27: The Kinetic Theory of Gases
- Page ID
- 11823
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 27.1: The Average Translational Kinetic Energy of a Gas
- This page explores the kinetic molecular theory of gases, detailing how gas behavior relates to particle properties such as velocity and temperature. It derives the ideal gas law (PV = nRT) and discusses mean-square speed, root-mean-square speed, and their connections to pressure, temperature, and molecular mass.
- 27.2: The Gaussian Distribution of One Component of the Molecular Velocity
- This page discusses the pressure of an ideal gas, emphasizing the importance of molecular speed distribution over uniform velocity assumptions. It introduces the Maxwell-Boltzmann distribution, requiring normalization for probability summation. The calculation of average velocity in the x-direction shows a zero average due to symmetry, while the average speed is non-zero. Ultimately, the velocity distribution appears Gaussian, despite differences in the overall molecular speed distribution.
- 27.3: The Distribution of Molecular Speeds is Given by the Maxwell-Boltzmann Distribution
- This page outlines the Boltzmann distribution and its relation to molecular velocity in gases, primarily the Maxwell-Boltzmann distribution. It explains how temperature influences molecular speeds, resulting in a broader distribution as energy increases. The derivation of average speeds, including \(\langle v \rangle\), most probable speed, and \(v_{rms}\), is discussed, highlighting the effect of mass on speed.
- 27.4: The Frequency of Collisions with a Wall
- This page explains the derivation of gas pressure through the analysis of gas molecule collisions with container walls. It introduces key concepts like collision volume and number density, while exploring motion in three dimensions to determine collision frequency per unit area (\(z_w\)).
- 27.5: The Maxwell-Boltzmann Distribution Has Been Verified Experimentally
- This page discusses the experimental validation of the Maxwell-Boltzmann distribution using a velocity selector. This device features spinning wheels with holes to allow gas particles of specific velocities to pass through. By adjusting the wheel speeds, researchers can accurately count particles at different velocities, demonstrating compliance with the Maxwell-Boltzmann distribution.
- 27.6: Mean Free Path
- This page discusses particle interactions in gases, focusing on collision energy, cross-section, collision frequency, and mean free path. It describes particle motion as a random walk affected by collisions, represented probabilistically. The text introduces Gaussian distribution for displacement and diffusion, emphasizing the diffusion constant and equation.
- 27.7: Rates of Gas-Phase Chemical Reactions
- This page explains how gas molecule collisions can lead to chemical reactions, emphasizing that the frequency and energy of collisions determine reaction occurrence. While many collisions happen, not all yield reactions. It introduces equations for calculating collision frequency based on molecular speeds, which influence the likelihood of reactions.