Carnahan StarlingEdit
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Carnahan–Starling equation of state
Carnahan–Starling refers to a landmark analytic model in statistical mechanics and thermodynamics for the behavior of a fluid composed of hard spheres. In its most common form, it provides a closed-form expression for the compressibility factor Z (the ratio of the actual pressure to the pressure of an ideal gas at the same density and temperature) as a function of the packing fraction η. For a monodisperse system of non-attracting spheres with diameter σ and number density ρ, the packing fraction is η = (π/6) ρ σ^3, and the Carnahan–Starling equation is
Z = (1 + η + η^2 − η^3) / (1 − η)^3.
This compact formula makes the Carnahan–Starling model a convenient reference in problems where short-range repulsion dominates the thermodynamics, and it is frequently used to characterize the equation of state of hard-sphere fluids.
Introductory overview
The Carnahan–Starling equation of state arose in the field of liquid-state theory as a practical and accurate description of the pressure–density relationship for a system of hard spheres. Hard-sphere models abstract away attractive forces and focus on purely repulsive interactions, making them a foundational reference in both theoretical studies and applied contexts. The equation’s relatively simple algebraic form belies a careful combination of theoretical insights, including considerations arising from the virial expansion and results associated with the Percus–Yevick theory for hard spheres. In practice, the Carnahan–Starling expression provides a remarkably good match to simulated and experimental data for mono-disperse hard-sphere fluids over a broad range of densities, and it serves as a baseline for more complex models.
Historical context
The equation is named after its developers, Carnahan and Starling, who introduced the form in the late 1960s. The work appeared in the context of efforts to understand the thermodynamics of dense fluids and to develop tractable equations of state that could be leveraged in statistical mechanics, chemical engineering, and soft-matter physics. The result quickly established itself as a standard reference because of its balance between analytical simplicity and empirical accuracy. For broader reading on the underlying topics, see statistical mechanics, thermodynamics, and the hard-sphere model.
Mathematical form and interpretation
The compressibility factor Z, defined by P/(ρ k_B T) (where P is pressure, ρ is number density, k_B is Boltzmann’s constant, and T is temperature), is expressed as a function of η in the Carnahan–Starling form.
η represents the fraction of volume occupied by the hard spheres in a given system, i.e., the packing fraction. It is a dimensionless measure of density for the hard-sphere reference system.
The key feature of the expression is its nontrivial η-dependence, capturing how excluded-volume effects grow as particles crowd the system. The denominator (1 − η)^3 reflects the rapid rise in pressure as the available free volume shrinks at higher densities, while the polynomial in the numerator modulates this growth to align with the virial behavior expected for hard spheres.
Although presented for a single species of spheres, the same general spirit informs extensions to mixtures, where sizes differ and the packing geometry becomes more involved.
Generalizations and related theories
Mixtures and polydispersity: Real fluids often contain particles of different sizes. An important generalization is the BMCSL equation of state (Boublík–Mansoori–Carnahan–Starling–Leland), which extends the Carnahan–Starling framework to mixtures of hard spheres with varying diameters. This family of equations has become a standard reference in modeling colloidal suspensions, polymer-coated particles, and other multispecies hard-sphere systems. See BMCSL equation of state for more detail.
Link to perturbation theory: The Carnahan–Starling reference system—hard spheres—serves as the baseline in various perturbation-theory approaches to fluids with attractions (for example, models that start from a hard-sphere reference and add attractive terms). In this role, its accuracy and simplicity help quantify the influence of non-hard interactions.
Related models and comparisons: The Percus–Yevick (PY) theory provides another approximate route to the hard-sphere equation of state. While PY can yield closed forms for certain thermodynamic routes, Carnahan–Starling is often favored for its accuracy in predicting the compressibility factor and related thermodynamic properties. Readers may compare these approaches when selecting a model for a given application.
Applicability, accuracy, and limitations
Domain of validity: The Carnahan–Starling equation is formulated for monodisperse hard-sphere fluids with purely repulsive interactions. It is particularly well-suited for systems where the repulsive core dominates the thermodynamics and where attractive forces are weak or can be treated separately.
Accuracy: Across a wide range of densities up to moderately high packing, the Carnahan–Starling form reproduces simulation data with high fidelity relative to many other analytic approximations. It is often cited as one of the most reliable simple equations of state for hard spheres.
Limitations: At very high densities approaching random close packing or crystalline order, discrepancies between the Carnahan–Starling prediction and actual behavior become more pronounced. Moreover, the model does not account for attractions or complex molecular shapes, so it is not appropriate for liquids where cohesive forces play a central role, nor for anisotropic or highly polydisperse systems without modifications.
Practical role: In engineering and physics, the Carnahan–Starling equation acts as a convenient baseline for calculations, a testing ground for perturbation theories, and a teaching tool for illustrating the consequences of excluded-volume effects in dense fluids.
Applications and influence
Hard-sphere reference in simulations and theory: The simple analytic form provides a quick, transparent way to estimate pressure and related thermodynamic quantities for hard-sphere systems, informing both computational studies and analytical work.
Colloids and soft matter: In contexts where colloidal particles can be approximated as hard spheres, the Carnahan–Starling equation informs understanding of osmotic pressure, phase behavior, and equations of state used in experimental interpretation.
Educational value: The equation is frequently presented in courses on statistical mechanics and thermodynamics as a paradigmatic example of how microscopic geometry (packing) translates into macroscopic pressure–density relationships.
See also