*Gary A. Glatzmaier*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus ...
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This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.Less

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.

*Greg M. Anderson and David A. Crerar*

- Published in print:
- 1993
- Published Online:
- November 2020
- ISBN:
- 9780195064643
- eISBN:
- 9780197560198
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195064643.003.0016
- Subject:
- Earth Sciences and Geography, Geochemistry

At this point we have introduced the activity as a ratio of fugacities (Chapter 11). The fugacity of a constituent, in turn, we saw was a quantity very much like a vapor pressure or partial ...
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At this point we have introduced the activity as a ratio of fugacities (Chapter 11). The fugacity of a constituent, in turn, we saw was a quantity very much like a vapor pressure or partial pressure, which is directly linked to the Gibbs free energy of that constituent, such that a ratio of fugacities leads directly to a difference in free energies. The fugacity was introduced as a means of dealing with gases and gaseous solutions, and it is measured by measuring gas volumes or densities. Nevertheless, there is nothing restricting its use to gaseous constituents, and we suggested that it is very useful to regard the fugacity as a state variable; as a property of any constituent of any system, solid, liquid, or gas, whether equilibrated with a gas or not, and whether measurable or not. This leads to the easiest approach to understanding activities. The activity of a constituent is the ratio of the fugacity of that constituent to its fugacity in some other state, which we called a reference state. We then showed through consideration of the Lewis Fugacity Rule, which is an extension of Dalton's Law, that for ideal solutions of condensed phases, the activity of a constituent equals its mole fraction, if the reference state is the pure constituent at the same P and T. Deviations from ideal behaviour are then conveniently handled by introducing Henryan and Raoultian activity coefficients. The utility of these relations would be quite sufficient for retaining the activity in our collection of thermodynamic parameters, but in fact the activity can be applied to a much wider range of conditions, simply by varying the choice of reference state. We now examine the various possible choices of this reference state, and the resulting equations and applications. In the most general sense, the fugacity and activity concepts satisfy the need to relate system compositions to free energy changes. That a single parameter, the activity, can do this for essentially any system is a tribute to its tremendous versatility.
Less

At this point we have introduced the activity as a ratio of fugacities (Chapter 11). The fugacity of a constituent, in turn, we saw was a quantity very much like a vapor pressure or partial pressure, which is directly linked to the Gibbs free energy of that constituent, such that a ratio of fugacities leads directly to a difference in free energies. The fugacity was introduced as a means of dealing with gases and gaseous solutions, and it is measured by measuring gas volumes or densities. Nevertheless, there is nothing restricting its use to gaseous constituents, and we suggested that it is very useful to regard the fugacity as a state variable; as a property of any constituent of any system, solid, liquid, or gas, whether equilibrated with a gas or not, and whether measurable or not. This leads to the easiest approach to understanding activities. The activity of a constituent is the ratio of the fugacity of that constituent to its fugacity in some other state, which we called a reference state. We then showed through consideration of the Lewis Fugacity Rule, which is an extension of Dalton's Law, that for ideal solutions of condensed phases, the activity of a constituent equals its mole fraction, if the reference state is the pure constituent at the same P and T. Deviations from ideal behaviour are then conveniently handled by introducing Henryan and Raoultian activity coefficients. The utility of these relations would be quite sufficient for retaining the activity in our collection of thermodynamic parameters, but in fact the activity can be applied to a much wider range of conditions, simply by varying the choice of reference state. We now examine the various possible choices of this reference state, and the resulting equations and applications. In the most general sense, the fugacity and activity concepts satisfy the need to relate system compositions to free energy changes. That a single parameter, the activity, can do this for essentially any system is a tribute to its tremendous versatility.