2.S: Gases (Summary)
- Page ID
- 84447
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
After mastering the material covered in this chapter, one will be able to:
- Understand the relationships demonstrated by and perform calculations using the empirical gas laws (Boyle’s Law, Charles’ Law, Gay-Lussac’s Law, and Avogadro’s Law, as well as the combined gas law.)
- Understand and be able to utilize the ideal gas law in applications important in chemistry.
- State the postulates of the Kinetic Molecular theory of gases.
- Utilize the Maxwell and Maxwell-Boltzmann distributions to describe the relationship between temperature and the distribution of molecular speeds.
- Derive an expression for pressure based on the predictions of the kinetic molecular theory for the collisions of gas molecules with the walls of a container.
- Derive and utilize an expression for the frequency with which molecules in a gas sample collide with other molecules.
- Derive and utilize an expression for the mean-free-path of molecules based on temperature, pressure, and collisional cross section.
- Explain how the van der Waals (and other) model(s) allow for deviations from ideal behavior of gas samples.
- Derive an expression for the Boyle temperature and interpret the results based on how a gas’s behavior approaches that of an ideal gas.
- Explain and utilize the Principle of Corresponding States.
Vocabulary and Concepts
- average
- Boyle temperature
- collisional cross section
- compression factor
- critical point
- critical temperature
- diffusion
- effusion
- empirical
- empirical gas laws
- frequency of collisions
- frequency of collisions with the wall
- gas law constant
- ideal gas law
- intermolecular potential
- isotherm
- Kinetic Molecular Theory
- Knudsen cell
- Leonard-Jones potential
- maximum probability
- Maxwell’s distribution
- Maxwell-Boltzmann distribution
- mean free path
- normalization constant
- number density
- principle of corresponding states
- reduced variables
- root-mean-square
- Second Virial Coefficient
- Taylor Series Expansion
- van der Waals’ equation
- Virial Equation
References
- Avogadro, A. (1811). Essay on a Manner of Determining the Relative Masses of the Elementary Molecules of Bodies, and the Proportions in Which They Enter into These Compounds. Journal de Physique, 73, 58-76.
- Bernoulli, D. (1738). Hydronamica.
- Clausius, R. (1857). Ueber die Art der Bewegung, welche wir Wärme nennen. Annalen der Physik, 176(3), 353–379. doi:10.1002/andp.18571760302
- Dieterici, C. (1899). Ann. Phys. Chem., 69, 685.
- Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 17(8), 549-560. doi:10.1002/andp.19053220806
- Encycolopedia, N. W. (2016). Amedeao Avogadro. Retrieved April 13, 2016, from New World Encycolpedia: http://www.newworldencyclopedia.org/...medeo_Avogadro
- Fazio, F. (1992). Using Robert Boyle's Original Data in the Physics and Chemistry Classrooms. Journal of College Science Teaching, 363-365.
- Guggenheim, E. A. (1945). Corresponding State for Perfect Liquids. Journal of Chemical Physics, 13, 253-261.
- Hunter, M. (2004). Robert Boyle (1627 - 91). Retrieved March 10, 2016, from The Robert Boyle Project: http://www.bbk.ac.uk/boyle/
- Johannes Diderik van der Waals - Biographical. (2014). Retrieved March 12, 2016, from Nobelprize.org: http://www.nobelprize.org/nobel_priz...waals-bio.html
- Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. Part 1. On the motions and collisions of perfectly elastic spheres. Phil. Mag., XIX, 19-32.
- Maxwell, J. C. (1873). Molecules. Nature, 417, 903-915. doi:10.1038/417903a
- Redlich, O., & Kwong, J. N. (1949). On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions. Chemical Reviews, 44(1), 233-244.
- van der Waals, J. D. (1913). The law of corresponding states for different substances. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, (pp. 971-981).
- van der Waals, J. D. (1967). The equation of state for gases and liquids. Nobel Lectures in Physics 1901 - 1921, 254-265.