16.S: The Properties of Gases (Summary)
- Page ID
- 550570
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We will begin by looking at various empirical equations of state that account for deviations from ideality using equations that are cubic in molar volume. Such equations can crudely account for the gas-liquid phase transition, which is one of the most readily observable consequences of intermolecular forces. We will see that by analyzing the behavior of these equations, a more fundamental relationship arises: the law of corresponding states, which shows that there is one universal equation of state that all gases/liquids obey (though regrettably, there is no analytical functional form for it).
After working with these equations of state, we will explore in detail how intermolecular forces give rise to nonideality. Using a virial expansion, we can concentrate on the potential energy between two particles, between 3 particles, and so on. The two-body interaction is embodies by the “second virial coefficient,” which gives a first-order correction to the molar volume calculated from the ideal gas law. We examine the two-body interactions by looking at several effective intermolecular potentials and analyze the attracive forces between molecules that arise from the interaction of molecular dipole moments and from electron correlation (i.e., London dispersion forces). The repulsive forces are not well-grounded theoretically, but we find that the behavior of gases is largely insensitive to the exact functional form of the repulsive potential, so we can use a number of effective repulsive potentials to obtain similar results. We find that we can relate the attractive forces to atomic/molecular parameters such as their dipole moments, polarizabilities, and ionization energies; these parameters are the most important contributors to the nonideality of a gas at low temperature and/or moderate density.
By the end of this chapter, you should know the following:
- The compressibility factor and how can it be used as a measure of nonideality
- How equations of state that are cubic in molar volume account for phase transitions
- The definitions of critical points and critical constants
- The law of corresponding states
- How intermolecular forces connect quantum mechanical properties of isolated properties to bulk properties of materials
- Virial expansion, and virial coefficients relate to \(n\)-body interactions
- The relationship between the second virial coefficient and compressibility
- How the second virial coefficient relates to intermolecular forces
- Explain the physical justification for the Lennard-Jones potential
Skill goals (what you should be able to do/calculate):
- Use the van der Waals equation calculate state variables of real gases
- Use the law of corresponding states to calculate the state of a real gas
- Read compressibility charts, phase diagrams, and other visualizations of physical data pertaining to real gases
- Classify the types of intermolecular forces between atoms and molecules
- Use the Lennard-Jones equation to calculate an intermolecular potential, and use molecular parameters to calculate the \(c_6\) coefficient
- Use the hard-sphere and square-well potentials to calculate virial coefficients