Equation of State of an Ideal Gas

6 Key excerpts on "Equation of State of an Ideal Gas"

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  eBook - ePub

  Engineering Thermodynamics

  Fundamental and Advanced Topics

  • Kavati Venkateswarlu(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  8    Properties of Gases and Gas Mixtures

  Learning Outcomes After learning this chapter, students should be able to

  • Explain the quantitative relationship between T, v, and P as described by the kinetic theory of gases and ideal gas models.
  • Interpret the relationship between partial pressures and the total pressure as described in Dalton’s law of partial pressure.
  • Determine the mole fractions of gases within and gas mixture and relate mole fraction to the partial pressure of a gas within a gas mixture.
  • Explain the relationship between kinetic energy and temperature of a gas; between temperature and the velocity of a gas; and between molar mass and the velocity of a gas.
  • Explain the deviation of ideal gas models with the behavior of real gases observed in nature.
  • Explain the general principles of the hard-sphere model and the Van der Waals model of gas.

  8.1     Ideal Gas Equation of State

  Equation of State

  It is defined as the functional relationship among the properties pressure p, specific volume ν, and temperature T, expressed as f(p,ν,T) = 0. The equation of state is useful for finding the properties of a gas; that is, if two of the properties are known, the third can be found. There are numerous equations of state, including simple and complex ones. The ideal gas equation of state is comparatively simple and it can accurately predict the behavior of substances in the gas phase.

  Robert Boyle proved experimentally that the pressure of gases is inversely proportional to their volume. Charles and Gay Lussac showed that the volume of a gas is directly proportional to its temperature at low pressures, which is expressed as

  p ν = RT

  (8.1)

  which is an ideal gas equation of state. R is a characteristic gas constant; its value is different for different gases.

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  eBook - ePub

  High-Pressure Fluid Phase Equilibria

  Phenomenology and Computation

  • Ulrich K Deiters, Thomas Kraska(Authors)
  • 2012(Publication Date)
  • Elsevier

    (Publisher)

  T ). These two functions are related by

       (7.1)

  7.2 The Ideal Gas

  Thermal equations of state go back to the compression and expansion experiments of Boyle and Mariotte at the end of the 17th Century, which can be summarized as

       (7.2)

  on condition that the amount of substance and the temperature are constant. About a 100 years later, Charles, Amontons, and Gay-Lussac discovered the relations

       (7.3)

  at constant pressure and amount of substance and

       (7.4)

  at constant volume and amount of substance. The combination of these empirical results gives the ideal-gas law, which is historically the first equation of state:

       (7.5)

  or

       (7.6)

  Here R = 8.314472 J K−1 mol−1 denotes the universal gas constant.

  Later, with the development of quantum mechanics and statistical thermodynamics, it became possible to derive the ideal-gas law from first principles. This gave an insight into various assumptions and approximations limiting its applicability. Ideal behavior results if the gas molecules do not interact with each other; in particular, this means

  • the absence of attractive forces between the molecules, • an infinitesimally small size of the molecules, • and no quantum mechanical restrictions with regard to wave-function symmetry or the number of accessible states.

  This explains why the ideal-gas law is valid for gases at low pressures (large molar volumes) only, for only under this condition the average distances between the molecules are so large that the interactions become negligible. Conversely, each gas can be made to behave like an ideal gas if it is expanded sufficiently. Therefore, the ideal-gas law is the limiting law for all equations of state:

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  The Structure and Evolution of Stars

  • J J Eldridge, Christopher A Tout;0(Authors)
  • 2018(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Chapter 3

  The Equation of State

  In most stars, the equation of state is not everywhere sufficiently simple that we may write pressure P = P(ρ) as a function of density ρ only. Generally we need two state variables, such as density and temperature T, and a description of the composition, such as the set of mass fractions X

  i

  of nuclide i. An equation of state might then be written in the form P = P(ρ, T, {X

  i

  }). The first two equations of stellar structure (2.2) and (2.10) are no longer closed and we shall need to include two more, the equation of heat transport and the equation of energy generation that form the subjects of Chapters 4 and 6 .

  It is often convenient to divide the pressure into contributions from the ions, electrons and radiation, but in a non-degenerate gas the electron contribution can be combined with the ion contribution in a single gas pressure,

  We begin our discussion with this case before examining the contribution of radiation, the importance of which increases with temperature. We then examine the quantum mechanical contribution of electron degeneracy, important in the dense material of the cores of giants and white dwarfs.

  3.1.Gas Pressure

  Cool but sufficiently tenuous stellar material is usually well described simply as an ideal gas. This is a reasonably good approximation in the Sun and a very good one outside its relatively dense core and we can write

  where is the gas constant and μ is the mean molecular weight of the ions and electrons combined. The gas constant

  is the ratio of Boltzmann’s constant k to the atomic mass unit mH . One mole of hydrogen, or protons, has a mass very close to 1 g so in the form commonly used for stellar evolution

  too. Formally mH is defined to be one-twelfth of the mass of a carbon-12 atom so that one mole of protons actually has a mass of 1.007 g. In our discussions in this chapter we shall ignore small differences from integer atomic masses to make the discussion simpler. They do not make very significant changes to the equation of state but should be taken properly into account in quantitative calculations. On the other hand the differences are essential when we consider nuclear fusion in Sec. 6.2

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  Compressors

  Selection and Sizing

  • Royce N. Brown(Author)
  • 2011(Publication Date)
  • Gulf Professional Publishing

    (Publisher)

  2 Basic Relationships

  Introduction

  This chapter presents some basic thermodynamic relationships that apply to all compressors. Equations that apply to a particular type of compressor will be covered in the chapter addressing that compressor. In most cases, the derivations will not be presented, as these are available in the literature. The references given are one possible source for additional background information.

