CW01-1
\nExercise 1.1 Kinetic Theory of Gases
\nA gas is considered to be isotropic (equal in all directions) as part of the assumptions
\nintrinsic to the KTG. It can be shown that for an isotropic distribution, the flux of particles
\n(number of particles passing through a surface per unit area per unit time) in one direction
\nis
\nF =
\n1
\n4
\n\u03c1hci
\nwhere \u03c1 is the number density (number of particles per unit volume) and hci is the mean
\nspeed of the particles. (This seems like a pretty trivial relation, except for that pesky factor
\nof 1\/4, which requires a bit of thought). This relation can be used to calculate the frequency
\nof collisions that an ideal gas makes with the walls of its container.
\nConsider n moles of ideal gas of molecular mass m at a temperature T confined in a cubical
\nbox of volume V . Define f as the average number of collisions per second with the gas
\nmolecules with the walls of the container.
\ni How does this frequency scale with temperature for a given gas if the volume and moles are
\nconstant?
\nii How does this frequency scale with volume for a given gas if the temperature and moles are
\nconstant?
\niii How does this frequency scale with molecular mass if the temperature, volume, and moles
\nare fixed?
\niv Derive an expression for f in terms of n, m, T, and V . Is this consistent with the above
\nobservations?
\nv What would be different about your expression in item iv if the container were not cubical?
\nvi Calculate the collision frequency f in seconds\u22121
\nfor Argon gas at STP in a 22.41397 L
\ncubical container.
\nExercise 1.2 Intensive Properties of a Gas
\nIntensive properties are typically directly related to the microscopic behavior (molecular
\nmotion) of the system. For a gas, the pressure, mass density, and root mean square speed
\nof the molecules are all intensive properties, and thus could be related. This problem asks
\nyou to prove that they are.
\ni Derive a symbolic relationship between the pressure, p, the mass density \u03c1m and the root
\nmean square speed, crms for an Ideal Gas
\nii Consider an Ideal Gas with a mass density of 17.86 g\/L at a pressure of 10.00 atm. What
\nis the root mean square speed of the molecules in the gas, in m\/s?
\nExercise 1.3 The Maxwell-Boltzmann Distribution
\nIn the notes, a discussion of the Maxwell-Boltzmann distribution for molecular speeds was
\nmade, and this was used to create an energy distribution. In many ways the energy of a
\nmolecule is a much more \u2018chemically significant\u2019 quantity than speed. Use the expression
\nfor the MB energy distribution, eq. 1.6.8, to derive the following quantities in analogy to
\nwhat was done in eq. 1.6.6 for molecular speeds.
\ni h\u000fi
\nii p
\nh\u000f
\n2i \u2261 \u000frms
\niii \u000fmp
\n\u2014 CW01 continues \u2014
\nCHM4411 pjbrucat 2022 page 1
\nCW01-2
\nExercise 2.1 The van der Waals Equation of State (vdW EOS)
\nSteam (gaseous water) has been used as the working substance in industrial-scale engines
\nfor well over a century, and is still used in that capacity in modern nuclear power plants
\n(amazing!). Steam is not a very ideal gas, as one might expect. If steam is treated as a van
\nder Waals gas, its \u2018best fit\u2019 parameters are:
\na = 5.464 L2
\n\u00b7atm\u00b7mol\u22122
\nb = 0.03049 L\u00b7mol\u22121
\nMeasurement of the mass density of steam at 776.4 K and 327.6 atm places it at 133.2 g\/L.
\ni What is the measured compression factor, Z, for steam using the experimental mass density
\nat 776.4 K and 327.6 atm?
\nii What is the compression factor, Z, for steam computed using the vdW EOS and the above
\na and b parameters at 776.4 K and 327.6 atm?
\niii What is the percent error of the vdW EOS prediction of Z?
\nExercise 2.2 Critical Behavior
\nConsider a gas that obeys the following Equation of State (EOS)
\np =
\nRT
\nV
\n\u2212
\nB
\nV
\n2 +
\nC
\nV
\n3
\nwhere B and C are positive empirical constants peculiar to a given gas.
\ni Explain and justify the condensability of this gas.
\nii Determine the critical parameters, Tc, pc and V c of the gas in terms of B and C.
\niii Determine the critical compression factor, Zc, for this gas.
\nExercise 2.3 Isobaric Thermal Expansivity
\nThe thermal expansion of a material may be characterized by the following intensive quantity
\n\u03b1 \u2261
\n1
\nV
\n\u0012
\n\u2202V
\n\u2202T \u0013
\np
\nThis quantity is related, naturally, to the equation of state of the material. Consider a gas
\nthat obeys the following EOS
\np =
\nRT
\nV
\n+
\n(a + bT)
\nV
\n2
\nwhere a and b are empirical constants peculiar to a given gas.
\ni Determine \u03b1 for this gas. (Hint: The EOS involves state variables, which must have exact
\ndifferentials.)
\nii Is it possible for the quantity, \u03b1, to be negative? Justify your answer.
\n\u2014 End of CW01 \u2014
\nCHM4411 pjbrucat 2022 page 2<\/p>\n","protected":false},"excerpt":{"rendered":"
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