5.E: The Second Law (Exercises)
- Page ID
- 84764
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Q5.1
What is the minimum amount of work needed to remove 10.0 J of energy from a freezer at -10.0 °C, depositing the energy into a room that is 22.4 °C?
Q5.2
Consider the isothermal, reversible expansion of 1.00 mol of a monatomic ideal gas (CV = 3/2 R) from 10.0 L to 25.0 L at 298 K. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.
Q5.3
Consider the isobaric, reversible expansion of 1.00 mol of a monatomic ideal gas (Cp = 5/2 R) from 10.0 L to 25.0 L at 1.00 atm. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.
Q5.4
Consider the isochoric, reversible temperature increase of 1.00 mol of a monatomic ideal gas (CV = 3/2 R) °Ccupying 25.0 L from 298 K to 345 K. Calculate \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for this process.
Q5.5
Consider the adiabatic expansion of 1.00 mol of a monatomic ideal gas (CV = 3/2 R) from 10.0 L at 273 K to a final volume of 45.0 L. Calculate \(\Delta T\), \(q\), \(w\), \(\Delta U\), \(\Delta H\), and \(\Delta S\) for the expansion.
Q5.6
15.0 g of ice (\(\Delta H_{fus} = 6.009\, kJ/mol\)) at 0 °C sits in a room that is at 21 °C. The ice melts to form liquid at 0 °C. Calculate the entropy change for the ice, the room, and the universe. Which has the largest magnitude?
Q5.7
15.0 g of liquid water (Cp = 75.38 J mol-1 °C-1) at 0 °C sits in a room that is at 21 °C. The liquid warms from 0 °C to 21 °C. Calculate the entropy change for the liquid, the room, and the universe. Which has the largest magnitude?
Q5.8
Calculate the entropy change for taking 12.0 g of H2O from the solid phase (Cp = 36.9 J mol-1 K-1) at -12.0 °C to liquid (Cp = 75.2 J mol-1 K-1) at 13.0 °C. The enthalpy of fusion for water is \(\Delta H_{fus} = 6.009 \,kJ/mol\).
Q5.9
Using Table T1, calculate the standard reaction entropies (\(\Delta S^o\)) for the following reactions at 298 K.
- \(CH_3CH_2OH(l) + 3 O_2(g) \rightarrow 2 CO_2(g) + 3 H_2O(l)\)
- \(C_{12}H_{22}O_{11}(s) + 12 O_2 \rightarrow 12 CO_2(g) + 11 H_2O(l)\)
- \(2 POCl_3(l) \rightarrow 2 PCl_3(l) + O_2(g)\)
- \(2 KBr(s) + Cl2(g) \rightarrow 2 KCl(s) + Br_2(l)\)
- \(SiH_4(g) + 2 Cl(g) \rightarrow SiCl_4(l) + 2 H_2(g)\)
Q5.10
1.00 mole of an ideal gas is taken through a cyclic process involving three steps:
- Isothermal expansion from V1 to V2 at T1
- Isochoric heating from, T1 to T2 at V2
- Adiabatic compression from V2 to V1
- Graph the process on a V-T diagram.
- Find \(q\), \(w\), \(\Delta U\), and \(\Delta S\) for each leg. (If you want, you can find \(\Delta H\) too!)
- Use the fact that \(\Delta S\) for the entire cycle must be zero (entropy being a state function and all …), determine the relationship between V1 and V2 in terms of Cv, T1 and T2.
Q5.11
2.00 moles of a monatomic ideal gas (CV = 3/2 R) initially exert a pressure of 1.00 atm at 300.0 K. The gas undergoes the following three steps, all of which are reversible:
- isothermal compression to a final pressure of 2.00 atm,
- Isobaric temperature increase to a final temperature of 400.0 K, and
- A return to the initial state along a pathway in which
\[p = a+bT\]
where \(a\) and \(b\) are constants. Sketch the cycle on a pressure-temperature plot, and calculate \(\Delta U\) and \(\Delta S\) for each of the legs. Are \(\Delta U\) and \(\Delta S\) zero for the sum of the three legs?
Q5.12
A 10.0 g piece of iron (C = 0.443 J/g °C) initially at 97.6 °C is placed in 50.0 g of water (C = 4.184 J/g °C) initially at 22.3 °C in an insulated container. The system is then allowed to come to thermal equilibrium. Assuming no heat flow to or from the surroundings, calculate
- the final temperature of the metal and water
- the change in entropy for the metal
- the change in entropy for the water
- the change in entropy for the universe
Q5.13
Considers a crystal of \(CHFClBr\) as having four energetically equivalent orientations for each molecule. What is the expected residual entropy at 0 K for 2.50 mol of the substance?
Q5.14
A sample of a certain solid is measured to have a constant pressure heat capacity of 0.436 J mol-1 K-1 at 10.0 K. Assuming the Debeye extrapolation model
\[ C_p(T) = aT^3\]
holds at low temperatures, calculate the molar entropy of the substance at 12.0 K.