  The equations are presented in their primitive form to keep them more universal. Consistent units must be used, as appropriate, at the time of application. The example problems will include conversion values for the units presented. The symbol g will be used for the universal gravity constant to maintain open form to the units.

  Gas and Vapor

  A gas is defined as the state of matter distinguished from solid and liquid states by very low density and viscosity, relatively great expansion and contraction with changes in pressure and temperature, and the ability to diffuse (readily distributing itself uniformly throughout any container).

  A vapor is defined as a substance that exists below its critical temperature and that may be liquefied by application of sufficient pressure. It may be defined more broadly as the gaseous state of any substance that is liquid or solid under ordinary conditions.

  Many of the common “gases” used in compressors for process plant service are actually vapors. In many cases, the material may change states during a portion of the compression cycle. Water is a good example, since a decrease in temperature at high pressure will cause a portion of the water to condense. This is a common occurrence in the first intercooler of a plant air compressor. Conversely, lowering the pressure in a reservoir of liquid refrigerant at a fixed temperature will cause the vapor quantity to increase.

  Perfect Gas Equation

  Jacques A. C. Charles and Joseph Gay-Lussac, working independently, found that gas pressure varied with the absolute temperature. If the volume was maintained constant, the pressure would vary in proportion to the absolute temperature [4]

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  eBook - ePub

  Practical Chemical Thermodynamics for Geoscientists

  • Bruce Fegley Jr.(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  P . We then get the equation

  (8-161)

  Integration of Eq. (8-161) gives

  (8-162)

  (8-163)

  If we now take the pressure P 1 as the standard-state pressure of one bar, G 1 is equal to G o , the standard Gibbs free energy, and Eq. (8-163) can be rearranged to give

  (8-164)

  Equation (8-163) gives the isothermal change in Gibbs free energy of an ideal gas between any two arbitrary pressures. The related Eq. (8-164) gives G for an ideal gas at an arbitrary pressure P relative to the standard-state pressure of one bar. Both equations, but Eq. (8-164) in particular, are important for chemical equilibria of ideal gases.

  Example 8-15. Astronomers are currently detecting planets around other stars. These planets are called extrasolar planets (exoplanets ). Imagine that an Earth-like exoplanet has been found with a surface temperature and pressure of 300 K and 10 bars and an atmosphere composed of air. Calculate G –

  Go

  per mole of air at 300 K and 10 bars assuming ideality. Equation (8-164) gives

  (8-165)

  By analogy with its major constituents N2 , O2 , and Ar, we could define the standard Gibbs free energy of air (

  Go

  ) as zero at the standard-state pressure of one bar. In this case the calculated value of G –

  Go

   = 5743.1 J mol−1 is the Gibbs free energy of air at 10 bars pressure. If we take a nonzero value of

  Go

  for air at one bar pressure, then G at 10 bars is 5743.1 J mol−1 larger.

  Equations (8-161) to (8-165) are very useful, but they only apply to ideal gases. We now want to derive analogous equations that apply to real gases. This is done using the concept of fugacity, which was developed in 1901 by the American physical chemist Gilbert N. Lewis (1875–1946).

  Fugacity (f)

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  Dense Phase Carbon Dioxide

  Food and Pharmaceutical Applications

  • Murat O. Balaban, Giovanna Ferrentino, Murat O. Balaban, Giovanna Ferrentino(Authors)
  • 2012(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  i pure species, the residual term is equal to zero.

  2.2.2 Calculation of

  ϕ

  The equation of state (EOS) is an analytical relationship between pressure P , temperature T and molar volume V :

  (2.31)  

  Starting from this equation, the calculation of volumetric and thermodynamic properties of a pure component or of a mixture is possible. Substituting this equation in Equation (2.7) or (2.8) , the fugacity ­coefficients and as a consequence equilibrium between different phases are also ­calculated.

  The equation of state models for the calculation of fugacities can be divided into classes on the basis of different criteria. One is based on the degree of the polynomial used in developing the equation of state in terms of volume. In this case it is possible to divide the EOS in cubic and ­non-cubic models.

  An interesting observation, that can be useful for the classification, is that a common feature of the equations of state is that it is possible to ­recognize separate contributions resulting from repulsive and attractive interactions.

  Following these criteria, the EOS can be separated into three families:

  (1) Family of virial equation of state. (2) Family of Van der Waals–type EOS where the contribution of repulsive and attractive forces is present. (3) Molecular-based equation of state.

  In the virial equation of state, the compressibility factor z is given as the power expansion of the density ρ :

  (2.32)  

  where B and C are the second and third virial coefficients that are a ­function of temperature for a pure fluid or a function of temperature and composition for a mixture.

  This theoretical equation was empirically modified by different authors often introducing a large number of constants. These modifications can be useful for the evaluation of pure-component properties, but their ­extension to mixtures is generally questionable.

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