaluated at fixed temperature. Typical values are
\[c=\left\{\begin{array}{l}
1500 m s^{-1}, \quad \text { in water } \\
300 m s^{-1}, \text {in air. }
\end{array}\right. \nonumber \]
6.6.3 Temperature-dependent: \(\rho = \rho \left(p,T\right)\)
To increase realism, we can allow for density to be governed by both pressure and temperature. We’ll look at two examples:
1. Perhaps the best-known equation of state is the ideal gas law:
\[\rho(p, T)=\frac{p}{R T},\label{eqn:2} \]
where \(R\) is the gas constant, equal to 287 J kg−1K−1 for dry air. The ideal gas law can be derived from the assumption that molecular collisions conserve kinetic energy (Curry and Webster 1998).
2. For liquids, the equation of state can only be determined empirically, i.e., density is measured over a range of pressures and temperatures and the results are fitted to some mathematical function by adjusting values of constants. For example, a useful approximation for liquid water has the form
\[\rho(p, T)=\frac{\sum_{n=0}^{5} A_{i} T^{n}}{1-p / p_{0}} \nonumber \]
with \(A_0\) = 999.842594, \(A_1\) = 6.793952 × 10−2, \(A_2\) = 9.09529 × 10−3 , \(A_3\) = 1.001685 × 10−4, \(A_4\) = −1.120083 × 10−6, \(A_5\) = 6.536332×10−9, \(p_0\) = 19652, \(T\) in Celsius and \(p\) in bars (Gill 1982). (If you’re serious about computing the density of water, I suggest downloading a software package such as seawater, described in subsection 6.6.4).
The thermal expansion coefficient quantifies the tendency for a material to expand when heated.
\[\alpha=-\frac{1}{\rho_{0}}\left(\frac{\partial \rho}{\partial T}\right)_{p, S}, \nonumber \]
with typical values
\[\alpha=\left\{\begin{array}{l}
1 \times 10^{-4} K^{-1}, \quad \text { in water } \\
3 \times 10^{-3} K^{-1}, \quad \text {in air. }
\end{array}\right. \nonumber \]
The thermal expansion coefficient can vary significantly and should only be treated as constant in nearly-uniform conditions.
6.6.4 Composition effects
The next step toward increased realism is to allow density to depend not only on pressure and temperature but also on the chemical composition of the fluid. We consider three examples.
1. The air in Earth’s atmosphere has almost uniform composition except for a variable fraction of water vapor. We define the specific humidity \(q\) to be the mass of water per unit mass of air, measured in parts per thousand or \(g/kg\). Accounting for humidity, the ideal gas law becomes
\[\rho(p, T)=\frac{p}{R T} \frac{1}{1+0.61 q}.\label{eqn:3} \]
The constant 0.61 is determined by the molecular masses of air and water(Gill 1982).
2. In salt water, density is affected by salinity, defined as the mass of salt per unit mass of water. Salinity can be measured in parts per thousand, \(g/kg\), commonly called practical salinity units (psu). Values range from zero in fresh water to 41 psu in the Red Sea; 35 psu is typical. The equation of state for seawater is entirely empirical and is too complicated to reproduce here. You can look it up in (for example) Gill (1982), or you can evaluate it using standard software such as the “seawater” package, currently available at
www.cmar.csiro.au/datacentre/ext_docs/seawater.htm
6.6.5 Linearized equations of state
The empirical equations of state for liquids are far too cumbersome for use in analytical calculations. Because many problems involve only small variations in density, it is useful to work with the equation of state in a linearized form. In the case of seawater, for example, we can assume that \(p\), \(T\) and \(S\) vary only slightly from some fixed values \(p_0\), \(T_0\) and \(S_0\) at which the density is \(\rho_0\). We can then represent the density dependence, \(\rho = \rho(p,T,S)\), as a first-order Taylor series expansion
\[\rho=\rho_{0}+\left(\frac{\partial \rho}{\partial p}\right)_{T, S}\left(p-p_{0}\right)+\left(\frac{\partial \rho}{\partial T}\right)_{p, S}\left(T-T_{0}\right)+\left(\frac{\partial \rho}{\partial S}\right)_{p, T}\left(S-S_{0}\right). \nonumber \]
The partial derivatives are taken to be constants. One of these is the inverse square of the sound speed as discussed above:
\[\left(\frac{\partial \rho}{\partial p}\right)_{T, S}=\frac{1}{c^{2}}, \nonumber \]
where \(c=c(p_0,T_0,S_0)\). A second is the thermal expansion coefficient \(\alpha\). We also use the saline density coefficient
\[\beta=\frac{1}{\rho_{0}}\left(\frac{\partial \rho}{\partial S}\right)_{p, T}, \nonumber \]
whose value in seawater remains fairly close to 7×10−4 psu−1. With these definitions, the linearized equation of state can be written as
\[\frac{\rho-\rho_{0}}{\rho_{0}}=-\alpha\left(T-T_{0}\right)+\beta\left(S-S_{0}\right)+\frac{1}{c^{2}}\left(p-p_{0}\right).\label{eqn:4} \]