1.E: Gases (Exercises)
- Page ID
- 339639
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Q2.1
Calculate the pressure exerted by 1.00 mol of Ar, N2, and CO2 as an ideal gas, a van der Waals gas, and a Redlich-Kwong gas, at 25 °C and 24.4 L.
Q2.2
The compression factor \(Z\) for CO2 at 0 °C and 100 atm is 0.2007. Calculate the volume of a 2.50 mole sample of CO2 at 0 °C and 100 atm.
Q2.3
Calculate the pressures for 1 mole each of the following gases under standard conditions (273 K and 1 atmosphere).
| \(Ar\) | \(N_2\) | \(CO_2\) | |
|---|---|---|---|
| ideal | |||
| van der Waals | |||
| Redlich-Kwong |
Q2.4
In a Knudsen cell, the effusion orifice is measured to be 0.50 mm2. If a sample of naphthalene is allowed to effuse for 1.0 hr at a temperature of 40.3 °C, the cell loses 0.0236 g. From this data, calculate the vapor pressure of naphthalene at this temperature.
Q2.5
Assuming it is a van der Waals gas, calculate the critical temperature, pressure and volume for \(CO_2\).
Q2.6
Find an expression in terms of van der Waals coefficients for the Boyle temperature. (Hint: use the viral expansion of the van der Waals equation to find an expression for the second viral coefficient!)
Q2.7
Consider a gas that follows the equation of state
\[p =\dfrac{RT}{V_m - b}\]
Using a virial expansion, find an expression for the second virial coefficient.
Q2.8
Consider a gas that obeys the equation of state
\[ p =\dfrac{nRT}{V_m - b}\]
where a and b are non-zero constants. Does this gas exhibit critical behavior? If so, find expressions for \(p_c\), \(V_c\), and \(T_c\) in terms of the constants \(a\), \(b\), and \(R\).
Q2.9
Consider a gas that obeys the equation of state
\[ p = \dfrac{nRT}{V- nB}-\dfrac{an}{V}\]
- Find an expression for the Boyle temperature in terms of the constant \(a\), \(b\), and \(R\).
- Does this gas exhibit critical behavior? If so, find expressions for \(p_c\), \(V_c\), and \(T_c\) in terms of the constants \(a\), \(b\), and \(R\).